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Rational expressions

An expression that is the quotient of two algebraic expressions (with denominator not 0) is called a fractional expression.  The most common fractional expressions are those that are the quotients of two polynomials; these are called rational expressions.  Since fractional expressions involve quotients, it is important to keep track of values of the variable that satisfy the requirement that no denominator be0. For example, x != -2 in the rational expression:

because replacing x with -2 makes the denominator equal 0.  Similarly, in

x!=-2 and x!= -4

The restrictions on the variable are found by determining the values that make the denominator equal to zero. In the second example above, finding the values of x that make (x + 2)(x + 4) = 0 requires using the property that ab = 0 if and only if a = 0 or b = 0,  as follows.

(x+2)(x+4)=0

x+2=0 or x+4=0

x=-2 or x=-4

Just as the fraction 6/8 is written in lowest terms as 3/4, rational expressions may also be written in lowest terms. This is done with the fundamental principle.

Write each expression in lowest terms.

Factor the numerator and denominator to get

By the fundamental  principle,

In the original expression p cannot be 0 or -4, because

So this result is valid only for values of p other than 0 and -4.   From now on, we shall always assume such restrictions when reducing rational expressions.

Now let's take a look at how our step-by-step fraction solver solves this problem:

Factor to get

The factors 2 - k and k - 2 have opposite signs.  Because of this, multiply numerator and denominator by -1,   as follows.

Since (k-2)*(-1)=-k+2   or 2- k ,

Working in an alternative way would lead to the equivalent result

Our fraction calculator can solve this and many similar problems. If you would like a similar problem to be generated, click on solve similar button:

Caution            

Probably the most common error made in algebra is the incorrect use of the fundamental principle to write a fraction in lowest terms, Remember, the fundamental principle requires a pair of common factors, one in the numerator and one in the denominator. For example,

On the other hand

cannot be simplified further by the fundamental principle, because the numerator cannot be factored. 

Math Topics

More solvers.

• Add Fractions
• Simplify Fractions

Solving Rational Equations · Examples

Listen up, fraction fans! In today’s lesson, you will learn and practice solving rational equations. As you will see, these are any equation involving a fraction, also known as a rational number in math talk!

By the end, you will know the difference between rational and irrational numbers and have two tricks for solving rational equations.

You could even tackle one of the tricky challenges to form a rational equation using the Pythagorean theorem , or to simplify an expression involving some radicals!

What is a Rational Equation? How to Solve Rational Equations Step 1: Eliminate the Denominators Step 2: Simplify the Equation Step 3: Solve the Equation Step 4: Check Solutions Practice & Challenges Question 1 Question 2 Challenge 1 Challenge 2 Worksheet To Sum Up (Pun Intended!)

What is a Rational Equation?

A rational equation is simply an equation involving a rational number.

A ratio -nal number can be written as a ratio of two integers – an irratio -nal number cannot.

Most of the numbers you know and love such as \(\Large\frac{2}{7}\), \(\Large\frac{1}{2}\) and \(-\Large\frac{20817}{43}\) are rational. Some common irrational numbers are π, \(\sqrt{2}\) and Euler’s number, e. These cannot be written as a fraction of integers.

Numberphile has an interesting video about All the Numbers , which categorizes number types, including rational and irrational numbers .

Technically speaking, basic equations like x+2=5 are rational because each term is a rational number. However, the rational equations you will solve today won’t be so easy!

An example of what you will more likely see in an exam is something like this:

Each term is shown as a fraction.

Rational equations can also include radicals:

Or other operations such as division:

Luckily, the technique you learn now will work for every type of rational equation!

How to Solve Rational Equations

The method to solve these equations is pretty much the same for every type of rational equation. You’ll see questions of varying difficulty in this lesson; don’t be afraid to tackle the challenges later on!

Step 1: The Denominator Elimination Round!

First, you need to deal with the elephant in the room: what should you do with the denominators!?

Solving rational equations is just like solving any other equation once you complete this step.

If it’s a simple case, where you have one fraction being equal to one other fraction, you can cross multiply .

Multiply both sides by the values of both denominators. In this example, both sides are multiplied by 3, then 5.

The 3 cancels with the left denominator and the 5 cancels with the right denominator, leaving you with 5(x+4)=3×2.

See why it’s called cross multiplying?

The product of the left denominator and right numerator equals the product of the right denominator and left numerator !

The more general way to deal with the denominators is to find their lowest common multiple (LCM) . This is the smallest number which all denominators divide neatly, leaving no remainder.

If you cannot find the LCM by inspection – if you cannot “just see it” – you need to factor every denominator like you would with a polynomial.

If you have more than one constant term, you may need to find their prime factors.

The LCM is the smallest combination of each denominator’s factors.

You’ll now see a worked example to illustrate!

Remember, you can only cross multiply when each side has only one fraction, so in this case, your first step is to find the LCM.

The only factors of 3x you know for certain are 3 and x. The only factor you know of x is just x, and 4 is a constant so you can use it as it is.

Write down each denominator’s polynomial factors into rows, with similar terms lined up in the same column.

You need to include both 3 and 4 because neither is a factor of the other. You don’t need both copies of x because x is a factor of itself! So the LCM is 12x.

You might find another example of finding the LCM with the same technique helpful.

You’re now ready to eliminate the denominators by multiplying both sides by the LCM.

Step 2: Simplify the Equation

Multiply each term by the LCM. Continuing from the last example, you have:

You now have a regular equation with no fractions, which should be familiar ground!

Step 3: Solve the Equation

Solving rational equations usually produces a simple polynomial equation. Hopefully, you’ve solved lots of these before!

You could complete the square, factor the terms by inspection, or use the quadratic formula.

This example can be solved by factoring the polynomial, having found that x+2 and x+4 are factors.

You could also solve the equation by completing the square:

Or by using the quadratic formula with a=1, b=6 and c=8:

Each way of solving the simplified rational equation is valid, but you will find that some are quicker than others!

Step 4: Check Every Solution

It is important to check that your solutions are complete, meaning you’ve found all of them and that they don’t give any weird numbers when substituted into the original equation.

In the worked example, you were left with a quadratic equation and found two distinct roots.

Quadratic equations either have two distinct solutions, one repeated solution, or no real solution so the solution x=-2 or x=-4 is complete.

You must be careful that none of the rational terms in the original equation have a zero in the denominator.

Do this by going back to the beginning and substituting your answers into the denominators!

The denominators in the worked example are 3x, x, and 4. Replacing x with -2 or -4 doesn’t give you zero in any of them, so you’re safe here!

A solution that gives a zero-denominator is not allowed. That’s because dividing by zero is “illegal” in math!

Any number divided by zero gives an error on a calculator. Ever wondered why that is?

This is your time to shine – try solving rational equations for yourself and, if you’re feeling confident, tackle the challenges too.

As they say, practice makes perfect! Use the worked example for guidance if you get stuck.

Find x in the following rational equation:

The equation is two equal fractions so you can cross-multiply. You could also simplify \(\Large\frac{15}{3}\normalsize\) to 5, but this does not change the final answer.

Each term is divisible by 9. Simplify the equation by dividing both sides by 9:

This form is called the difference of two squares because it can be factored like this:

So the solution is x=±3.

These must be all the solutions because quadratic equations have a maximum of two distinct real roots.

Neither denominator in the original rational equation has an x term, so substituting any value for x makes no difference to their values – there is no chance of them being zero!

This means the solutions x=3 and x=-3 are valid.

Solve the following rational equation:

There are three fractions so you cannot cross-multiply.

See that the second denominator is the difference of two squares?

Multiply each term by the LCM and simplify.

So its solution is -5, right?… STOP RIGHT THERE! Don’t forget, we can’t divide by zero!

If you put x=±5 into the original equation, at least one of the denominators is always zero, so the original equation has no solutions.

Challenge 1

Can you spot the mistake in the following example? Hint: there has been some cheating with radicals!

If you need a refresher on radicals , check out our lesson on multiplying them. That will get you on the right track!

The mistake is that radicals cannot be subtracted like normal terms.

Instead, you must square both sides of the equation to remove the radical. Similar terms can then be combined as usual.

Still confused? You can find lots of interactive questions on Lumen Learning . Radicals often pop up in rational equations, so getting comfortable with radicals is super helpful for exam success!

Challenge 2

Find the value of x, by using the Pythagorean theorem on the following right-angled triangle:

If you need a refresher on the Pythagorean theorem or are interested in the man himself, check out our lesson. Do the worksheets and you’ll be acing triangle questions in no time !

The Pythagorean theorem states that:

Where c is the length of the hypotenuse, and a and b are the other side lengths.

This gives the rational equation:

Simplifying, you find:

The LCM is 36 so the denominators are removed by dividing each term by this:

It’s always fun when different areas of math link together!

To Sum Up (Pun Intended!)

In today’s lesson on solving rational equations, you first saw the difference between rational and irrational numbers.

Rational numbers are “nice” because they can be written as a fraction of integers. Remember that all integers are rational because they can be written with a denominator of 1!

Irrational numbers are a little more abstract. They include weird but incredibly beautiful numbers like π and e, which cannot be written as a fraction of integers.

Rational equations are solved by eliminating the denominator in every term, then simplifying and solving as normal.

Denominators can be removed by cross-multiplication if there is only one fraction on either side or by finding the LCM if the equation is more complicated.

Don’t be shy, leave a comment below if you have any questions or need help!

Still curious about rational numbers, or eager for an extra challenge? Check out our lesson on the rational root theorem , which combines algebra and equation solving.

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8.6: Solve Rational Equations

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Learning Objectives

By the end of this section, you will be able to:

• Solve rational equations
• Solve a rational equation for a specific variable

Before you get started, take this readiness quiz.

If you miss a problem, go back to the section listed and review the material.

• Solve: \(\frac{1}{6}x+\frac{1}{2}=\frac{1}{3}\). If you missed this problem, review Exercise 2.5.1 .
• Solve: \(n^2−5n−36=0\). If you missed this problem, review Exercise 7.6.13 .
• Solve for y in terms of x: 5x+2y=10 for y. If you missed this problem, review Exercise 2.6.22 .

After defining the terms expression and equation early in Foundations , we have used them throughout this book. We have simplified many kinds of expressions and solved many kinds of equations . We have simplified many rational expressions so far in this chapter. Now we will solve rational equations.

The definition of a rational equation is similar to the definition of equation we used in Foundations .

Definition: RATIONAL EQUATION

A rational equation is two rational expressions connected by an equal sign.

You must make sure to know the difference between rational expressions and rational equations. The equation contains an equal sign.

\[\begin{array}{cc} {\textbf{Rational Expression}}&{\textbf{Rational Equation}}\\ {\frac{1}{8}x+\frac{1}{2}}&{\frac{1}{8}x+\frac{1}{2}=\frac{1}{4}}\\ {\frac{y+6}{y^2−36}}&{\frac{y+6}{y^2−36}=y+1}\\ {\frac{1}{n−3}+\frac{1}{n+4}}&{\frac{1}{n−3}+\frac{1}{n+4}=\frac{15}{n^2+n−12}}\\ \nonumber \end{array}\]

Solve Rational Equations

We have already solved linear equations that contained fractions. We found the LCD of all the fractions in the equation and then multiplied both sides of the equation by the LCD to “clear” the fractions.

Here is an example we did when we worked with linear equations:

We will use the same strategy to solve rational equations. We will multiply both sides of the equation by the LCD. Then we will have an equation that does not contain rational expressions and thus is much easier for us to solve.

But because the original equation may have a variable in a denominator we must be careful that we don’t end up with a solution that would make a denominator equal to zero.

So before we begin solving a rational equation, we examine it first to find the values that would make any denominators zero. That way, when we solve a rational equation we will know if there are any algebraic solutions we must discard.

An algebraic solution to a rational equation that would cause any of the rational expressions to be undefined is called an extraneous solution .

Definition: EXTRANEOUS Solution TO A RATIONAL EQUATION

An extraneous solution to a rational equation is an algebraic solution that would cause any of the expressions in the original equation to be undefined.

We note any possible extraneous solutions, c , by writing \(x \ne c\) next to the equation.

How to Solve Equations with Rational Expressions

Example \(\PageIndex{1}\)

Solve: \(\frac{1}{x}+\frac{1}{3}=\frac{5}{6}\).

Example \(\PageIndex{2}\)

Solve: \(\frac{1}{y}+\frac{2}{3}=\frac{1}{5}\).

\(−\frac{15}{7}\)

Example \(\PageIndex{3}\)

Solve: \(\frac{2}{3}+\frac{1}{5}=\frac{1}{x}\).

\(\frac{15}{13}\)

The steps of this method are shown below.

Definition: SOLVE EQUATIONS WITH RATIONAL EXPRESSIONS.

• Note any value of the variable that would make any denominator zero.
• Find the least common denominator of all denominators in the equation.
• Clear the fractions by multiplying both sides of the equation by the LCD.
• Solve the resulting equation.
• If any values found in Step 1 are algebraic solutions, discard them.
• Check any remaining solutions in the original equation.

We always start by noting the values that would cause any denominators to be zero.

Example \(\PageIndex{4}\)

Solve: \(1−\frac{5}{y}=−\frac{6}{y^2}\).

Example \(\PageIndex{5}\)

Solve: \(1−\frac{2}{a}=\frac{15}{a^2}\).

5, −3

Example \(\PageIndex{6}\)

Solve: \(1−\frac{4}{b}=\frac{12}{b^2}\).

6, −2

Example \(\PageIndex{7}\)

Solve: \(\frac{5}{3u−2}=\frac{3}{2u}\).

Example \(\PageIndex{8}\)

Solve: \(\frac{1}{x−1}=\frac{2}{3x}\).

Example \(\PageIndex{9}\)

Solve: \(\frac{3}{5n+1}=\frac{2}{3n}\).

When one of the denominators is a quadratic, remember to factor it first to find the LCD.

Example \(\PageIndex{10}\)

Solve: \(\frac{2}{p+2}+\frac{4}{p−2}=\frac{p−1}{p^2−4}\).

Example \(\PageIndex{11}\)

Solve: \(\frac{2}{x+1}+\frac{1}{x−1}=\frac{1}{x^2−1}\).

\(\frac{2}{3}\)

Example \(\PageIndex{12}\)

Solve: \(\frac{5}{y+3}+\frac{2}{y−3}=\frac{5}{y^2−9}\)

Example \(\PageIndex{13}\)

Solve: \(\frac{4}{q−4}−\frac{3}{q−3}=1\).

Example \(\PageIndex{14}\)

Solve: \(\frac{2}{x+5}−\frac{1}{x−1}=1\).

−1, −2

Example \(\PageIndex{15}\)

Solve: \(\frac{3}{x+8}−\frac{2}{x−2}=1\).

−2, −3

Example \(\PageIndex{16}\)

Solve: \(\frac{m+11}{m^2−5m+4}=\frac{5}{m−4}−\frac{3}{m−1}\).

Example \(\PageIndex{17}\)

Solve: \(\frac{x+13}{x^2−7x+10}=\frac{6}{x−5}−\frac{4}{x−2}\).

no solution

Example \(\PageIndex{18}\)

Solve: \(\frac{y−14}{y^2+3y−4}=\frac{2}{y+4}+\frac{7}{y−1}\).

The equation we solved in Example had only one algebraic solution, but it was an extraneous solution. That left us with no solution to the equation. Some equations have no solution.

Example \(\PageIndex{19}\)

Solve: \(\frac{n}{12}+\frac{n+3}{3n}=\frac{1}{n}\).

Example \(\PageIndex{20}\)

Solve: \(\frac{x}{18}+\frac{x+6}{9x}=\frac{2}{3x}\).

Example \(\PageIndex{21}\)

Solve: \(\frac{y+5}{5y}+\frac{y}{15}=\frac{1}{y}\).

Example \(\PageIndex{22}\)

Solve: \(\frac{y}{y+6}=\frac{72}{y^2−36}+4\).

Example \(\PageIndex{23}\)

Solve: \(\frac{x}{x+4}=\frac{32}{x^2−16}+5\).

−4, 3

Example \(\PageIndex{24}\)

Solve: \(\frac{y}{y+8}=\frac{128}{y^2−64}+9\).

Example \(\PageIndex{25}\)

Solve: \(\frac{x}{2x−2}−\frac{2}{3x+3}=\frac{5x^2−2x+9}{12x^2−12}\).

Example \(\PageIndex{26}\)

Solve: \(\frac{y}{5y−10}−\frac{5}{3y+6}=\frac{2y^2−19y+54}{15y^2−60}\).

Example \(\PageIndex{27}\)

Solve: \(\frac{z^2}{z+8}−\frac{3}{4z−8}=\frac{3z^2−16z−68}{z^2+8z−64}\).

Solve a Rational Equation for a Specific Variable

When we solved linear equations, we learned how to solve a formula for a specific variable. Many formulas used in business, science, economics, and other fields use rational equations to model the relation between two or more variables. We will now see how to solve a rational equation for a specific variable.

We’ll start with a formula relating distance, rate, and time. We have used it many times before, but not usually in this form.

Example \(\PageIndex{28}\)

Solve: \(\frac{D}{T}=R\) for T.

Example \(\PageIndex{29}\)

Solve: \(\frac{A}{L}=W\) for L.

\(L=\frac{A}{W}\)

Example \(\PageIndex{30}\)

Solve: \(\frac{F}{A}=M\) for A.

\(A=\frac{F}{M}\)

Example uses the formula for slope that we used to get the point-slope form of an equation of a line.

Example \(\PageIndex{31}\)

Solve: \(m=\frac{x−2}{y−3}\) for y.

Example \(\PageIndex{32}\)

Solve: \(\frac{y−2}{x+1}=\frac{2}{3}\) for x.

\(x=\frac{3y−8}{2}\)

Example \(\PageIndex{33}\)

Solve: \(x=\frac{y}{1−y}\) for y.

\(y=\frac{x}{1+x}\)

Be sure to follow all the steps in Example . It may look like a very simple formula, but we cannot solve it instantly for either denominator.

Example \(\PageIndex{34}\)

Solve \(\frac{1}{c}+\frac{1}{m}=1\) for c.

Notice that even though we excluded c=0 and m=0 from the original equation, we must also now state that \(m \ne 1\).

Example \(\PageIndex{35}\)

Solve: \(\frac{1}{a}+\frac{1}{b}=c\) for a.

\(a=\frac{b}{cb−1}\)

Example \(\PageIndex{36}\)

Solve: \(\frac{2}{x}+\frac{1}{3}=\frac{1}{y}\) for y.

\(y=\frac{3x}{6+x}\)

Key Concepts

7.4 Solve Rational Equations

Learning objectives.

By the end of this section, you will be able to:

• Solve rational equations
• Use rational functions
• Solve a rational equation for a specific variable

Be Prepared 7.10

Before you get started, take this readiness quiz.

Solve: 1 6 x + 1 2 = 1 3 . 1 6 x + 1 2 = 1 3 . If you missed this problem, review Example 2.9 .

Be Prepared 7.11

Solve: n 2 − 5 n − 36 = 0 . n 2 − 5 n − 36 = 0 . If you missed this problem, review Example 6.45 .

Be Prepared 7.12

Solve the formula 5 x + 2 y = 10 5 x + 2 y = 10 for y . y . If you missed this problem, review Example 2.31 .

After defining the terms ‘expression’ and ‘equation’ earlier, we have used them throughout this book. We have simplified many kinds of expressions and solved many kinds of equations . We have simplified many rational expressions so far in this chapter. Now we will solve a rational equation .

Rational Equation

A rational equation is an equation that contains a rational expression.

You must make sure to know the difference between rational expressions and rational equations. The equation contains an equal sign.

Solve Rational Equations

We have already solved linear equations that contained fractions. We found the LCD of all the fractions in the equation and then multiplied both sides of the equation by the LCD to “clear” the fractions.

We will use the same strategy to solve rational equations. We will multiply both sides of the equation by the LCD. Then, we will have an equation that does not contain rational expressions and thus is much easier for us to solve. But because the original equation may have a variable in a denominator, we must be careful that we don’t end up with a solution that would make a denominator equal to zero.

So before we begin solving a rational equation, we examine it first to find the values that would make any denominators zero. That way, when we solve a rational equation we will know if there are any algebraic solutions we must discard.

An algebraic solution to a rational equation that would cause any of the rational expressions to be undefined is called an extraneous solution to a rational equation .

Extraneous Solution to a Rational Equation

An extraneous solution to a rational equation is an algebraic solution that would cause any of the expressions in the original equation to be undefined.

We note any possible extraneous solutions, c , by writing x ≠ c x ≠ c next to the equation.

Example 7.33

How to solve a rational equation.

Solve: 1 x + 1 3 = 5 6 . 1 x + 1 3 = 5 6 .

Try It 7.65

Solve: 1 y + 2 3 = 1 5 . 1 y + 2 3 = 1 5 .

Try It 7.66

Solve: 2 3 + 1 5 = 1 x . 2 3 + 1 5 = 1 x .

The steps of this method are shown.

Solve equations with rational expressions.

• Step 1. Note any value of the variable that would make any denominator zero.
• Step 2. Find the least common denominator of all denominators in the equation.
• Step 3. Clear the fractions by multiplying both sides of the equation by the LCD.
• Step 4. Solve the resulting equation.
• If any values found in Step 1 are algebraic solutions, discard them.
• Check any remaining solutions in the original equation.

We always start by noting the values that would cause any denominators to be zero.

Example 7.34

How to solve a rational equation using the zero product property.

Solve: 1 − 5 y = − 6 y 2 . 1 − 5 y = − 6 y 2 .

Try It 7.67

Solve: 1 − 2 x = 15 x 2 . 1 − 2 x = 15 x 2 .

Try It 7.68

Solve: 1 − 4 y = 12 y 2 . 1 − 4 y = 12 y 2 .

In the next example, the last denominators is a difference of squares. Remember to factor it first to find the LCD.

Example 7.35

Solve: 2 x + 2 + 4 x − 2 = x − 1 x 2 − 4 . 2 x + 2 + 4 x − 2 = x − 1 x 2 − 4 .

Try It 7.69

Solve: 2 x + 1 + 1 x − 1 = 1 x 2 − 1 . 2 x + 1 + 1 x − 1 = 1 x 2 − 1 .

Try It 7.70

Solve: 5 y + 3 + 2 y − 3 = 5 y 2 − 9 . 5 y + 3 + 2 y − 3 = 5 y 2 − 9 .

In the next example, the first denominator is a trinomial . Remember to factor it first to find the LCD.

Example 7.36

Solve: m + 11 m 2 − 5 m + 4 = 5 m − 4 − 3 m − 1 . m + 11 m 2 − 5 m + 4 = 5 m − 4 − 3 m − 1 .

Try It 7.71

Solve: x + 13 x 2 − 7 x + 10 = 6 x − 5 − 4 x − 2 . x + 13 x 2 − 7 x + 10 = 6 x − 5 − 4 x − 2 .

Try It 7.72

Solve: y − 6 y 2 + 3 y − 4 = 2 y + 4 + 7 y − 1 . y − 6 y 2 + 3 y − 4 = 2 y + 4 + 7 y − 1 .

The equation we solved in the previous example had only one algebraic solution, but it was an extraneous solution. That left us with no solution to the equation. In the next example we get two algebraic solutions. Here one or both could be extraneous solutions.

Example 7.37

Solve: y y + 6 = 72 y 2 − 36 + 4 . y y + 6 = 72 y 2 − 36 + 4 .

Try It 7.73

Solve: x x + 4 = 32 x 2 − 16 + 5 . x x + 4 = 32 x 2 − 16 + 5 .

Try It 7.74

Solve: y y + 8 = 128 y 2 − 64 + 9 . y y + 8 = 128 y 2 − 64 + 9 .

In some cases, all the algebraic solutions are extraneous.

Example 7.38

Solve: x 2 x − 2 − 2 3 x + 3 = 5 x 2 − 2 x + 9 12 x 2 − 12 . x 2 x − 2 − 2 3 x + 3 = 5 x 2 − 2 x + 9 12 x 2 − 12 .

Try It 7.75

Solve: y 5 y − 10 − 5 3 y + 6 = 2 y 2 − 19 y + 54 15 y 2 − 60 . y 5 y − 10 − 5 3 y + 6 = 2 y 2 − 19 y + 54 15 y 2 − 60 .

Try It 7.76

Solve: z 2 z + 8 − 3 4 z − 8 = 3 z 2 − 16 z − 16 8 z 2 + 16 z − 64 . z 2 z + 8 − 3 4 z − 8 = 3 z 2 − 16 z − 16 8 z 2 + 16 z − 64 .

Example 7.39

Solve: 4 3 x 2 − 10 x + 3 + 3 3 x 2 + 2 x − 1 = 2 x 2 − 2 x − 3 . 4 3 x 2 − 10 x + 3 + 3 3 x 2 + 2 x − 1 = 2 x 2 − 2 x − 3 .

Try It 7.77

Solve: 15 x 2 + x − 6 − 3 x − 2 = 2 x + 3 . 15 x 2 + x − 6 − 3 x − 2 = 2 x + 3 .

Try It 7.78

Solve: 5 x 2 + 2 x − 3 − 3 x 2 + x − 2 = 1 x 2 + 5 x + 6 . 5 x 2 + 2 x − 3 − 3 x 2 + x − 2 = 1 x 2 + 5 x + 6 .

Use Rational Functions

Working with functions that are defined by rational expressions often lead to rational equations. Again, we use the same techniques to solve them.

Example 7.40

For rational function, f ( x ) = 2 x − 6 x 2 − 8 x + 15 , f ( x ) = 2 x − 6 x 2 − 8 x + 15 , ⓐ find the domain of the function, ⓑ solve f ( x ) = 1 , f ( x ) = 1 , and ⓒ find the points on the graph at this function value.

ⓐ The domain of a rational function is all real numbers except those that make the rational expression undefined. So to find them, we will set the denominator equal to zero and solve.

However, x = 3 x = 3 is outside the domain of this function, so we discard that root as extraneous.

ⓒ The value of the function is 1 when x = 7 . x = 7 . So the points on the graph of this function when f ( x ) = 1 f ( x ) = 1 is ( 7 , 1 ) ) ( 7 , 1 ) )

Try It 7.79

For rational function, f ( x ) = 8 − x x 2 − 7 x + 12 , f ( x ) = 8 − x x 2 − 7 x + 12 , ⓐ find the domain of the function ⓑ solve f ( x ) = 3 f ( x ) = 3 ⓒ find the points on the graph at this function value.

Try It 7.80

For rational function, f ( x ) = x − 1 x 2 − 6 x + 5 , f ( x ) = x − 1 x 2 − 6 x + 5 , ⓐ find the domain of the function ⓑ solve f ( x ) = 4 f ( x ) = 4 ⓒ find the points on the graph at this function value.

Solve a Rational Equation for a Specific Variable

When we solved linear equations, we learned how to solve a formula for a specific variable. Many formulas used in business, science, economics, and other fields use rational equations to model the relation between two or more variables. We will now see how to solve a rational equation for a specific variable.

When we developed the point-slope formula from our slope formula, we cleared the fractions by multiplying by the LCD.

m = y − y 1 x − x 1 Multiply both sides of the equation by x − x 1 . m ( x − x 1 ) = ( y − y 1 x − x 1 ) ( x − x 1 ) Simplify. m ( x − x 1 ) = y − y 1 Rewrite the equation with the y terms on the left. y − y 1 = m ( x − x 1 ) m = y − y 1 x − x 1 Multiply both sides of the equation by x − x 1 . m ( x − x 1 ) = ( y − y 1 x − x 1 ) ( x − x 1 ) Simplify. m ( x − x 1 ) = y − y 1 Rewrite the equation with the y terms on the left. y − y 1 = m ( x − x 1 )

In the next example, we will use the same technique with the formula for slope that we used to get the point-slope form of an equation of a line through the point ( 2 , 3 ) . ( 2 , 3 ) . We will add one more step to solve for y .

Example 7.41

Solve: m = y − 2 x − 3 m = y − 2 x − 3 for y . y .

Try It 7.81

Solve: m = y − 5 x − 4 m = y − 5 x − 4 for y . y .

Try It 7.82

Solve: m = y − 1 x + 5 m = y − 1 x + 5 for y . y .

Remember to multiply both sides by the LCD in the next example.

Example 7.42

Solve: 1 c + 1 m = 1 1 c + 1 m = 1 for c .

Try It 7.83

Solve: 1 a + 1 b = c 1 a + 1 b = c for a .

Try It 7.84

Solve: 2 x + 1 3 = 1 y 2 x + 1 3 = 1 y for y .

Access this online resource for additional instruction and practice with equations with rational expressions.

• Equations with Rational Expressions

Section 7.4 Exercises

Practice makes perfect.

In the following exercises, solve each rational equation.

1 a + 2 5 = 1 2 1 a + 2 5 = 1 2

6 3 − 2 d = 4 9 6 3 − 2 d = 4 9

4 5 + 1 4 = 2 v 4 5 + 1 4 = 2 v

3 8 + 2 y = 1 4 3 8 + 2 y = 1 4

1 − 2 m = 8 m 2 1 − 2 m = 8 m 2

1 + 4 n = 21 n 2 1 + 4 n = 21 n 2

1 + 9 p = −20 p 2 1 + 9 p = −20 p 2

1 − 7 q = −6 q 2 1 − 7 q = −6 q 2

5 3 v − 2 = 7 4 v 5 3 v − 2 = 7 4 v

8 2 w + 1 = 3 w 8 2 w + 1 = 3 w

3 x + 4 + 7 x − 4 = 8 x 2 − 16 3 x + 4 + 7 x − 4 = 8 x 2 − 16

5 y − 9 + 1 y + 9 = 18 y 2 − 81 5 y − 9 + 1 y + 9 = 18 y 2 − 81

8 z − 10 − 7 z + 10 = 5 z 2 − 100 8 z − 10 − 7 z + 10 = 5 z 2 − 100

9 a + 11 − 6 a − 11 = 6 a 2 − 121 9 a + 11 − 6 a − 11 = 6 a 2 − 121

−10 q − 2 − 7 q + 4 = 1 −10 q − 2 − 7 q + 4 = 1

2 s + 7 − 3 s − 3 = 1 2 s + 7 − 3 s − 3 = 1

v − 10 v 2 − 5 v + 4 = 3 v − 1 − 6 v − 4 v − 10 v 2 − 5 v + 4 = 3 v − 1 − 6 v − 4

w + 8 w 2 − 11 w + 28 = 5 w − 7 + 2 w − 4 w + 8 w 2 − 11 w + 28 = 5 w − 7 + 2 w − 4

x − 10 x 2 + 8 x + 12 = 3 x + 2 + 4 x + 6 x − 10 x 2 + 8 x + 12 = 3 x + 2 + 4 x + 6

y − 5 y 2 − 4 y − 5 = 1 y + 1 + 1 y − 5 y − 5 y 2 − 4 y − 5 = 1 y + 1 + 1 y − 5

b + 3 3 b + b 24 = 1 b b + 3 3 b + b 24 = 1 b

c + 3 12 c + c 36 = 1 4 c c + 3 12 c + c 36 = 1 4 c

d d + 3 = 18 d 2 − 9 + 4 d d + 3 = 18 d 2 − 9 + 4

m m + 5 = 50 m 2 − 25 + 6 m m + 5 = 50 m 2 − 25 + 6

n n + 2 − 3 = 8 n 2 − 4 n n + 2 − 3 = 8 n 2 − 4

p p + 7 − 8 = 98 p 2 − 49 p p + 7 − 8 = 98 p 2 − 49

q 3 q − 9 − 3 4 q + 12 = 7 q 2 + 6 q + 63 24 q 2 − 216 q 3 q − 9 − 3 4 q + 12 = 7 q 2 + 6 q + 63 24 q 2 − 216

r 3 r − 15 − 1 4 r + 20 = 3 r 2 + 17 r + 40 12 r 2 − 300 r 3 r − 15 − 1 4 r + 20 = 3 r 2 + 17 r + 40 12 r 2 − 300

s 2 s + 6 − 2 5 s + 5 = 5 s 2 − 3 s − 7 10 s 2 + 40 s + 30 s 2 s + 6 − 2 5 s + 5 = 5 s 2 − 3 s − 7 10 s 2 + 40 s + 30

t 6 t − 12 − 5 2 t + 10 = t 2 − 23 t + 70 12 t 2 + 36 t − 120 t 6 t − 12 − 5 2 t + 10 = t 2 − 23 t + 70 12 t 2 + 36 t − 120

2 x 2 + 2 x − 8 − 1 x 2 + 9 x + 20 = 4 x 2 + 3 x − 10 2 x 2 + 2 x − 8 − 1 x 2 + 9 x + 20 = 4 x 2 + 3 x − 10

5 x 2 + 4 x + 3 + 2 x 2 + x − 6 = 3 x 2 − x − 2 5 x 2 + 4 x + 3 + 2 x 2 + x − 6 = 3 x 2 − x − 2

3 x 2 − 5 x − 6 + 3 x 2 − 7 x + 6 = 6 x 2 − 1 3 x 2 − 5 x − 6 + 3 x 2 − 7 x + 6 = 6 x 2 − 1

2 x 2 + 2 x − 3 + 3 x 2 + 4 x + 3 = 6 x 2 − 1 2 x 2 + 2 x − 3 + 3 x 2 + 4 x + 3 = 6 x 2 − 1

Solve Rational Equations that Involve Functions

For rational function, f ( x ) = x − 2 x 2 + 6 x + 8 , f ( x ) = x − 2 x 2 + 6 x + 8 , ⓐ find the domain of the function ⓑ solve f ( x ) = 5 f ( x ) = 5 ⓒ find the points on the graph at this function value.

For rational function, f ( x ) = x + 1 x 2 − 2 x − 3 , f ( x ) = x + 1 x 2 − 2 x − 3 , ⓐ find the domain of the function ⓑ solve f ( x ) = 1 f ( x ) = 1 ⓒ find the points on the graph at this function value.

For rational function, f ( x ) = 2 − x x 2 − 7 x + 10 , f ( x ) = 2 − x x 2 − 7 x + 10 , ⓐ find the domain of the function ⓑ solve f ( x ) = 2 f ( x ) = 2 ⓒ find the points on the graph at this function value.

For rational function, f ( x ) = 5 − x x 2 + 5 x + 6 , f ( x ) = 5 − x x 2 + 5 x + 6 , ⓐ find the domain of the function ⓑ solve f ( x ) = 3 f ( x ) = 3 ⓒ the points on the graph at this function value.

In the following exercises, solve.

C r = 2 π C r = 2 π for r . r .

I r = P I r = P for r . r .

v + 3 w − 1 = 1 2 v + 3 w − 1 = 1 2 for w . w .

x + 5 2 − y = 4 3 x + 5 2 − y = 4 3 for y . y .

a = b + 3 c − 2 a = b + 3 c − 2 for c . c .

m = n 2 − n m = n 2 − n for n . n .

1 p + 2 q = 4 1 p + 2 q = 4 for p . p .

3 s + 1 t = 2 3 s + 1 t = 2 for s . s .

2 v + 1 5 = 3 w 2 v + 1 5 = 3 w for w . w .

6 x + 2 3 = 1 y 6 x + 2 3 = 1 y for y . y .

m + 3 n − 2 = 4 5 m + 3 n − 2 = 4 5 for n . n .

r = s 3 − t r = s 3 − t for t . t .

E c = m 2 E c = m 2 for c . c .

R T = W R T = W for T . T .

3 x − 5 y = 1 4 3 x − 5 y = 1 4 for y . y .

c = 2 a + b 5 c = 2 a + b 5 for a . a .

Writing Exercises

Your class mate is having trouble in this section. Write down the steps you would use to explain how to solve a rational equation.

Alek thinks the equation y y + 6 = 72 y 2 − 36 + 4 y y + 6 = 72 y 2 − 36 + 4 has two solutions, y = −6 y = −6 and y = 4 . y = 4 . Explain why Alek is wrong.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ On a scale of 1 − 10 , 1 − 10 , how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

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Access for free at https://openstax.org/books/intermediate-algebra-2e/pages/1-introduction

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• Book title: Intermediate Algebra 2e
• Publication date: May 6, 2020
• Location: Houston, Texas
• Book URL: https://openstax.org/books/intermediate-algebra-2e/pages/1-introduction
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How to Solve Rational Numbers Word Problems

Indeed, addressing word problems with rational numbers can seem overwhelming at first. But with a structured step-by-step approach, a pinch of patience, and a splash of practice, it's possible to tackle these challenges. Let's delve into the intricate world of rational numbers and explore a comprehensive approach to dealing with word problems centered on these number types.

A Step-by-step Guide to Solve Rational Numbers Word Problems

Here is a step-by-step guide to solving rational numbers word problems:

Step 1: Dissect the Problem

Reading the word problem, your primary goal is to understand its essence. Look for key phrases and values that hint towards the type of problem. Jot down the rational numbers involved, and underline or highlight them if you need to. The specifics of the problem will guide your entire solution process.

Step 2: Identify the Unknowns

Now, ascertain what the problem is asking you to solve. This could be a specific value, a relationship, or perhaps even a series of values. Label these unknowns with symbols, most commonly ‘\(x\)’ or ‘\(y\)’, to facilitate further calculations.

Step 3: Translate into Mathematical Language

Convert the word problem into a mathematical equation using the information given. Treat phrases like “is the same as” as equals \((=)\), “added to” as plus \((+)\), “minus” or “less” as subtract \((-)\), “multiplied by” as times \((x)\), “divided by” as division \((÷)\). Remember, rational numbers can be expressed as fractions or decimals.

Step 4: Formulate the Equation(s)

Based on the translation, form the equation or equations necessary to solve the problem. Ensure that each equation correctly represents the situations or conditions described in the problem. If the word problem deals with ratios or proportions, your equation will likely contain fractions.

Step 5: Solve the Equation(s)

Employ your mathematical acumen to solve the formulated equations. Remember, the solution process may involve adding, subtracting, multiplying, or dividing both sides of the equation by the same (non-zero) number. Keep the golden rule of equations in mind: What you do to one side, you must do to the other.

Step 6: Verify Your Solution

Substitute your solution back into the original equations to validate them. If they hold true, congratulations – you’ve found your solution! If not, retrace your steps to spot any errors in formulation or calculation.

Step 7: Answer the Question

Finally, remember to answer the question asked in the word problem. You’ve done all the mathematical heavy lifting; now, it’s just about presenting your answer in a complete sentence or as per the question’s requirements.

by: Effortless Math Team about 9 months ago (category: Articles )

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Illustrative Mathematics Grade 7, Unit 5, Lesson 14: Solving Problems with Rational Numbers

Learning Targets:

• I can represent situations with expressions that include rational numbers.
• I can solve problems using the four operations with rational numbers.

Related Pages Illustrative Math Grade 7

Lesson 14: Solving Problems with Rational Numbers

Let’s use all four operations with signed numbers to solve problems.

Illustrative Math Unit 7.5, Lesson 14 (printable worksheets)

Lesson 14 Summary

Lesson 14.1 Which One Doesn’t Belong: Equations

Which equation doesn’t belong? 1/2 x = -50 -60t = 30 x + 90 = -100 -0.01 = -0.001x

Lesson 14.2 Draining and Filling a Tank

A tank of water is being drained. Due to a problem, the sensor does not start working until some time into the draining process. The sensor starts its recording at time zero when there are 770 liters in the tank.

• Given that the drain empties the tank at a constant rate of 14 liters per minute, complete the table:
• Later, someone wants to use the data to find out how long the tank had been draining before the sensor started. Complete this table:
• If the sensor started working 15 minutes into the tank draining, how much was in the tank to begin with?

Lesson 14.3 Buying and Selling Power

A utility company charges $0.12 per kilowatt-hour for energy a customer uses. They give a credit of $0.025 for every kilowatt-hour of electricity a customer with a solar panel generates that they don’t use themselves. A customer has a charge of $82.04 and a credit of -$4.10 on this month’s bill.

• What is the amount due this month?
• How many kilowatt-hours did they use?
• How many kilowatt-hours did they generate that they didn’t use themselves?

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• Write an expression that uses addition, subtraction, multiplication, and division and only negative numbers that has the same value.

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Rational Equations Word Problems Lesson

• Demonstrate an understanding of how to solve a word problem
• Demonstrate an understanding of how to solve an equation with rational expressions
• Learn how to solve word problems that involve rational equations

How to Solve a Word Problem with Rational Equations

Six-step method for solving word problems with rational expressions.

• Read the problem carefully and determine what you are asked to find
• Assign a variable to represent the unknown
• Write out an equation that describes the given situation
• Solve the equation
• State the answer using a nice clear sentence
• Check the result by reading back through the problem

Solving a Proportion Problem

Motion word problems with rational expressions, rate of work word problems, skills check:.

Solve each word problem.

Working alone, it takes Steve 11 hours to complete a restoration project on a truck. Jacob can perform the same task in 110 hours. How long would it take if they worked together?

Please choose the best answer.

Jamie’s hot tub has an outlet pipe that can empty the hot tub in 6 minutes. Additionally, her hot tub has an inlet pipe that can fill the hot tub in 3 minutes. If both pipes were turned on, how long would it take to fill a completely empty hot tub?

On her drive from Port Smith to Maryland, Stephanie averaged 51 miles per hour. If she had been able to average 60 miles per hour, she would have reached Maryland 3 hours earlier. What is the driving distance between Port Smith and Maryland?

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Precalculus

Course: precalculus   >   unit 4.

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Rational equations word problem: combined rates (example 2)

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Computer Science > Computation and Language

Title: large language models are unconscious of unreasonability in math problems.

Abstract: Large language models (LLMs) demonstrate substantial capabilities in solving math problems. However, they tend to produce hallucinations when given questions containing unreasonable errors. In this paper, we study the behavior of LLMs when faced with unreasonable math problems and further explore their potential to address these problems. First, we construct the Unreasonable Math Problem (UMP) benchmark to examine the error detection ability of LLMs. Experiments show that LLMs are able to detect unreasonable errors, but still fail in generating non-hallucinatory content. In order to improve their ability of error detection and correction, we further design a strategic prompt template called Critical Calculation and Conclusion(CCC). With CCC, LLMs can better self-evaluate and detect unreasonable errors in math questions, making them more reliable and safe in practical application scenarios.

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March 12, 2024

The Simplest Math Problem Could Be Unsolvable

The Collatz conjecture has plagued mathematicians for decades—so much so that professors warn their students away from it

By Manon Bischoff

Mathematicians have been hoping for a flash of insight to solve the Collatz conjecture.

James Brey/Getty Images

At first glance, the problem seems ridiculously simple. And yet experts have been searching for a solution in vain for decades. According to mathematician Jeffrey Lagarias, number theorist Shizuo Kakutani told him that during the cold war, “for about a month everybody at Yale [University] worked on it, with no result. A similar phenomenon happened when I mentioned it at the University of Chicago. A joke was made that this problem was part of a conspiracy to slow down mathematical research in the U.S.”

The Collatz conjecture—the vexing puzzle Kakutani described—is one of those supposedly simple problems that people tend to get lost in. For this reason, experienced professors often warn their ambitious students not to get bogged down in it and lose sight of their actual research.

The conjecture itself can be formulated so simply that even primary school students understand it. Take a natural number. If it is odd, multiply it by 3 and add 1; if it is even, divide it by 2. Proceed in the same way with the result x : if x is odd, you calculate 3 x + 1; otherwise calculate x / 2. Repeat these instructions as many times as possible, and, according to the conjecture, you will always end up with the number 1.

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For example: If you start with 5, you have to calculate 5 x 3 + 1, which results in 16. Because 16 is an even number, you have to halve it, which gives you 8. Then 8 / 2 = 4, which, when divided by 2, is 2—and 2 / 2 = 1. The process of iterative calculation brings you to the end after five steps.

Of course, you can also continue calculating with 1, which gives you 4, then 2 and then 1 again. The calculation rule leads you into an inescapable loop. Therefore 1 is seen as the end point of the procedure.

Following iterative calculations, you can begin with any of the numbers above and will ultimately reach 1.

Credit: Keenan Pepper/Public domain via Wikimedia Commons

It’s really fun to go through the iterative calculation rule for different numbers and look at the resulting sequences. If you start with 6: 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1. Or 42: 42 → 21 → 64 → 32 → 16 → 8 → 4 → 2 → 1. No matter which number you start with, you always seem to end up with 1. There are some numbers, such as 27, where it takes quite a long time (27 → 82 → 41 → 124 → 62 → 31 → 94 → 47 → 142 → 71 → 214 → 107 → 322 → 161 → 484 → 242 → 121 → 364 → 182 → 91 → 274 → ...), but so far the result has always been 1. (Admittedly, you have to be patient with the starting number 27, which requires 111 steps.)

But strangely there is still no mathematical proof that the Collatz conjecture is true. And that absence has mystified mathematicians for years.

The origin of the Collatz conjecture is uncertain, which is why this hypothesis is known by many different names. Experts speak of the Syracuse problem, the Ulam problem, the 3 n + 1 conjecture, the Hasse algorithm or the Kakutani problem.

German mathematician Lothar Collatz became interested in iterative functions during his mathematics studies and investigated them. In the early 1930s he also published specialist articles on the subject , but the explicit calculation rule for the problem named after him was not among them. In the 1950s and 1960s the Collatz conjecture finally gained notoriety when mathematicians Helmut Hasse and Shizuo Kakutani, among others, disseminated it to various universities, including Syracuse University.

Like a siren song, this seemingly simple conjecture captivated the experts. For decades they have been looking for proof that after repeating the Collatz procedure a finite number of times, you end up with 1. The reason for this persistence is not just the simplicity of the problem: the Collatz conjecture is related to other important questions in mathematics. For example, such iterative functions appear in dynamic systems, such as models that describe the orbits of planets. The conjecture is also related to the Riemann conjecture, one of the oldest problems in number theory.

Empirical Evidence for the Collatz Conjecture

In 2019 and 2020 researchers checked all numbers below 2 68 , or about 3 x 10 20 numbers in the sequence, in a collaborative computer science project . All numbers in that set fulfill the Collatz conjecture as initial values. But that doesn’t mean that there isn’t an outlier somewhere. There could be a starting value that, after repeated Collatz procedures, yields ever larger values that eventually rise to infinity. This scenario seems unlikely, however, if the problem is examined statistically.

An odd number n is increased to 3 n + 1 after the first step of the iteration, but the result is inevitably even and is therefore halved in the following step. In half of all cases, the halving produces an odd number, which must therefore be increased to 3 n + 1 again, whereupon an even result is obtained again. If the result of the second step is even again, however, you have to divide the new number by 2 twice in every fourth case. In every eighth case, you must divide it by 2 three times, and so on.

In order to evaluate the long-term behavior of this sequence of numbers , Lagarias calculated the geometric mean from these considerations in 1985 and obtained the following result: ( 3 / 2 ) 1/2 x ( 3 ⁄ 4 ) 1/4 x ( 3 ⁄ 8 ) 1/8 · ... = 3 ⁄ 4 . This shows that the sequence elements shrink by an average factor of 3 ⁄ 4 at each step of the iterative calculation rule. It is therefore extremely unlikely that there is a starting value that grows to infinity as a result of the procedure.

There could be a starting value, however, that ends in a loop that is not 4 → 2 → 1. That loop could include significantly more numbers, such that 1 would never be reached.

Such “nontrivial” loops can be found, for example, if you also allow negative integers for the Collatz conjecture: in this case, the iterative calculation rule can end not only at –2 → –1 → –2 → ... but also at –5 → –14 → –7 → –20 → –10 → –5 → ... or –17 → –50 → ... → –17 →.... If we restrict ourselves to natural numbers, no nontrivial loops are known to date—which does not mean that they do not exist. Experts have now been able to show that such a loop in the Collatz problem, however, would have to consist of at least 186 billion numbers .

The length of the Collatz sequences for all numbers from 1 to 9,999 varies greatly.

Credit: Cirne/Public domain via Wikimedia Commons

Even if that sounds unlikely, it doesn’t have to be. In mathematics there are many examples where certain laws only break down after many iterations are considered. For instance,the prime number theorem overestimates the number of primes for only about 10 316 numbers. After that point, the prime number set underestimates the actual number of primes.

Something similar could occur with the Collatz conjecture: perhaps there is a huge number hidden deep in the number line that breaks the pattern observed so far.

A Proof for Almost All Numbers

Mathematicians have been searching for a conclusive proof for decades. The greatest progress was made in 2019 by Fields Medalist Terence Tao of the University of California, Los Angeles, when he proved that almost all starting values of natural numbers eventually end up at a value close to 1.

“Almost all” has a precise mathematical meaning: if you randomly select a natural number as a starting value, it has a 100 percent probability of ending up at 1. ( A zero-probability event, however, is not necessarily an impossible one .) That’s “about as close as one can get to the Collatz conjecture without actually solving it,” Tao said in a talk he gave in 2020 . Unfortunately, Tao’s method cannot generalize to all figures because it is based on statistical considerations.

All other approaches have led to a dead end as well. Perhaps that means the Collatz conjecture is wrong. “Maybe we should be spending more energy looking for counterexamples than we’re currently spending,” said mathematician Alex Kontorovich of Rutgers University in a video on the Veritasium YouTube channel .

Perhaps the Collatz conjecture will be determined true or false in the coming years. But there is another possibility: perhaps it truly is a problem that cannot be proven with available mathematical tools. In fact, in 1987 the late mathematician John Horton Conway investigated a generalization of the Collatz conjecture and found that iterative functions have properties that are unprovable. Perhaps this also applies to the Collatz conjecture. As simple as it may seem, it could be doomed to remain unsolved forever.

This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.

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Ronna McDaniel, TV News and the Trump Problem

The former republican national committee chairwoman was hired by nbc and then let go after an outcry..

This transcript was created using speech recognition software. While it has been reviewed by human transcribers, it may contain errors. Please review the episode audio before quoting from this transcript and email [email protected] with any questions.

From “The New York Times,” I’m Michael Barbaro. This is “The Daily.”

[MUSIC PLAYING]

Today, the saga of Ronna McDaniel and NBC and what it reveals about the state of television news headed into the 2024 presidential race. Jim Rutenberg, a “Times” writer at large, is our guest.

It’s Monday, April 1.

Jim, NBC News just went through a very public, a very searing drama over the past week, that we wanted you to make sense of in your unique capacity as a longtime media and political reporter at “The Times.” This is your sweet spot. You were, I believe, born to dissect this story for us.

Oh, brother.

Well, on the one hand, this is a very small moment for a major network like NBC. They hire, as a contributor, not an anchor, not a correspondent, as a contributor, Ronna McDaniel, the former RNC chairwoman. It blows up in a mini scandal at the network.

But to me, it represents a much larger issue that’s been there since that moment Donald J. Trump took his shiny gold escalator down to announce his presidential run in 2015. This struggle by the news media to figure out, especially on television, how do we capture him, cover him for all of his lies, all the challenges he poses to Democratic norms, yet not alienate some 74, 75 million American voters who still follow him, still believe in him, and still want to hear his reality reflected in the news that they’re listening to?

Right. Which is about as gnarly a conundrum as anyone has ever dealt with in the news media.

Well, it’s proven so far unsolvable.

Well, let’s use the story of what actually happened with Ronna McDaniel and NBC to illustrate your point. And I think that means describing precisely what happened in this situation.

The story starts out so simply. It’s such a basic thing that television networks do. As elections get underway, they want people who will reflect the two parties.

They want talking heads. They want insiders. They want them on their payroll so they can rely on them whenever they need them. And they want them to be high level so they can speak with great knowledge about the two major candidates.

Right. And rather than needing to beg these people to come on their show at 6 o’clock, when they might be busy and it’s not their full-time job, they go off and they basically put them on retainer for a bunch of money.

Yeah. And in this case, here’s this perfect scenario because quite recently, Ronna McDaniel, the chairwoman of the Republican National Committee through the Trump era, most of it, is now out on the market. She’s actually recently been forced out of the party. And all the networks are interested because here’s the consummate insider from Trump world ready to get snatched up under contract for the next election and can really represent this movement that they’ve been trying to capture.

So NBC’S key news executives move pretty aggressively, pretty swiftly, and they sign her up for a $300,000 a year contributor’s contract.

Nice money if you can get it.

Not at millions of dollars that they pay their anchors, but a very nice contract. I’ll take it. You’ll take it. In the eyes of NBC execs she was perfect because she can be on “Meet the Press” as a panelist. She can help as they figure out some of their coverage. They have 24 hours a day to fill and here’s an official from the RNC. You can almost imagine the question that would be asked to her. It’s 10:00 PM on election night. Ronna, what are the Trump people thinking right now? They’re looking at the same numbers you are.

That was good, but that’s exactly it. And we all know it, right? This is television in our current era.

So last Friday, NBC makes what should be a routine announcement, but one they’re very proud of, that they’ve hired Ronna McDaniel. And in a statement, they say it couldn’t be a more important moment to have a voice like Ronna’s on the team. So all’s good, right? Except for there’s a fly in the ointment.

Because it turns out that Ronna McDaniel has been slated to appear on “Meet the Press,” not as a paid NBC contributor, but as a former recently ousted RNC chair with the “Meet The Press” host, Kristen Welker, who’s preparing to have a real tough interview with Ronna McDaniel. Because of course, Ronna McDaniel was chair of the party and at Trump’s side as he tried to refuse his election loss. So this was supposed to be a showdown interview.

From NBC News in Washington, the longest-running show in television history. This is “Meet The Press” with Kristen Welker.

And here, all of a sudden, Kristin Welker is thrown for a loop.

In full disclosure to our viewers, this interview was scheduled weeks before it was announced that McDaniel would become a paid NBC News contributor.

Because now, she’s actually interviewing a member of the family who’s on the same payroll.

Right. Suddenly, she’s interviewing a colleague.

This will be a news interview, and I was not involved in her hiring.

So what happens during the interview?

So Welker is prepared for a tough interview, and that’s exactly what she does.

Can you say, as you sit here today, did Joe Biden win the election fair and square?

He won. He’s the legitimate president.

Did he win fair and square?

Fair and square, he won. It’s certified. It’s done.

She presses her on the key question that a lot of Republicans get asked these days — do you accept Joe Biden was the winner of the election?

But, I do think, Kristen —

Ronna, why has it taken you until now to say that? Why has it taken you until now to be able to say that?

I’m going to push back a little.

McDaniel gets defensive at times.

Because I do think it’s fair to say there were problems in 2020. And to say that does not mean he’s not the legitimate president.

But, Ronna, when you say that, it suggests that there was something wrong with the election. And you know that the election was the most heavily scrutinized. Chris Krebs —

It’s a really combative interview.

I want to turn now to your actions in the aftermath of the 2020 election.

And Welker actually really does go deeply into McDaniel’s record in those weeks before January 6.

On November 17, you and Donald Trump were recorded pushing two Republican Michigan election officials not to certify the results of the election. And on the call —

For instance, she presses McDaniel on McDaniel’s role in an attempt to convince a couple county commissioner level canvassers in Michigan to not certify Biden’s victory.

Our call that night was to say, are you OK? Vote your conscience. Not pushing them to do anything.

McDaniel says, look, I was just telling them to vote their conscience. They should do whatever they think is right.

But you said, do not sign it. If you can go home tonight, do not sign it. How can people read that as anything other than a pressure campaign?

And Welker’s not going to just let her off the hook. Welker presses her on Trump’s own comments about January 6 and Trump’s efforts recently to gloss over some of the violence, and to say that those who have been arrested, he’ll free them.

Do you support that?

I want to be very clear. The violence that happened on January 6 is unacceptable.

And this is a frankly fascinating moment because you can hear McDaniel starting to, if not quite reverse some of her positions, though in some cases she does that, at least really soften her language. It’s almost as if she’s switching uniforms from the RNC one to an NBC one or almost like breaking from a role she was playing.

Ronna, why not speak out earlier? Why just speak out about that now?

When you’re the RNC chair, you kind of take one for the whole team, right? Now, I get to be a little bit more myself.

She says, hey, you know what? Sometimes as RNC chair, you just have to take it for the team sometimes.

Right. What she’s really saying is I did things as chairwoman of the Republican National committee that now that I no longer have that job, I can candidly say, I wished I hadn’t done, which is very honest. But it’s also another way of saying I’m two faced, or I was playing a part.

Ronna McDaniel, thank you very much for being here this morning.

Then something extraordinary happens. And I have to say, I’ve never seen a moment like this in decades of watching television news and covering television news.

Welcome back. The panel is here. Chuck Todd, NBC News chief political analyst.

Welker brings her regular panel on, including Chuck Todd, now the senior NBC political analyst.

Chuck, let’s dive right in. What were your takeaways?

And he launches right into what he calls —

Look, let me deal with the elephant in the room.

The elephant being this hiring of McDaniel.

I think our bosses owe you an apology for putting you in this situation.

And he proceeds, on NBC’S air, to lace into management for, as he describes it, putting Welker in this crazy awkward position.

Because I don’t know what to believe. She is now a paid contributor by NBC News. I have no idea whether any answer she gave to you was because she didn’t want to mess up her contract.

And Todd is very hung up on this idea that when she was speaking for the party, she would say one thing. And now that she’s on the payroll at NBC, she’s saying another thing.

She has credibility issues that she still has to deal with. Is she speaking for herself, or is she speaking on behalf of who’s paying her?

Todd is basically saying, how are we supposed to know which one to believe.

What can we believe?

It is important for this network and for always to have a wide aperture. Having ideological diversity on this panel is something I prided myself on.

And what he’s effectively saying is that his bosses should have never hired her in this capacity.

I understand the motivation, but this execution, I think, was poor.

Someone said to me last night we live in complicated times. Thank you guys for being here. I really appreciate it.

Now, let’s just note here, this isn’t just any player at NBC. Chuck Todd is obviously a major news name at the network. And him doing this appears to just open the floodgates across the entire NBC News brand, especially on its sister cable network, MSNBC.

And where I said I’d never seen anything like what I saw on “Meet the Press” that morning, I’d never seen anything like this either. Because now, the entire MSNBC lineup is in open rebellion. I mean, from the minute that the sun comes up. There is Joe Scarborough and Mika Brzezinski.

We weren’t asked our opinion of the hiring. But if we were, we would have strongly objected to it.

They’re on fire over this.

believe NBC News should seek out conservative Republican voices, but it should be conservative Republicans, not a person who used her position of power to be an anti-democracy election denier.

But it rolls out across the entire schedule.

Because Ronna McDaniel has been a major peddler of the big lie.

The fact that Ms. McDaniel is on the payroll at NBC News, to me that is inexplicable. I mean, you wouldn’t hire a mobster to work at a DA’s office.

Rachel Maddow devotes an entire half hour.

It’s not about just being associated with Donald Trump and his time in the Republican Party. It’s not even about lying or not lying. It’s about our system of government.

Thumbing their noses at our bosses and basically accusing them of abetting a traitorous figure in American history. I mean, just extraordinary stuff. It’s television history.

And let’s face it, we journalists, our bosses, we can be seen as crybabies, and we’re paid complaining. Yeah, that’s what we’re paid to do. But in this case, the NBC executives cannot ignore this, because in the outcry, there’s a very clear point that they’re all making. Ronna McDaniel is not just a voice from the other side. She was a fundamental part of Trump’s efforts to deny his election loss.

This is not inviting the other side. This is someone who’s on the wrong side —

Of history.

Of history, of these moments that we’ve covered and are still covering.

And I think it’s fair to say that at this point, everyone understands that Ronna McDaniel’s time at NBC News is going to be very short lived. Yeah, basically, after all this, the executives at NBC have to face facts it’s over. And on Tuesday night, they release a statement to the staff saying as much.

They don’t cite the questions about red lines or what Ronna McDaniel represented or didn’t represent. They just say we need to have a unified newsroom. We want cohesion. This isn’t working.

I think in the end, she was a paid contributor for four days.

Yeah, one of the shortest tenures in television news history. And look, in one respect, by their standards, this is kind of a pretty small contract, a few hundred thousand dollars they may have to pay out. But it was way more costly because they hired her. They brought her on board because they wanted to appeal to these tens of millions of Americans who still love Donald J. Trump.

And what happens now is that this entire thing is blown up in their face, and those very same people now see a network that, in their view, in the view of Republicans across the country, this network will not accept any Republicans. So it becomes more about that. And Fox News, NBC’S longtime rival, goes wall to wall with this.

Now, NBC News just caved to the breathless demands from their far left, frankly, emotionally unhinged host.

I mean, I had it on my desk all day. And every minute I looked at that screen, it was pounding on these liberals at NBC News driving this Republican out.

It’s the shortest tenure in TV history, I think. But why? Well, because she supports Donald Trump, period.

So in a way, this leaves NBC worse off with that Trump Republican audience they had wanted to court than maybe even they were before. It’s like a boomerang with a grenade on it.

Yeah, it completely explodes in their face. And that’s why to me, the whole episode is so representative of this eight-year conundrum for the news media, especially on television. They still haven’t been able to crack the code for how to handle the Trump movement, the Trump candidacy, and what it has wrought on the American political system and American journalism.

We’ll be right back.

Jim, put into context this painful episode of NBC into that larger conundrum you just diagnosed that the media has faced when it comes to Trump.

Well, Michael, it’s been there from the very beginning, from the very beginning of his political rise. The media was on this kind of seesaw. They go back and forth over how to cover him. Sometimes they want to cover him quite aggressively because he’s such a challenging candidate. He was bursting so many norms.

But at other times, there was this instinct to understand his appeal, for the same reason. He’s such an unusual candidate. So there was a great desire to really understand his voters. And frankly, to speak to his voters, because they’re part of the audience. And we all lived it, right?

But just let me take you back anyway because everything’s fresh again with perspective. And so if you go back, let’s look at when he first ran. The networks, if you recall, saw him as almost like a novelty candidate.

He was going to spice up what was expected to be a boring campaign between the usual suspects. And he was a ratings magnet. And the networks, they just couldn’t get enough of it. And they allowed him, at times, to really shatter their own norms.

Welcome back to “Meet the Press,” sir.

Good morning, Chuck.

Good morning. Let me start —

He was able to just call into the studio and riff with the likes of George Stephanopoulos and Chuck Todd.

What does it have to do with Hillary?

She can’t talk about me because nobody respects women more than Donald Trump.

And CNN gave him a lot of unmitigated airtime, if you recall during the campaign. They would run the press conferences.

It’s the largest winery on the East Coast. I own it 100 percent.

And let him promote his Trump steaks and his Trump wine.

Trump steaks. Where are the steaks? Do we have steaks?

I mean, it got that crazy. But again, the ratings were huge. And then he wins. And because they had previously given him all that airtime, they’ve, in retrospect, sort of given him a political gift, and more than that now have a journalistic imperative to really address him in a different way, to cover him as they would have covered any other candidate, which, let’s face it, they weren’t doing initially. So there’s this extra motivation to make up for lost ground and maybe for some journalistic omissions.

Right. Kind of correct for the lack of a rigorous journalistic filter in the campaign.

Exactly. And the big thing that this will be remembered for is we’re going to call a lie a lie.

I don’t want to sugarcoat this because facts matter, and the fact is President Trump lies.

Trump lies. We’re going to say it’s a lie.

And I think we can’t just mince around it because they are lies. And so we need to call them what they are.

We’re no longer going to use euphemisms or looser language we’re. Going to call it for what it is.

Trump lies in tweets. He spreads false information at rallies. He lies when he doesn’t need to. He lies when the truth is more than enough for him.

CNN was running chyrons. They would fact check Trump and call lies lies on the screen while Trump is talking. They were challenging Trump to his face —

One of the statements that you made in the tail end of the campaign in the midterms that —

Here we go.

That — well, if you don’t mind, Mr. President, that this caravan was an invasion.

— in these crazy press conferences —

They’re are hundreds of miles away, though. They’re hundreds and hundreds of miles away. That’s not an invasion.

Honestly, I think you should let me run the country. You run CNN. And if you did it well, your ratings —

Well, let me ask — if I may ask one other question. Mr. President, if I may ask another question. Are you worried —

That’s enough. That’s enough.

And Trump is giving it right back.

I tell you what, CNN should be ashamed of itself having you working for them. You are a rude, terrible person. You shouldn’t be working for CNN.

Very combative.

So this was this incredibly fraught moment for the American press. You’ve got tens of millions of Trump supporters seeing what’s really basic fact checking. These look like attacks to Trump supporters. Trump, in turn, is calling the press, the reporters are enemies of the people. So it’s a terrible dynamic.

And when January 6 happens, it’s so obviously out of control. And what the traditional press that follows, traditional journalistic rules has to do is make it clear that the claims that Trump is making about a stolen election are just so abjectly false that they don’t warrant a single minute of real consideration once the reporting has been done to show how false they are. And I think that American journalism really emerged from that feeling strongly about its own values and its own place in society.

But then there’s still tens of millions of Trump voters, and they don’t feel so good about the coverage. And they don’t agree that January 6 was an insurrection. And so we enter yet another period, where the press is going to have to now maybe rethink some things.

In what way?

Well, there’s a kind of quiet period after January 6. Trump is off of social media. The smoke is literally dissipating from the air in Washington. And news executives are kind of standing there on the proverbial battlefield, taking a new look at their situation.

And they’re seeing that in this clearer light, they’ve got some new problems, perhaps none more important for their entire business models than that their ratings are quickly crashing. And part of that diminishment is that a huge part of the country, that Trump-loving part of the audience, is really now severed from him from their coverage.

They see the press as actually, in some cases, being complicit in stealing an election. And so these news executives, again, especially on television, which is so ratings dependent, they’ve got a problem. So after presumably learning all these lessons about journalism and how to confront power, there’s a first subtle and then much less subtle rethinking.

Maybe we need to pull back from that approach. And maybe we need to take some new lessons and switch it up a little bit and reverse some of what we did. And one of the best examples of this is none other than CNN.

It had come under new management, was being led by a guy named Chris Licht, a veteran of cable news, but also Stephen Colbert’s late night show in his last job. And his new job under this new management is we’re going to recalibrate a little bit. So Chris Licht proceeds to try to bring the network back to the center.

And how does he do that?

Well, we see some key personalities who represented the Trump combat era start losing air time and some of them lose their jobs. There’s talk of, we want more Republicans on the air. There was a famous magazine article about Chris Licht’s balancing act here.

And Chris Licht says to a reporter, Tim Alberta of the “Atlantic” magazine, look, a lot in the media, including at his own network, quote unquote, “put on a jersey, took a side.” They took a side. And he says, I think we understand that jersey cannot go back on him. Because he says in the end of the day, by the way, it didn’t even work. We didn’t change anyone’s mind.

He’s saying that confrontational approach that defined the four years Trump was in office, that was a reaction to the feeling that TV news had failed to properly treat Trump with sufficient skepticism, that that actually was a failure both of journalism and of the TV news business. Is that what he’s saying?

Yeah. On the business side, it’s easier call, right? You want a bigger audience, and you’re not getting the bigger audience. But he’s making a journalistic argument as well that if the job is to convey the truth and take it to the people, and they take that into account as they make their own voting decisions and formulate their own opinions about American politics, if tens of millions of people who do believe that election was stolen are completely tuning you out because now they see you as a political combatant, you’re not achieving your ultimate goal as a journalist.

And what does Licht’s “don’t put a jersey back on” approach look like on CNN for its viewers?

Well, It didn’t look good. People might remember this, but the most glaring example —

Please welcome, the front runner for the Republican nomination for president, Donald Trump.

— was when he held a town hall meeting featuring Donald J. Trump, now candidate Trump, before an audience packed with Trump’s fans.

You look at what happened during that election. Unless you’re a very stupid person, you see what happens. A lot of the people —

Trump let loose a string of falsehoods.

Most people understand what happened. It was a rigged election.

The audience is pro-Trump audience, was cheering him on.

Are you ready? Are you ready? Can I talk?

Yeah, what’s your answer?

Can I? Do you mind?

I would like for you to answer the question.

OK. It’s very simple to answer.

That’s why I asked it.

It’s very simple. You’re a nasty person, I’ll tell you that.

And during, the CNN anchor hosting this, Kaitlan Collins, on CNN’s own air, it was a disaster.

It felt like a callback to the unlearned lessons of 2016.

Yeah. And in this case, CNN’s staff was up in arms.

Big shakeup in the cable news industry as CNN makes another change at the top.

Chris Licht is officially out at CNN after a chaotic run as chairman and CEO.

And Chris Licht didn’t survive it.

The chief executive’s departure comes as he faced criticism in recent weeks after the network hosted a town hall with Donald Trump and the network’s ratings started to drop.

But I want to say that the CNN leadership still, even after that, as they brought new leadership in, said, this is still the path we’re going to go on. Maybe that didn’t work out, but we’re still here. This is still what we have to do.

Right. And this idea is very much in the water of TV news, that this is the right overall direction.

Yeah. This is, by no means, isolated to CNN. This is throughout the traditional news business. These conversations are happening everywhere. But CNN was living it at that point.

And this, of course, is how we get to NBC deciding to hire Ronna McDaniel.

Right. Because they’re picking up — right where that conversation leaves off, they’re having the same conversation. But for NBC, you could argue this tension between journalistic values and audience. It’s even more pressing. Because even though MSNBC is a niche cable network, NBC News is part of an old-fashioned broadcast network. It’s on television stations throughout the country.

And in fact, those networks, they still have 6:30 newscasts. And believe it or not, millions of people still watch those every night. Maybe not as many as they used to, but there’s still some six or seven million people tuning in to nightly news. That’s important.

Right. We should say that kind of number is sometimes double or triple that of the cable news prime time shows that get all the attention.

On their best nights. So this is big business still. And that business is based on broad — it’s called broadcast for a reason. That’s based on broad audiences. So NBC had a business imperative, and they argue they had a journalistic imperative.

So given all of that, Jim, I think the big messy question here is, when it comes to NBC, did they make a tactical error around hiring the wrong Republican which blew up? Or did they make an even larger error in thinking that the way you handle Trump and his supporters is to work this hard to reach them, when they might not even be reachable?

The best way to answer that question is to tell you what they’re saying right now, NBC management. What the management saying is, yes, this was a tactical error. This was clearly the wrong Republican. We get it.

But they’re saying, we are going to — and they said this in their statement, announcing that they were severing ties with McDaniel. They said, we’re going to redouble our efforts to represent a broad spectrum of the American votership. And that’s what they meant was that we’re going to still try to reach these Trump voters with people who can relate to them and they can relate to.

But the question is, how do you even do that when so many of his supporters believe a lie? How is NBC, how is CNN, how are any of these TV networks, if they have decided that this is their mission, how are they supposed to speak to people who believe something fundamentally untrue as a core part of their political identity?

That’s the catch-22. How do you get that Trump movement person who’s also an insider, when the litmus test to be an insider in the Trump movement is to believe in the denialism or at least say you do? So that’s a real journalistic problem. And the thing that we haven’t really touched here is, what are these networks doing day in and day out?

They’re not producing reported pieces, which I think it’s a little easier. You just report the news. You go out into the world. You talk to people, and then you present it to the world as a nuanced portrait of the country. This thing is true. This thing is false. Again, in many cases, pretty straightforward. But their bread and butter is talking heads. It’s live. It’s not edited. It’s not that much reported.

So their whole business model especially, again, on cable, which has 24 hours to fill, is talking heads. And if you want the perspective from the Trump movement, journalistically, especially when it comes to denialism, but when it comes to some other major subjects in American life, you’re walking into a place where they’re going to say things that aren’t true, that don’t pass your journalistic standards, the most basic standards of journalism.

Right. So you’re saying if TV sticks with this model, the kind of low cost, lots of talk approach to news, then they are going to have to solve the riddle of who to bring on, who represents Trump’s America if they want that audience. And now they’ve got this red line that they’ve established, that that person can’t be someone who denies the 2020 election reality. But like you just said, that’s the litmus test for being in Trump’s orbit.

So this doesn’t really look like a conundrum. This looks like a bit of a crisis for TV news because it may end up meaning that they can’t hire that person that they need for this model, which means that perhaps a network like NBC does need to wave goodbye to a big segment of these viewers and these eyeballs who support Trump.

I mean, on the one hand, they are not ready to do that, and they would never concede that that’s something they’re ready to do. The problem is barring some kind of change in their news model, there’s no solution to this.

But why bar changes to their news model, I guess, is the question. Because over the years, it’s gotten more and more expensive to produce news, the news that I’m talking about, like recorded packages and what we refer to as reporting. Just go out and report the news.

Don’t gab about it. Just what’s going on, what’s true, what’s false. That’s actually very expensive in television. And they don’t have the kind of money they used to have. So the talking heads is their way to do programming at a level where they can afford it.

They do some packages. “60 Minutes” still does incredible work. NBC does packages, but the lion’s share of what they do is what we’re talking about. And that’s not going to change because the economics aren’t there.

So then a final option, of course, to borrow something Chris Licht said, is that a network like NBC perhaps doesn’t put a jersey on, but accepts the reality that a lot of the world sees them wearing a jersey.

Yeah. I mean, nobody wants to be seen as wearing a jersey in our business. No one wants to be wearing a jersey on our business. But maybe what they really have to accept is that we’re just sticking to the true facts, and that may look like we’re wearing a jersey, but we’re not. And that may, at times, look like it’s lining up more with the Democrats, but we’re not.

If Trump is lying about a stolen election, that’s not siding against him. That’s siding for the truth, and that’s what we’re doing. Easier said than done. And I don’t think any of these concepts are new.

I think there have been attempts to do that, but it’s the world they’re in. And it’s the only option they really have. We’re going to tell you the truth, even if it means that we’re going to lose a big part of the country.

Well, Jim, thank you very much.

Thank you, Michael.

Here’s what else you need to know today.

[PROTESTERS CHANTING]

Over the weekend, thousands of protesters took to the streets of Tel Aviv and Jerusalem in some of the largest domestic demonstrations against the government of Prime Minister Benjamin Netanyahu since Israel invaded Gaza in the fall.

[NON-ENGLISH SPEECH]

Some of the protesters called on Netanyahu to reach a cease fire deal that would free the hostages taken by Hamas on October 7. Others called for early elections that would remove Netanyahu from office.

During a news conference on Sunday, Netanyahu rejected calls for early elections, saying they would paralyze his government at a crucial moment in the war.

Today’s episode was produced by Rob Szypko, Rikki Novetsky, and Alex Stern, with help from Stella Tan.

It was edited by Brendan Klinkenberg with help from Rachel Quester and Paige Cowett. Contains original music by Marion Lozano, Dan Powell, and Rowan Niemisto and was engineered by Chris Wood. Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly.

That’s it for “The Daily.” I’m Michael Barbaro. See you tomorrow.

• April 2, 2024   •   29:32 Kids Are Missing School at an Alarming Rate
• April 1, 2024   •   36:14 Ronna McDaniel, TV News and the Trump Problem
• March 29, 2024   •   48:42 Hamas Took Her, and Still Has Her Husband
• March 28, 2024   •   33:40 The Newest Tech Start-Up Billionaire? Donald Trump.
• March 27, 2024   •   28:06 Democrats’ Plan to Save the Republican House Speaker
• March 26, 2024   •   29:13 The United States vs. the iPhone
• March 25, 2024   •   25:59 A Terrorist Attack in Russia
• March 24, 2024   •   21:39 The Sunday Read: ‘My Goldendoodle Spent a Week at Some Luxury Dog ‘Hotels.’ I Tagged Along.’
• March 22, 2024   •   35:30 Chuck Schumer on His Campaign to Oust Israel’s Leader
• March 21, 2024   •   27:18 The Caitlin Clark Phenomenon
• March 20, 2024   •   25:58 The Bombshell Case That Will Transform the Housing Market
• March 19, 2024   •   27:29 Trump’s Plan to Take Away Biden’s Biggest Advantage

Hosted by Michael Barbaro

Featuring Jim Rutenberg

Produced by Rob Szypko ,  Rikki Novetsky and Alex Stern

With Stella Tan

Edited by Brendan Klinkenberg ,  Rachel Quester and Paige Cowett

Original music by Marion Lozano ,  Dan Powell and Rowan Niemisto

Engineered by Chris Wood

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Ronna McDaniel’s time at NBC was short. The former Republican National Committee chairwoman was hired as an on-air political commentator but released just days later after an on-air revolt by the network’s leading stars.

Jim Rutenberg, a writer at large for The Times, discusses the saga and what it might reveal about the state of television news heading into the 2024 presidential race.

On today’s episode

Jim Rutenberg , a writer at large for The New York Times.

Background reading

Ms. McDaniel’s appointment had been immediately criticized by reporters at the network and by viewers on social media.

The former Republican Party leader tried to downplay her role in efforts to overturn the 2020 election. A review of the record shows she was involved in some key episodes .

There are a lot of ways to listen to The Daily. Here’s how.

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The Daily is made by Rachel Quester, Lynsea Garrison, Clare Toeniskoetter, Paige Cowett, Michael Simon Johnson, Brad Fisher, Chris Wood, Jessica Cheung, Stella Tan, Alexandra Leigh Young, Lisa Chow, Eric Krupke, Marc Georges, Luke Vander Ploeg, M.J. Davis Lin, Dan Powell, Sydney Harper, Mike Benoist, Liz O. Baylen, Asthaa Chaturvedi, Rachelle Bonja, Diana Nguyen, Marion Lozano, Corey Schreppel, Rob Szypko, Elisheba Ittoop, Mooj Zadie, Patricia Willens, Rowan Niemisto, Jody Becker, Rikki Novetsky, John Ketchum, Nina Feldman, Will Reid, Carlos Prieto, Ben Calhoun, Susan Lee, Lexie Diao, Mary Wilson, Alex Stern, Dan Farrell, Sophia Lanman, Shannon Lin, Diane Wong, Devon Taylor, Alyssa Moxley, Summer Thomad, Olivia Natt, Daniel Ramirez and Brendan Klinkenberg.

Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly. Special thanks to Sam Dolnick, Paula Szuchman, Lisa Tobin, Larissa Anderson, Julia Simon, Sofia Milan, Mahima Chablani, Elizabeth Davis-Moorer, Jeffrey Miranda, Renan Borelli, Maddy Masiello, Isabella Anderson and Nina Lassam.

Jim Rutenberg is a writer at large for The Times and The New York Times Magazine and writes most often about media and politics. More about Jim Rutenberg

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How can Australia solve the math teacher shortage? It can start by training more existing teachers to teach math

I magine if you enrolled your child in swimming lessons but instead of a qualified swimming instructor, they were taught freestyle technique by a soccer coach.

Something similar is happening in classrooms around Australia every day. As part of the ongoing teacher shortage, there are significant numbers of teachers teaching " out-of-field ." This means they are teaching subjects they are not qualified to teach.

One of the subjects where out-of-field teaching is particularly common is math.

A 2021 report on Australia's teaching workforce found that 40% of those teaching high school mathematics are out-of-field (English and science were 28% and 29%, respectively).

Another 2021 study of students in Year 8 found they were more likely to be taught by teachers who had specialist training in both math and math education if they went to a school in an affluent area rather than a disadvantaged one (54% compared with 31%).

Our new report looks at how we can fix this situation by training more existing teachers in math education.

Why is this a problem?

Mathematics is one of the key parts of school education. But we are seeing worrying signs students are not receiving the math education they need.

The 2021 study of Year 8 students showed those taught by teachers with a university degree majoring in math had markedly higher results, compared with those taught by out-of-field teachers.

We also know math skills are desperately needed in the broader workforce. The burgeoning worlds of big data and artificial intelligence rely on mathematical and statistical thinking, formulae and algorithms. Math has also been identified as a national skill shortages priority area .

What do we do about this?

There have been repeated efforts to address teacher shortages, including trying to retain existing mathematics teachers, having specialist teachers teaching across multiple schools and higher salaries . There is also a push to train more teachers from scratch, which of course will take many years to implement.

There is one strategy, however, that has not yet been given much attention by policy makers: upgrading current teachers' math and statistics knowledge and their skills in how to teach these subjects.

They already have training and expertise in how to teach and a commitment to the profession. Specific training in math will mean they can move from being out-of-field to "in-field".

How to give teachers this training

A new report commissioned by mathematics and statistics organizations in Australia (including the Australian Mathematical Sciences Institute) looks at what is currently available in Australia to train teachers in math.

It identified 12 different courses to give existing teachers math teaching skills. They varied in terms of location, duration (from six months to 18 months full-time) and aims.

For example, some were only targeted at teachers who want to teach math in the junior and middle years of high school. Some taught university-level math and others taught school-level math. Some had government funding support; others could cost students more than A$37,000.

Overall, we found the current system is confusing for teachers to navigate. There are complex differences between states about what qualifies a teacher to be "in-field" for a subject area.

In the current incentive environment, we found these courses cater to a very small number of teachers. For example, in 2024 in New South Wales this year there are only about 50 government-sponsored places available.

This is not adequate. Pre-COVID, it was estimated we were losing more than 1,000 equivalent full-time math teachers per year to attrition and retirement and new graduates were at best in the low hundreds.

But we don't know exactly how many extra teachers need to be trained in math. One of the key recommendations of the report is for accurate national data of every teacher's content specializations.

We need a national approach

The report also recommends a national strategy to train more existing teachers to be math teachers. This would replace the current piecemeal approach.

It would involve a standard training regime across Australia with government and school-system incentives for people to take up extra training in math.

There is international evidence to show a major upskilling program like this could work.

In Ireland, where the same problem was identified, the government funds a scheme run by a group of universities. Since 2012, teachers have been able to get a formal qualification (a professional diploma). Between 2009 and 2018 the percentage of out-of-field math teaching in Ireland dropped from 48% to 25%.

To develop a similar scheme here in Australia, we would need coordination between federal and state governments and universities. Based on the Irish experience, it would also require several million dollars in funding.

But with students receiving crucial math lessons every day by teachers who are not trained to teach math, the need is urgent.

The report mentioned in this article was commissioned by the Australian Mathematical Sciences Institute, the Australian Mathematical Society, the Statistical Society of Australia, the Mathematics Education Research Group of Australasia and the Actuaries Institute.

This article is republished from The Conversation under a Creative Commons license. Read the original article .

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