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Direct Proportion

Here we will learn about direct proportion, including what direct proportion is and how to solve direct proportion problems. We will also look at solving word problems involving direct proportion.

There are also direct proportion worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is direct proportion?

Direct proportion is a type of proportionality relationship. For direct proportion, as one value increases, so does the other value and conversely, as one value decreases, so does the other value.

The symbol \textbf{∝} represents a proportional relationship .

If y is directly proportional to x, we can write this relationship as:

Direct proportion is useful in numerous real life situations such as exchange rates, conversion between units, and fuel prices.

What is direct proportion?

Direct proportion formula

The direct proportion formula allows us to express the relationship between two variables, using an equivalence relationship; the formula contains an equals symbol (=) instead of the proportionality symbol (\propto).

When y is directly proportional to x, the value of y \div x is a constant value. This is known as the constant of proportionality and we use the letter k to denote this number.

Given that k=y\div{x}, we can rearrange this formula to make y the subject, and hence obtain the standard format of the direct proportion formula:

Step-by-step guide: Direct proportion formula

Graphs representing a direct proportion between two variables

Proportional relationships can also be represented graphically. 

If we sketched the straight line graph for the equation y=kx, the line must go through the origin (0,0) as the value of the y -intercept is 0, and the gradient of the line is equal to the value of k.

Step-by-step guide: y=mx+c.

Direct proportion image 1

Note, the value of y can be proportional to other powers of x including x^{2}, x^{3}, or even \sqrt{x}. Each of these has a different algebraic and graphical representation.

Step-by-step guide: Directly proportional graphs / inversely proportional graphs

How to use direct proportion

In order to work out an unknown value given a directly proportional relationship:

Write down the direct proportion formula.

Determine the value of \textbf{k} .

  • Substitute \textbf{k} and the known value into the direct proportion formula .

Solve the equation.

Explain how to use direct proportion

Explain how to use direct proportion

Direct proportion worksheet

Get your free direct proportion worksheet of 20+ questions and answers. Includes reasoning and applied questions.

Direct proportion examples

Example 1: y is directly proportional to x (table).

Given that y is directly proportional to x, calculate the value for y when x=6.

Direct proportion example 1

As y \propto x we can state the formula y=kx.

2 Determine the value of \textbf{k} .

From the table, we can see that when x=2, y=5. By substituting these values into the formula, we get the value of k.

3 Substitute \textbf{k} and the known value into the direct proportion formula.

Substituting k=2.5 into the formula, we have:

To find the value for y when x=6, substituting x=6 into the equation, we have

4 Solve the equation.

Example 2: y is directly proportional to x (table)

Let y\propto{x}. Calculate the value for x when y=32.

Direct proportion example 2

As y\propto{x}, we can state the formula y=kx.

The constant of proportionality cannot be calculated at the point (0,0) as by substituting this into the formula, we have k=0 \div 0 which is not mathematically possible as we cannot divide by 0.

We therefore have to use the other coordinate (5,8) and substitute these values into y=kx to calculate k.

When x=5 and y=8,

\begin{aligned} 8&=k\times{5}\\\\ k&=8\div{5}\\\\ k&=1.6 \end{aligned}

Substitute \textbf{k} and the known value into the direct proportion formula.

We now have y=1.6x . Substituting y=32 into the equation, we can calculate the value for x:

32=1.6 \times x

Example 3: y is directly proportional to x 2 (table)

Let y be directly proportional to x^2. Calculate the missing value p.

Direct proportion example 3

We are now given the relationship that y\propto{x}^{2}. This means that the constant of proportionality k=y\div{x}^{2}. Rearranging this formula by multiplying both sides by x^{2}, we get:

We can use either of the two coordinates (2,12) or (3,27) to find k as we know the values for x and y for each point.

Substituting (2,12) into y=kx^2, we have

\begin{aligned} 12&=k \times 2^2 \\\\ 12&=k \times 4 \\\\ 12 \div 4 &=k \\\\ k&=3 \end{aligned}

As k=3, the direct proportion formula can be written as y=3x^{2}.

When x=4 and y=p , we have

p=3\times{4}^{2}

Example 4: y is directly proportional to x (worded problem)

A florist uses ribbon to trim bunches of flowers.

80 \ cm of ribbon is needed to make the trimmings on 2 bunches of flowers.

The florist has 9 bunches of flowers to trim today.

How much ribbon will be needed?

As each bunch of flowers requires a specific length of ribbon, the length of ribbon required (r) is directly proportional to the number of bunches of flowers (b).

Writing this relationship between r and b, we have r \propto b.

This means our direct proportion formula is r=kb where k represents the constant of proportionality.

We currently know that 80cm of ribbon is needed to trim 2 bunches of flowers. This means that when r=80, \ b=2. Substituting these values into the above formula, we have:

\begin{aligned} 80&=k\times{2}\\k&=80 \div 2\\k&=40 \end{aligned}

Now, r=40b. As we need to find out how much ribbon is needed for 9 bunches of flowers, we substitute b=9 into the formula to get:

r=40\times{9}

Example 5: y is directly proportional to x 2 (worded problem)

The surface area A of a cube is directly proportional to the square of the side length L. A cube with side length 3cm has a surface area of 54cm^{2}.

Determine the side length of a cube with a surface area of 600cm^{2}.

As A is directly proportional to L^2, we can state that A \propto L^2 and write the formula:

In the question, when A=54, \ L=3. Substituting these values into the formula, we have

\begin{aligned} 54&=k\times{3}^{2}\\\\ 54&=k\times{9}\\\\ k&=54\div{9}\\\\ k&=6 \end{aligned}

We have A=6L^{2}. We need to calculate the side length L when the surface area A=600. Substituting these into the formula, we have:

600=6\times{L}^{2}

Example 6: y is directly proportional to the cube root of x (worded problem)

Let D represent the density of an object, and V=x^3 represent the volume of the object in terms of the side length x. A block of ice has a density of 0.9g/cm^3 and a volume of 10.8cm^{3}.

Given that D \propto V, determine the side length of a block of ice with the same mass, but a density of 0.92g/cm^{3}. Write your answer to 2 decimal places.

As D\propto{V} where V=x^3 we can state the direct proportion formula D=kV.

As D=0.9 when V=10.8, substituting these into the formula above, we have

\begin{aligned} 0.9&=k\times{10.8}\\\\ 0.9\div{10.8}&=k\\\\ k&=\frac{9}{108}=\frac{1}{12} \end{aligned}

We now have D=\frac{1}{12}V. As we want to determine the side length of the new block of ice and V=x^{3}, we can substitute this into the equation, along with the known density D=0.92. We therefore have the following:

0.92=\frac{1}{12}x^{3}

Common misconceptions

  • Direct proportion can be non-linear

If we represented the direct proportion formula y=kx using a graph, the line would be straight, going through the origin with gradient k. There are many nonlinear direct proportion relationships such as y=kx^2 (a quadratic graph), y=kx^3 (a cubic graph), or y= \sqrt{x} (a radical graph). These are nonlinear functions as each graph is not a straight line, but x and y are still directly proportional to one another.

  • The \textbf{y} -intercept is not equal to \bf{0}

Take the general equation of a straight line y=mx+c. The values of x and y are directly proportional if and only if c=0 as the gradient m describes the rate of change between the two variables ( m could be described as the constant of proportionality here). If c ≠ 0, the two variables are not directly proportional.

  • Summative relationship rather than a multiplicative relationship

When using a table, the next value along for x is 3 more than the current value. The value for y is calculated by adding 3 to this value. This uses a summative relationship, rather than the required multiplicative relationship between two variables (the multiplier being the constant of proportionality, k ).

  • Mixing up direct and inverse proportion

For direct proportion, the constant of proportionality k is the ratio of the two variables such as k=y \div x. For inverse proportion, k is the product of the two variables, such as k=xy. Step-by-step guide: Inverse proportion (coming soon)

  • Take care with writing money

Money is used in many direct proportion word problems. If an answer is 5.3 you may be tempted to write it as £5.3, but the correct way of writing it would be £5.30.

Related lessons on direct proportion

Direct proportion is part of our series of lessons to support revision on proportion . You may find it helpful to start with the main proportion lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

  • Direct and indirect proportion
  • Inverse proportion
  • Inverse proportion formula

Practice direct proportion questions

1. y is directly proportional to x. Find the missing value.

Direct proportion practice question 1

y\propto{x} and so y=kx. At (3,7),

So y=\frac{7}{3}x

When x=15, \ y=\frac{7}{3}\times{15}=7\times{5}=35

2. y is directly proportional to x. Find the missing value.

Direct proportion practice question 2

y\propto{x} and so y=kx. At (2,6),

3. Given that y \propto x^2 , calculate the value of w.

Direct proportion practice question 3

y\propto x^2 and so y=kx^{2}. At (6,18),

So y=0.5x^{2}

4. 3 pens cost £2.07. Find the cost of 13 pens.

As the cost c is directly proportional to the number of pens p, we can state c\propto{p}. This means that c=kp where k is the constant of proportionality. As c=2.07 when p=3,

Each pen costs £0.69.

Now p=13, so

13 pens cost £8.97.

5. The area of a circle is directly proportional to the square of the radius. Circle X has an area of 113.097cm^2 and a radius of 6cm. Circle Y has a radius of 4cm. Calculate the area of circle Y.

Let A represent the area of a circle, and r represent the radius of the circle, then A \propto r^2 so A=kr^{2}.

When A=113.097, \ r=6 and so

Now, A=3.14158\dot{3} \times r^2 and we know for circle Y, \ r=4.

Substituting this into the equation, we have:

6. The number of hours of sleep, h, is directly proportional to the square root of the number of marks achieved in an exam, m. The total marks in the exam is 100.

Identical twins Mark and Guy each sit the same exam. Mark had 6.5 hours of sleep and scored 64 marks in the exam. How many hours of sleep did Guy have to achieve 81 marks in the exam?

7.3125 hours

12.462 hours (3dp)

8.227 hours (3dp)

5.\bar{7} hours

The proportional relationship between the number of hours and the number of marks in the exam is written as h\propto\sqrt{m} and so h=k\sqrt{m}.

When h=6.5, \ m=64 and so

Now, h=\frac{13}{16}\sqrt{m}. As Guy scored 81 marks in the exam, m=81 and so we can substitute this into the equation to get:

Guy had 7.3125 hours of sleep.

Direct proportion GCSE questions

1. Crisps cost £3 for 12 packets.

Find the cost of 20 packets of crisps.

2. A force, F, of 128 Newtons, is applied to an area, A, of 2.8m^{2}.

Given that F \propto A, calculate the force required to apply the same amount of pressure over an area of 1.6m^{2}.

Write your answer to 2 decimal places.

3. Here are the ingredients needed to make 15 brownies

50 \ g chocolate

125 \ g margarine

225 \ g sugar

50 \ g plain flour

1 teaspoon baking powder

Orla wants to make 36 brownies.

She only has 320 \ g of margarine and 500 \ g of sugar.

She has plenty of the other ingredients

Does she have enough?

Orla has enough margarine as 320 \ g is more than 300 \ g

Orla does NOT have enough sugar as 500 \ g is less than 540 \ g

4. A windup toy car is being held stationary on a table, ready to move. The displacement S is directly proportional to the square of the time T the object is moving.

(a) Using the information in the table, determine the displacement of the windup toy after 10 seconds.

State the direct proportion equation used in your solution.

Direct proportion gcse question 4

(b) Hence determine how many seconds the windup toy has moved when he has travelled 144m.

Learning checklist

You have now learned how to:

  • Solve problems involving direct proportion

The next lessons are

  • Compound measures
  • Best buy maths

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Direct & Inverse Proportions/Variations

In these lessons, we will learn how to solve direct proportions (variations) and inverse proportions (inverse variations) problems. (Note: Some texts may refer to inverse proportions/variations as indirect proportions/variations.)

Related Pages: Direct Variations Proportion Word Problems More Algebra Lessons

The following diagram gives the steps to solve ratios and direct proportion word problems. Scroll down the page for examples and step-by-step solutions.

Ratios and Proportions

Direct Proportions/Variations

Knowing that the ratio does not change allows you to form an equation to find the value of an unknown variable.

Example : If two pencils cost $1.50, how many pencils can you buy with $9.00?

How To Solve Directly Proportional Questions?

Example 1: F is directly proportional to x. When F is 6, x is 4. Find the value of F when x is 5. Example 2: A is directly proportional to the square of B. When A is 10, B is 2. Find the value of A when B is 3.

How To Use Direct Proportion?

How To Solve Word Problems Using Proportions?

This video shows how to solve word problems by writing a proportion and solving 1. A recipe uses 5 cups of flour for every 2 cups of sugar. If I want to make a recipe using 8 cups of flour, how much sugar do I use? 2. A syrup is made by dissolving 2 cups of sugar in 2/3 cups of boiling water. How many cups of sugar should be used for 2 cups of boiling water? 3. A school buys 8 gallons of juice for 100 kids. how many gallons do they need for 175 kids?

Solving More Word Problems Using Proportions

1. On a map, two cities are 2 5/8 inches apart. If 3/8 inches on the map represents 25 miles, how far apart are the cities (in miles)? 2. Solve for the sides of similar triangles using proportions

Inverse Proportions/Variations Or Indirect Proportions

Two values x and y are inversely proportional to each other when their product xy is a constant (always remains the same). This means that when x increases y will decrease, and vice versa, by an amount such that xy remains the same.

Knowing that the product does not change also allows you to form an equation to find the value of an unknown variable

Example : It takes 4 men 6 hours to repair a road. How long will it take 8 men to do the job if they work at the same rate?

Solution : The number of men is inversely proportional to the time taken to do the job. Let t be the time taken for the 8 men to finish the job. 4 × 6 = 8 × t 24 = 8t t = 3 hours

Usually, you will be able to decide from the question whether the values are directly proportional or inversely proportional.

How To Solve Inverse Proportion Questions?

This video shows how to solve inverse proportion questions. It goes through a couple of examples and ends with some practice questions Example 1: A is inversely proportional to B. When A is 10, B is 2. Find the value of A when B is 8 Example 2: F is inversely proportional to the square of x. When A is 20, B is 3. Find the value of F when x is 5.

How To Use Inverse Proportion To Work Out Problems?

How to use a more advanced form of inverse proportion where the use of square numbers is involved.

More examples to explain direct proportions / variations and inverse proportions / variations

How to solve Inverse Proportion Math Problems on pressure and volume?

In math, an inverse proportion is when an increase in one quantity results in a decrease in another quantity. This video will show how to solve an inverse proportion math problem. Example : The pressure in a piston is 2.0 atm at 25°C and the volume is 4.0L. If the pressure is increased to 6.0 atm at the same temperature, what will be the volume?

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Direct Proportion

Direct proportion is a mathematical comparison between two numbers where the ratio of the two numbers is equal to a constant value. The proportion definition says that when two ratios are equivalent, they are in proportion. The symbol used to relate the proportions is "∝". Let us learn more about direct proportion in this article.

Direct Proportion Definition

The definition of direct proportion states that "When the relationship between two quantities is such that if we increase one, the other will also increase, and if we decrease one the other quantity will also decrease, then the two quantities are said to be in a direct proportion". For example, if there are two quantities x and y where x = number of candies and y = total money spent. If we buy more candies, we will have to pay more money, and we buy fewer candies then we will be paying less money. So, here we can say that x and y are directly proportional to each other. It is represented as x ∝ y. Direct proportion is also known as direct variation.

Some real-life examples of direct proportionality are given below:

  • The number of food items is directly proportional to the total money spent.
  • Work done is directly proportional to the number of workers.
  • Speed is in direct proportion to the distance w.r.t a fixed time.

Direct Proportion Formula

The direct proportion formula says if the quantity y is in direct proportion to quantity x, then we can say y = kx, for a constant k. y = kx is also the general form of the direct proportion equation.

direct proportion formula

  • k is the constant of proportionality .
  • y increases as x increases.
  • y decreases as x decreases.

Direct Proportion Graph

The graph of direct proportion is a straight line with an upward slope . Look at the image given below. There are two points marked on the x-axis and two on the y-axis, where (x) 1 < (x) 2 and (y) 2 < (y) 2 . If we increase the value of x from (x) 1 to (x) 2 , we observe that the value of y is also increased from (y) 1 to (y) 2 . Thus, the line y=kx represents direct proportionality graphically.

Direct proportion graph

Direct Proportion Vs Inverse Proportion

There are two types of proportionality that can be established based on the relation between the two given quantities. Those are direct proportion and inverse proportional. Two quantities are directly proportional to each other when an increase or decrease in one leads to an increase or decrease in the other. While on the other hand, two quantities are said to be in inverse proportion if an increase in one quantity leads to a decrease in the other, and vice-versa. The graph of direct proportion is a straight line while the inverse proportion graph is a curve. Look at the image given below to understand the difference between direct proportion and inverse proportion.

Direct proportion vs. inverse proportion

Topics Related to Direct Proportion

Check these interesting articles related to the concept of direct proportion.

  • Constant of Proportionality
  • Inversely Proportional
  • Percent Proportion

Direct Proportion Examples

Example 1: Let us assume that y varies directly with x, and y = 36 when x = 6. Using the direct proportion formula, find the value of y when x = 80?

Using the direct proportion formula, y = kx Substitute the given x and y values, and solve for k. 36 = k × 6 k = 36/6 = 6 The direct proportion equation is: y = 6x Now, substitute x = 80 and find y. y = 6 × 80 = 480

Answer: The value of y is 480.

Example 2: If the cost of 8 pounds of apples is $10, what will be the cost of 32 pounds of apples?

It is given that, Weight of apples = 8 lb Cost of 8 lb apples = $10 Let us consider the weight by x parameter and cost by y parameter. To find the cost of 32 lb apples, we will use the direct proportion formula. y=kx 10 = k × 8 (on substituting the values) k = 5/4 Now putting the value of k = 5/4 when x = 32 we have, The cost of 32 lb apples = 5/4 × 32 y =5×8 y = 40

Answer: The cost of 32 lb apples is $40 .

Example 3: Henry gets $300 for 50 hours of work. How many hours has he worked if he got $258?

Solution: Let the amount received by Henry be treated as y and the number of hours he worked as x. Substitute the given x and y values in the direct proportion formula, we get, 300 = k × 50

⇒ k=300/50 k = 6 The equation is: y = 6x. Now, substitute y = 258 and find x. 258 = 6 × x

⇒ x = 258/6 = 43 hours Therefore, if Henry got $258, he worked for 43 hours.

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steps in solving problem involving direct proportion

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Direct Proportion Practice Questions

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FAQs on Direct Proportion

What is direct proportion in maths.

Two quantities are said to be in direct proportion if an increase in one also leads to an increase in the other quantity, and vice-versa. For example, if a ∝ b, this implies if 'a' increases, 'b' will also increase, and if 'a' decreases, 'b' will also decrease.

What Is the Symbol ∝ Denotes in Direct Proportion Formula?

In the direct proportion formula, the proportionate symbol ∝ denotes the relationship between two quantities. It is expressed as y ∝ x, and can be written in an equation as y = kx, for a constant k.

What is Direct Proportion and Inverse Proportion?

Direct proportion, as the name suggests, indicates that an increase in one quantity will also increase the value of the other quantity and a decrease in one quantity will also decrease the value of the other quantity. While inverse proportion shows an inverse relationship between the two given quantities. It means an increase in one will decrease the value of the other quantity and vice-versa.

How do you Represent the Direct Proportional Formula?

The direct proportional formula depicts the relationship between two quantities and can be understood by the steps given below:

  • Identify the two quantities which vary in the given problem.
  • Identify the variation as the direct variation .
  • Direct proportion formula: y ∝ kx.

What is a Direct Proportion Equation?

The equation of direct proportionality is y = kx, where x and y are the given quantities and k is any constant value. Some examples of direct proportional equations are y = 3x, m = 10n, 10p = q, etc.

How to Solve Direct Proportion Problems?

To solve direct proportion word problems, follow the steps given below:

  • Make sure that the variation is directly proportional.
  • Form an equation in terms of y = kx and find the value of k base on the given values of x and y.
  • Find the unknown value by putting the values of x and the known variable.

How to Show Relationship Between Two Quantities Using Direct Proportion Formula?

The directly proportional relationship between two quantities can figure out using the following key points.

  • Identify the two quantities given in the problem.
  • If x/y is constant then the quantities have a directly proportional relationship.

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Mathematics LibreTexts

5.2: Applications of Proportionality

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  • Page ID 143063

In the previous section, we studied proportions, and used them to solve problems involving ratios. In this section, we continue our study of proportions, and investigate two different types of proportionality.

In this section, you will learn to:

Recognize direct proportionality relationships, and use them to answer questions involving direct proportionality

  • Compute and interpret the constant of proportionality in context
  • Recognize inverse proportionality relationships, and use them to answer questions involving inverse proportionality

There are two main types of proportionality. We will learn about the more common one first.

Direct Proportionality

Definition: directly proportional.

Two quantities are directly proportional if, as one quantity increases, the other quantity also increases at the same rate.

Direct proportionality describes all of the proportion problems we've seen before. Here is another example that shows how direct proportionality works, and introduces the next important notion.

Example \(\PageIndex{1}\)

At an hourly wage job, you work for \(5\) hours, and get paid \(\$83.75\). How much money will you earn if you work \(7\) hours? (Assume that you are not making overtime pay, or any other sort of special pay rate.)

This is a situation described by direct proportionality. Since this is an hourly wage job, and we're told that there is no overtime pay or other special pay rate, we can safely assume that we make the same amount per hour in this job. In other words, the rate of pay will be the same no matter how many hours we work. That means that if the number of hours worked increases, the amount you're paid increases at the same rate, no matter the number of hours worked.

We can solve this using a proportion, using the same techniques as in the last section. We will set up the following proportion equation: \[\frac{\$83.75}{5 \text{ hours}} = \frac{\$x}{7 \text{ hours}}\]

Notice that we have picked a variable, \(x\), to denote the answer we are trying to find -- the number of dollars earned for working \(7\) hours. Also notice how we've labeled our units, and have made sure that the corresponding quantities are together. If you do this every time -- label your units, and make sure corresponding quantities stay together -- you can solve any direct proportion problem.

Now for the process that will actually help us solve for \(x\). First, rewrite the equation without labels: \[\frac{83.75}{5} = \frac{x}{7}\]

Next, apply Cross Multiplication \[5x = 83.75 \times 7\]

Next, simplify the right side (using a calculator): \[5x = 586.25\]

Finally, apply Division undoes Multiplication to find \(x\): \[x = \frac{586.25}{5} = 117.25\]

That means that if you work \(7\) hours, you will make \(\$117.25\). Think for a moment to see if that's reasonable: it's more, but not too much more, than you made working for \(5\) hours. So, it seems like a sensible answer.

The approach above works just fine to find the desired answer. However, what if you wanted to know how much you'd make working for \(3\) hours? Or \(4\) hours? Or \(10\) hours? You could just reproduce the work above each time, you wanted. But you may also find the following approach quicker:

Definition: Constant of Proportionality

In a situation involving directly proportional quantities, the constant of proportionality is the common ratio that describes the comparison of any two corresponding quantities. In other words, it is the constant rate of change between the two quantities.

Let's see how to find a constant of proportionality, and how to interpret it.

You're in the same situation as the previous example: you work for \(5\) hours, and earn \($83.75\). What is the constant of proportionality in this example, and what does it mean in context?

The constant of proportionality is the common ratio that describes the comparison of any two corresponding quantities. In this situation, we actually have two sets of corresponding quantities, one of which we found in the previous example. You know that you'll make \($83.75\) working for \(5\) hours, and we calculated above that you'll make \(\$117.25\) if you work for \(7\) hours. Let's look at these two ratios:

\[\frac{\$83.75}{5 \text{ hours}} \quad \text{ and } \quad \frac{\$117.25}{7 \text{ hours}}\]

Both of these ratios are fractions, and we can simply divide the top by the bottom to reduce them to a single number. Using a calculator, we can see that

\[\frac{\$83.75}{5 \text{ hours}} = \$83.75 \div 5 \text{ hours} = \$16.75 \text{ per hour}\]

\[\frac{\$117.25}{7 \text{ hours}} = \$117.25 \div 7 \text{ hours} = \$16.75 \text{ per hour}\]

These are the same answer! Do you see why? We originally found our value for \(x\) in the previous example by setting the two ratios equal. So, they must give the same answer when divided.

This shared rate -- \(\$16.75\) per hour -- is the constant of proportionality in this situation. It is the shared value of all ratios described by this problem, where the top of the ratio is money earned, and the bottom is hours worked.

What does this constant of proportionality mean in this situation? In this case, the constant of proportionality is your hourly pay rate. In other words, it's how much you make per hour.

A few more comments on the example above: now that you know this rate, it's quite simple to find how much money you'll make if you work \(3, 4,\) or \(10\) hours. You just multiply your hourly rate by the number of hours worked. For example, if you worked \(4\) hours, you could calculate:

\[\underset{\text{hours}}{4} \times \$16.75 \text{ per hour} = \$67.00\]

This means you would make \(\$67.00\) working for \(4\) hours. Notice that if we reverse the process -- in other words, if we try to extract the constant of proportionality knowing that we make \(\$67.00\) in \(4\) hours, we get:

\[\frac{\$67.00}{4 \text{ hours}} = \$67.00 \div 4 \text{ hours} = \$16.75 \text{ per hour}\]

It's the same rate we found before. This is why it's called a constant of proportionality -- it stays the same, even as the corresponding quantities change.

Constants of proportionality will change in meaning depending on the context of the problem. For example, you might ask: If \(5\) people eat a total of \(10\) slices of pizza, how many slices would \(7\) people eat? In this case, the constant of proportionality could be found by: \[\frac{10 \text{ slices}}{5 \text{ people}} = 2 \text{ slices per person}\]

In this case, a correct interpretation would be: "The constant of proportionality is \(2\) slices per person, which means that each person eats \(2\) slices of pizza." When asked for an interpretation, you should write a sentence similar to the previous -- your goal is the explain the meaning of the constant of proportionality in context of the situation. You will need to read carefully and use critical thinking to deduce a meaningful interpretation. As with many questions in this class, there are multiple good answers to these types of questions!

Problems that involve rates, ratios, scale models, etc. can be solved with proportions. When solving a real-world problem using a proportion, be consistent with the units.

Inverse Proportionality

Let's start this section with a question to illustrate the main concept. Before reading ahead, try to answer this question on your own:

Example \(\PageIndex{2}\)

Suppose it takes \(6\) sanitation workers, all working simultaneously, \(4\) hours to pick up the trash and recycling in a given neighborhood. How many sanitation workers would it take to pick up the trash and recycling in the same neighborhood in \(3\) hours? (Assume that all sanitation workers work at the same rate, and can work independently.)

Did you try to answer the question yourself? What did you come up with? If you're like many students who carefully read the previous sections, you might have written this down:

\[\frac{6 \text{ sanitation workers}}{4 \text{ hours}} = \frac{x \text{ sanitation workers}}{3 \text{ hours}}\]

Then you'd apply Cross Multiplication to get:

\[4x = 18\]

and then you'd use Division which undoes Multiplication to get

\[x = \frac{18}{4} = 4.5\]

Now, the numerical answer \(4.5\) is a bit nonsensical, because it's talking about a number of people. So you could round up to \(5\), and say "It would take 5 sanitation workers to pick up the trash and recycling in \(3\) hours."

But wait a second: this does not make sense! Think about it: if it takes \(6\) workers \(4\) hours to accomplish this task, shouldn't it take \(5\) workers more time than \(4\) hours? After all, there is the same amount of work to be done, but fewer people to do it! So the answer "5 workers" cannot possibly be correct. We expect a number of workers that is larger than 6 to get the task done in a shorter amount of time.

We can learn two things from the previous discussion:

  • It is important to evaluate whether or not an answer to a question makes sense in context by asking: What sort of answer would I expect to get? Does my answer seem reasonable?
  • Not all problems can be solved using direct proportionality!

The good news is that this type of problem can be solved in a relatively simple way. We define the main concept in this section to see how these problems work.

Definition: Inversely Proportional

Two quantities are inversely proportional if, as one quantity increases, the other quantity decreases at the same rate.

Note how similar this definition is to the previous definition of direct proportionality. The only difference here is that one quantity increases while the other decreases:

In order to determine what type of problem you're working on, you'll need to think critically about the quantities involved, and use clues from your experience and the context of the problem to determine how the quantities are related. Things like the previous problem -- when a group of people are working together to accomplish a specific task -- are one of the primary examples of inverse proportionality. Let's see the same example again, and this time find the correct answer.

The way to approach this is to find the number of worker hours needed to accomplish the task of picking up the trash and recycling in this neighborhood. A worker hour is defined to be an hour of work done by a worker, and that number will remain constant no matter the number of workers used.

To find the number of worker hours needed for this particular neighborhood, we simply multiply the known number of workers by the known number of hours: \[\underset{\text{workers}}{6} \times \underset{\text{hours}}{4} = \underset{\text{worker hours}}{24}\]

That means that it will require \(24\) worker hours to pick up the trash and recycling in this neighborhood.

To find the number of workers needed to pick up the trash and recycling in \(3\) hours, we divide the number of worker hours by the number of hours to find the number of workers:

\[\frac{24 \text{ worker hours}}{3 \text{ hours}} = \underset{\text{worker hours}}{24} \div \underset{\text{hours}}{3} = \underset{\text{workers}}{8}\]

This means it will take \(8\) workers \(3\) hours to pick up the trash and recycling. This makes sense -- it's larger than \(6\), which was the number of workers needed to accomplish the task in \(4\) hours.

All inverse proportionality problems work this way -- multiply the two known corresponding quantities, and then divide to find the answer. As always, label your units, and check to see if your answers make sense!

Solve Similar Figure Applications

When you shrink or enlarge a photo on a phone or tablet, figure out a distance on a map, or use a pattern to build a bookcase or sew a dress, you are working with similar figures . If two figures have exactly the same shape, but different sizes, they are said to be similar. One is a scale model of the other. All their corresponding angles have the same measures and their corresponding sides are in the same ratio.

Definition: SIMILAR FIGURES

Two figures are similar if the measures of their corresponding angles are equal and their corresponding sides are in the same ratio.

For example, the two triangles in Figure are similar. Each side of ΔABC is 4 times the length of the corresponding side of ΔXYZ.

The above image shows the steps to solve the proportion 1 divided by 12.54 equals 325 divided by p. What are you asked to find? How many Mexican pesos did he get? Assign a variable. Let p equal the number of pesos. Write a sentence that gives the information to find it. If one dollar US is equal to 12.54 pesos, then 325 dollars is how many pesos. Translate into a proportion, be careful of the units. Dollars divided pesos equals dollars divided by pesos to get 1 divided by 12.54 equals 325 divided by p. Multiply both sides by the LCD, 12.54 p to get 1 divided by 12.54 p times 1 divided by 12.54 equals 12.54 p times 325 divided by p. Remove common factors from both sides. Cross out 12.54 from the left side of the equation. Cross out p from the right side of the equation. Simplify to get p equals 4075.5 in the original proportion. Check. Is the answer reasonable? Yes, $100 would be $1254 pesos. $325 is a little more than 3 times this amount, so our answer of 4075.5 pesos makes sense. Substitute p equals 4075.5 in the original proportion. Use a calculator. We now have 1 divided by 12.54 equals 325 divided by p. Next, 1 divided by 12.54 equals 325 divided by 4075.5 to get 0.07874 equals 0.07874. The answer checks.

This is summed up in the Property of Similar Triangles.

Definition: PROPERTY OF SIMILAR TRIANGLES

If ΔABC is similar to ΔXYZ

The above figure shows to similar triangles. The larger triangle labeled A B C. The length of A to B is c, The length of B to C is a. The length of C to A is b. The larger triangle is labeled X Y Z. The length of X to Y is z. The length of Y to Z is x. The length of X to Z is y. To the right of the triangles, it states that measure of corresponding angle A is equal to the measure of corresponding angle X, measure of corresponding angle B is equal to the measure of corresponding angle Y, and measure of corresponding angle C is equal to the measure of corresponding angle Z. Therefore, a divided by x equals b divided by y equals c divided by z.

To solve applications with similar figures we will follow the Problem-Solving Strategy for Geometry Applications we used earlier.

Definition: SOLVE GEOMETRY APPLICATIONS.

  • Read the problem and make all the words and ideas are understood. Draw the figure and label it with the given information.
  • Identify what we are looking for.
  • Name what we are looking for by choosing a variable to represent it.
  • Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
  • Solve the equation using good algebra techniques.
  • Check the answer in the problem and make sure it makes sense.
  • Answer the question with a complete sentence.

Example \(\PageIndex{8}\)

ΔABC is similar to ΔXYZ

The above image shows two similar triangles. Two sides are given for each triangle. The larger triangle is labeled A B C. The length of A to B is 4. The length from B to C is a. The length from C to A is 3.2. The smaller triangle is labeled X Y Z. The length from X to Y is 3. The length from Y to Z is 4.5. The length from Z to X is y.

TRy it \(\PageIndex{15}\)

ΔABC is similar to ΔXYZ. The lengths of two sides of each triangle are given in the figure.

The above image shows two similar triangles. The smaller triangle is labeled A B C. The length of two sides is given for the smaller triangle A B C. The length from A to B is 17. The length from B to C is a. The length from C to D is 15. The larger triangle is labeled X Y Z. The length is given for two sides. The length from X to Y is 25.5. The length from Y to Z is 12. The length from Z to X is y.

Find the length of side a

TRy it \(\PageIndex{16}\)

The next example shows how similar triangles are used with maps.

Example \(\PageIndex{9}\)

On a map, San Francisco, Las Vegas, and Los Angeles form a triangle whose sides are shown in the figure below. If the actual distance from Los Angeles to Las Vegas is 270 miles find the distance from Los Angeles to San Francisco.

The above image shows two similar triangles and how they are used with maps. The smaller triangle on the left shows San Francisco, Las Vegas and Los Angeles on the three points. San Francisco to Los Angeles is 1.3 inches. Los Angeles to Las Vegas is 1 inch. Las Vegas to San Francisco is 2.1 inches. The second larger triangle shows the same points. The distance from San Francisco to Los Angeles is x. The distance from Los Angeles to Las Vegas is 270 miles. The distance from Las Vegas to San Francisco is not noted.

TRY IT \(\PageIndex{17}\)

On the map, Seattle, Portland, and Boise form a triangle whose sides are shown in the figure below. If the actual distance from Seattle to Boise is 400 miles, find the distance from Seattle to Portland.

The above image is a triangle with one side labeled “Seattle, 4.5 inches”. The other side is labeled “Portland 3.5 inches”. The third side is labeled 1.5 inches. The vertex is labeled “Boise.”

Try it \(\PageIndex{18}\)

Using the map above, find the distance from Portland to Boise.

We can use similar figures to find heights that we cannot directly measure.

Example \(\PageIndex{10}\)

Tyler is 6 feet tall. Late one afternoon, his shadow was 8 feet long. At the same time, the shadow of a tree was 24 feet long. Find the height of the tree.

Try it \(\PageIndex{19}\)

A telephone pole casts a shadow that is 50 feet long. Nearby, an 8 foot tall traffic sign casts a shadow that is 10 feet long. How tall is the telephone pole?

Try it \(\PageIndex{20}\)

A pine tree casts a shadow of 80 feet next to a 30-foot tall building which casts a 40 feet shadow. How tall is the pine tree?

Make sure that when you are asked to interpret something, you write a complete sentence describing the meaning of your numerical answer in the context of the problem.

  • How many gallons of gas will you need to go 400 miles?
  • Is this situation described by direct or inverse proportionality, and why? Give a one-sentence answer.
  • What is the constant of proportionality in this situation, and how would you interpret it?
  • How many professors would it take to grade the same exams in 4 hours?
  • A family drinks 2 gallons of milk every 9 days. How many gallons of milk will they use in 2 weeks? Be careful with units here! (Round to one decimal place.)
  • At a rate of 30 miles per hour, a certain trip takes 2 hours. How long would the same trip take at 40 miles per hour? (Round to one decimal place or give a fractional answer.)
  • Think of a real-world example of direct proportionality that is different than ones we've covered in this section. Give a 2-3 sentence description of the quantities involved, and why you think they are directly proportional. If you use a source, please cite it by providing a URL.
  • Think of a real-world example of inverse proportionality that is different than ones we've covered in this section. Give a 2-3 sentence description of the quantities involved, and why you think they are inversely proportional. If you use a source, please cite it by providing a URL.

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R10a – Solving problems involving direct proportion

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  • A knowledge of the four operations from lessons N2a , N2b , N2c , N2d , N2e and N2f is assumed.
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An illustration of quantities in direct proportion

Adjust the dark grey slider to select the cost of 1 metre of rope. Now in this example, the cost of a piece of rope is directly proportional to its length . The longer the rope, the more it will cost. What happens to the cost if you double the length of rope? What if you triple the length? Investigate by adjusting the blue and green sliders.

Double number line activity

Solving problems involving direct proportion.

steps in solving problem involving direct proportion

Direct Proportion

Two quantities a and b are said to be in direct proportion if they increase or decrease together. In other words, the ratio of their corresponding values remains constant. This means that,

where k is a positive number, then the quantities a and b are said to vary directly.

In such a case if the values b 1 , b 2 of b corresponding to the values a 1 , a 2 of a, respectively then it becomes;

a 1/ /b 1 = a 2 /b 2

The direct proportion is also known as direct variation .

Directly Proportion Symbol

The symbol used to represent the direct proportion is “ ∝” .

Consider the statement,

a is directly proportional to b

This can be written using the symbol as:

Consider the other statement, a = 2b

In this case, it shows that a is proportional to b, and the value of one variable can be found if the value of other variable is given.

For example: 

Therefore, a = 2 x 7 = 14

Similarly, if you take the value of “a” as 14, you will find the value of b

Therefore, b=7

Inverse Proportion

The value is said to inversely proportional when one value increases, and the other decreases. The proportionality symbol is used in a different way. Consider an example; we know that the more workers on a job would reduce the time to complete the task. It is represented as:

Number of workers ∝ (1/ Time taken to complete the job)

Inverse Proportion Definition

Two quantities a and b are said to be in inverse proportion if an increase in quantity a, there will be a decrease in quantity b, and vice-versa. In other words, the product of their corresponding values should remain constant.  Sometimes, it is also known as inverse variation

That is, if ab = k, then a and b are said to vary inversely. In this case, if b 1 , b 2 are the values of b corresponding to the values a 1 , a 2 of a, respectively then a 1 b 1 = a 2 b 2 or a 1 /a 2 = b 2 /b 1

The statement ‘a is inversely proportional to b is written as

Here, an equation is given that involves the inverse proportions that can be used to calculate the other values.

Here a is inversely proportional to b

If one value is given, the other value can be easily found.

a= 25/10 = 2.5

Similarly, if a = 2.5, the value of b can be obtained.

b= 25/2.5 = 10

How to Write Direct and Indirect Proportion Equation?

If we have to write a proportionality whether it is direct or indirect in an equation, follow the below steps:

  • Step 1: First, write down the proportional symbol
  • Step 2: Convert it as an equation using the constant of proportionality
  • Step 3: Find the constant of proportionality from the given information
  • Step 4: After finding the constant of proportionality, substitute in an equation

Direct and Inverse Proportion Solved Examples

Below are examples to understand the concept of direct and inverse proportion in a better way.

A train is moving at a uniform speed of 75 kilometres/hour.

(i)How many kilometres are covered by train in 20 minutes?

(ii) Find the time required to cover a distance of 250 kilometres.

Let the distance travelled (in km) in 20 minutes be a and time taken (in minutes) to cover 250 km be b.

We know that 1 hour = 60 minutes

Since the speed of the train is uniform, therefore, the distance covered would be directly proportional to time.

(i) We have 75 /60 = a /20

or (75 /60) 20 = a

So, the train will cover a distance of 25 kilometres in 20 minutes.

(ii) Also, 75/60=250/ b or

b=(250 x 60)/ 75

b = 200 minutes or 3 hours 20 minutes.

Therefore, 3 hours 20 minutes is required to cover a distance of 250 kilometres.

Alternatively, when a is known, then one can determine b, using the relation

a/20 =250/ b

The value f is directly proportional to g. When f = 20, g = 10. Find an equation relating f and g.

Given, f ∝ g

or we can write,

f = kg, where k is the constant proportionality.

20 = k x 10

Therefore, the required equation is;

Practice Questions

  • A car is moving at a speed of 60 Km/hr. How far will it travel in 30 minutes?
  • If the cost of 10 pens is Rs.120, how much will it cost for 15 pens?
  • If 15 workers can do work in 12 days, then how many workers will complete the same work in 6 days?

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WORD PROBLEMS ON DIRECT PROPORTION

Direct proportion is a situation where an increases in one quantity causes a corresponding  increases  in the other quantity, 

Direct proportion is a situation where an decreases in one quantity causes a corresponding decreases in the other quantity.

Problem 1 :

A dozen bananas costs $20. What is the price of 48 bananas?

Let x be the price of 48 bananas.

As number of banana increases, the price of banana will also increase.

It comes under direct proportion.

steps in solving problem involving direct proportion

Doing cross multiplication,

12  ⋅  x = 48  ⋅  20

x = (48  ⋅  20)/12

Therefore, the price of 48 bananas is $80.

Problem 2 :

A group of 21 students paid $840 as the entry fee for a magic show.  How many students entered the magic show if the total amount paid was $ 1,680?

Let x be the number of students entered the magic show.

As number of student increases, the entry fees will also increase.  It comes under direct proportion.

steps in solving problem involving direct proportion

21  ⋅  1680 = x  ⋅  840

35280 = x  ⋅  840

35250/840 = x

Therefore, 42 students entered the magic show.

Problem 3 :

A birthday party is arranged in third floor of a hotel. 120 people take 8 trips in a lift to go to the party hall. If 12 trips were made how many people would have attended the party?

Let the number of people have attended the party be x.

As number of trips increases, number of people also increases.

steps in solving problem involving direct proportion

120  ⋅  12 = 8  ⋅  x

1440 = 8  ⋅  x

Therefore, 180 people attended the party in 12 trips.

Problem 4 :

The shadow of a pole with the height of 8 m is 6 m. if the shadow of another pole measured at the same time is 30 m, find the height of the pole?

Let x be the required height of the pole.

As length of shadow increases, height of the pole also increases.

steps in solving problem involving direct proportion

8  ⋅  30 = 6  ⋅  x

240 = 6  ⋅  x

Therefore, the height of the pole is 40 m.

Problem 5 :

A postman can sort out 738 letters in 6 hours. How many letters can be sorted in 9 hours?

Let x be the required number of letters.

As required time to sort in hours is increases, so, the number of letters is also increases.

steps in solving problem involving direct proportion

738  ⋅  9 = 6  ⋅  x

6642 = 6  ⋅  x

Therefore,  1107 letters can be sorted in 9 hours.

Problem 6 :

If half a meter of cloth costs $15. Find the cost of 8 1/3 meters of the same cloth.

Let x be the cost of 8  1/3 m of the same cloth.

Length of cloth is increases. So, cost of cloth is also increases.

steps in solving problem involving direct proportion

15  ⋅  (25/3) = x  ⋅  1/2

125 = x  ⋅  1/2

Therefore, the cost of 8 ⅓  m of cloth is $250.

Problem 7 :

The weight of 72 books is 9 kg. What is the weight of 40 such books?

Weight of 72 books = 9 kg.

72 books ----> 9 kg

40 books ----> x kg

Since the number of books increases, the weight of books will also increase. It comes under direct proportion.

Doing cross multiplication, we get

72x = 9(40)

x = 9(40)/72

Weight of 40 books is 5 kg.

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Direct and Inverse Proportion: Worksheets with Answers

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Home / TEAS Test Review Guide / Solve Problems Involving Proportions: TEAS

Solve Problems Involving Proportions: TEAS

Basic terms and terminology relating to solving problems involving proportions, ratios and their meaning, calculations using ratio and proportion, the conversion of percentages into ratios and converting ratios into percentages, the conversion of fractions into ratios and converting ratios into fractions.

steps in solving problem involving direct proportion

  • Ratio: The relationship of two numbers
  • Proportion: Two ratios that are equal to each other

steps in solving problem involving direct proportion

Simply stated, a ratio is the relationship of two numbers and proportions are two ratios that are equal to each other.

The picture above is a ratio; this ratio could indicate that there are 4 boys for every 3 girls, that there are 4 pears for every 3 oranges or that there are $ 4 in the piggy bank for every 3 dollars in the drawer. As you can see with these examples, ratios give limited information. For example, a ratio does not tell you how many pears or oranges you actually have; a ratio does not tell you how many boys and how many girls there actually are; and, a ratio does not tell you how much money you have in the piggy bank or in the drawer.

Ratios are read as " 4 is to 3".

In order to determine how many pears or oranges you actually have, how many boys and how many girls there actually are and how much money you have in the piggy bank or in the drawer, you would have to perform ratio and proportion to answer these questions.

The different ways to express ratios are:

When comparing ratios, they should be written as fractions. The fractions must be equal. If they are not equal they are NOT considered a ratio. For example, the ratios 3/8 and 6/16 are equal and equivalent.

Proportions are two ratios that are equal to each other and these ratio and proportion problems are calculated and solved as shown below.

If there is $12 in the drawer and the ratio of money in the drawer compared to money in the piggy bank is 4 :3, how much money is in the piggy bank?

4/3 = 12x OR 12x = 4/3

12×3/4 = 36/4 = $9

Answer: There is $9 in the drawer.

If you have 8 oranges and the ratio of oranges compared to pears is 4 : 3, how many pears do you have?

4:3 = 8:x OR 4/3 = 8x OR 8x = 4/3

8×3/4 = 24/4 = 6

Answer: You have 6 pears.

If there are 24 boys in the group and there are 4 boys to 3 girls in the group, how many girls are in the group?

4:3 = 24:x OR 4/3 = 24x OR 24x = 4/3

24×3/4 = 72/4 = 18

Answer: There are 18 girls in this group.

Here are some ratio and proportion problems that entail different measurement systems and converting between different measurement systems:

Knowing that there are 2.2 pounds in one kilogram, how many kilograms will you weigh when your current weight is 156 pounds?

1.2 pounds : 1 kilogram = 156 pounds : x kilograms

2.2/1 = 156/x

2.2x/2.2 = 156/2.2

x = 156/2.2

x = 70.9 kg

Answer: You weigh 70.9 kilograms when you weigh 156 pounds.

Knowing that there are 2.2 pounds in one kilogram, how many pounds will you weigh when your current weight is 65 kilograms?

2.2 pounds : 1 kilogram = x pounds : 65 kilograms

2.2/1 = x/65

2.2x/2.2 = 65/2.2

x = 65 x 2.2

x = 143 pounds

Answer: You weigh 143 pounds when you weigh 65 kilograms.

Knowing that there are 60 drops in 1 teaspoon, how many teaspoons are in 74 drops?

60 drops : 1 teaspoon = 74 drops : x teaspoons

60/1 = 74/x

60x/60 = 74/60 = x

74/60 = 1.2 teaspoons

Answer: There are 1.2 teaspoons in 74 drops

Proportions are often used in the calculation of dosages and solutions in pharmacology and the preparation of medications by nurses, pharmacists and pharmacy technicians as well as us, in our everyday lives.

These different measurement systems will be fully discussed below in the section entitled "Measurement and Data: M 2; Objective 5 : Converting Within and Between Standard and Metric Systems", however for the moment, we would like you to see some of the most commonly used measurement system conversion factors.

The most frequently used conversions are shown below. It is suggested that you memorize these.

  • 1 gr =60 mg
  • 1 kg = 2.2 lb.
  • 1 mg = 1,000 mcg
  • 1 g = 1,000 mg
  • 1 kg = 1,000 g
  • 1 tbsp. = 3 tsp
  • 1 tbsp. = 15 mL
  • 1 tsp = 5 mL
  • 1 l = 1,000 mL
  • 1 oz. = 30 mL
  • 1L = 1000 cc

The conversion of percentages into ratios can also be done.

The method for this is to place the percentage number and then : (colon) and then 100. A ratio is read as 12 is to 100 when you see 12 : 100, for example.

Here are some examples:

  • 12% = 12 : 100 or 12 is to 100
  • 120% = 120 : 100 or 120 is to 100
  • 220% = 220 : 100 or 220 is to 100
  • 2222% = 2222 : 100 or 2222 is to 100

Converting ratios into percentages is based on, again, the fact that ratios reflect parts of 100.

  • 23 : 100 = 23% or 23 is to 100
  • 567 : 100 = 567% or 567 is to 100
  • 1,222 : 100 = 1,222% or 1,222 is to 100
  • 32,678 : 100 = 32,678% or 32,678 is to 100
  • 1 : 100 = 1% or 1 is to 100

As stated above, ratios can be expressed in three different ways as follows:

The conversion of fractions into ratios is done in the following manner. The numerator becomes the first number before the colon and the denominator is the number after the colon.

The ratio is 2 : 10 or 2 is to 10

The ratio is 23 : 56 or 23 is to 56

The ratio is 19 : 45 or 19 is to 45

The ratio is 2 : 99 or 2 is to 99

The ratio is 16 : 789 or 16 is to 789

The ratio is 1 : 1 or 1 is to 1

The ratio is 100 : 100 or 100 : 100

Here are some examples of converting percentages into word ratios:

The ratio is 123 : 100

The ratio is 34 : 100

The ratio is 1 : 100

The ratio is 100 : 100

The ratio is 1,222 : 100

RELATED TEAS NUMBERS & ALGEBRA CONTENT :

  • Converting Among Non Negative Fractions, Decimals, and Percentages
  • Arithmetic Operations with Rational Numbers
  • Comparing and Ordering Rational Numbers
  • Solve Equations with One Variable
  • Solve One or Multi-Step Problems with Rational Numbers
  • Solve Problems Involving Percentages
  • Applying Estimation Strategies and Rounding Rules for Real-World Problems
  • Solve Problems Involving Proportions (Currently here)
  • Solve Problems Involving Ratios and Rates of Change
  • Translating Phrases and Sentences into Expressions, Equations and Inequalities
  • Recent Posts

Alene Burke, RN, MSN

Grade 6 Mathematics Module: Solving Problems Involving Different Types of Proportion

This Self-Learning Module (SLM) is prepared so that you, our dear learners, can continue your studies and learn while at home. Activities, questions, directions, exercises, and discussions are carefully stated for you to understand each lesson.

Each SLM is composed of different parts. Each part shall guide you step-by-step as you discover and understand the lesson prepared for you.

Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will tell you if you need to proceed on completing this module or if you need to ask your facilitator or your teacher’s assistance for better understanding of the lesson. At the end of each module, you need to answer the post-test to self-check your learning. Answer keys are provided for each activity and test. We trust that you will be honest in using these.

Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use a separate sheet of paper in answering the exercises and tests. And read the instructions carefully before performing each task.

If you have any questions in using this SLM or any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator.

This module was designed and written with you in mind. It is here to help you master the skills in finding the missing term in a proportion. The scope of this module permits it to be used in many different learning situations. The language used recognizes your diverse vocabulary level. The lessons are arranged to follow the standard sequence of the course. But the order in which you read them can be changed to correspond with the textbook you are now using.

The module contains lessons on:

1. finding a missing term in a proportion (direct, inverse, and partitive); and

2. solving problems involving direct proportion, partitive proportion and inverse proportion in different contexts such as distance, rate, and time using appropriate strategies and tools.

The module is divided into three lessons, namely:

  • Lesson 1 – Solving Word Problems Involving Direct Proportion
  • Lesson 2 – Solving Word Problems Involving Inverse Proportion
  • Lesson 3 – Solving Word Problems Involving Partitive Proportion

After going through this module, you are expected to:

1. find a missing term in a proportion (direct, inverse, and partitive);

2. use models, tables, and illustrations to understand, identify and differentiate direct, inverse, and partitive proportions; and,

3. solve problems involving direct proportion, partitive proportion and inverse proportion in different contexts such as distance, rate, and time using appropriate strategies and tools.

Grade 6 Mathematics Quarter 2 Self-Learning Module: Solving Problems Involving Different Types of Proportion

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IMAGES

  1. Solving Direct Proportion Problems

    steps in solving problem involving direct proportion

  2. Algebraic direct proportion (Problem solving Part 1)

    steps in solving problem involving direct proportion

  3. Direct Proportion Problem Solving Lesson

    steps in solving problem involving direct proportion

  4. R10a

    steps in solving problem involving direct proportion

  5. Direct Proportion

    steps in solving problem involving direct proportion

  6. R10a

    steps in solving problem involving direct proportion

VIDEO

  1. 10th Maths

  2. Solving Word Problems Involving Proportion @MathTeacherGon

  3. G9 MATH Q1 Week 5

  4. Direct Proportion

  5. DIRECT EXTERIOR TANGENTS

  6. how to solve word problem involving direct proportion, (math 6

COMMENTS

  1. Direct Proportion

    Step-by-step guide: y=mx+c. Note, the value of y y can be proportional to other powers of x x including x^ {2}, x^ {3}, x2,x3, or even \sqrt {x}. x. Each of these has a different algebraic and graphical representation. Step-by-step guide: Directly proportional graphs / inversely proportional graphs How to use direct proportion

  2. Direct & Inverse Proportions (Indirect Proportions) with solutions

    The following diagram gives the steps to solve ratios and direct proportion word problems. Scroll down the page for examples and step-by-step solutions. Direct Proportions/Variations Two values x and y are directly proportional to each other when the ratio x : y or is a constant (i.e. always remains the same).

  3. Direct Proportion Calculator

    Step by Step Work Examples of How to Solve Direct Proportion Problems: 1. Identify the two variables involved in the proportion. These will be referred to as x and y. 2. Write down the proportion. The written proportion should be in the form "x is directly proportional to y". 3. Find a pair of values that illustrates the proportion.

  4. Direct Proportion

    ALGEBRA Relevant for … Solving several examples of direct proportion. See examples Summary of direct proportion Direct proportion is the relationship between two variables, which have a ratio that is equal to a constant value.

  5. Direct Proportion

    The direct proportion formula says if the quantity y is in direct proportion to quantity x, then we can say y = kx, for a constant k. y = kx is also the general form of the direct proportion equation. where, k is the constant of proportionality. y increases as x increases. y decreases as x decreases. Direct Proportion Graph

  6. Direct proportion

    Key points When two variables are directly proportional , as one increases the other also increases at the same rate (proportionally). So if one doubles, the other also doubles. Direct...

  7. Direct Proportion

    Direct proportion or direct variation is the relation between two quantities where the ratio of the two is equal to a constant value. It is represented by the proportional symbol, ∝. In fact, the same symbol is used to represent inversely proportional, the matter of the fact that the other quantity is inverted here.

  8. Direct and inverse proportion

    There are four steps to do this: write the proportional relationship convert to an equation using a constant of proportionality use given information to find the constant of proportionality...

  9. Solving Problems Involving Direct, Partitive and Inverse Proportions

    This video is all about solving problems involving direct proportion, partitive proportion, and inverse proportion in different contexts such as distance, ra...

  10. 5.2: Applications of Proportionality

    In the previous section, we studied proportions, and used them to solve problems involving ratios. In this section, we continue our study of proportions, and investigate two different types of proportionality. ... and make sure corresponding quantities stay together -- you can solve any direct proportion problem. Now for the process that will ...

  11. direct proportions

    Solve problems from Pre Algebra to Calculus step-by-step . step-by-step. direct proportions. en. Related Symbolab blog posts. My Notebook, the Symbolab way. Math notebooks have been around for hundreds of years. You write down problems, solutions and notes to go back...

  12. R10a

    An illustration of quantities in direct proportion. Adjust the dark grey slider to select the cost of 1 metre of rope. Now in this example, the cost of a piece of rope is directly proportional to its length. The longer the rope, the more it will cost.

  13. Solving Direct Proportion Problems

    In this video we look at how to solving direct proportion problems using two methods: 1) cross multiplication and 2) solving for the constant of proportional...

  14. Direct and Inverse Proportion

    Maths Math Article Direct And Inverse Proportion Direct and Inverse Proportion A direct and inverse proportion are used to show how the quantities and amount are related to each other. They are also mentioned as directly proportional or inversely proportional. The symbol used to denote the proportionality is ' ∝ '.

  15. Proportion

    A proportion is a statement that two ratios are equal. Proportions are usually expressed mathematically as two equal ratios written as fractions, such as. 1 2 = 2 4. A proportion is denoted by an ...

  16. Direct and Inverse Proportion Practice Questions

    variation, proportionality. Practice Questions. Previous: Pythagoras Practice Questions. Next: Probability Practice Questions. The Corbettmaths Practice Questions on Direct and Inverse Proportion.

  17. Solving Word Problems Involving Direct Proportion, Inverse ...

    In this lesson, you will learn the different proportion problems. The set-up of the different proportion problems and the steps on how to solve these problem...

  18. WORD PROBLEMS ON DIRECT PROPORTION

    Therefore, 1107 letters can be sorted in 9 hours. Problem 6 : If half a meter of cloth costs $15. Find the cost of 8 1/3 meters of the same cloth. Solution : Let x be the cost of 8 1/3 m of the same cloth. Length of cloth is increases. So, cost of cloth is also increases. It comes under direct proportion.

  19. Direct and Inverse Proportion: Worksheets with Answers

    Mathster keyboard_arrow_up. Mathster is a fantastic resource for creating online and paper-based assessments and homeworks. They have kindly allowed me to create 3 editable versions of each worksheet, complete with answers. Worksheet Name. 1. 2. 3. Direct Proportionality - Basics. 1.

  20. Solving Problems Involving Proportions: TEAS

    60x/60 = 74/60 = x. 74/60 = 1.2 teaspoons. Answer: There are 1.2 teaspoons in 74 drops. Proportions are often used in the calculation of dosages and solutions in pharmacology and the preparation of medications by nurses, pharmacists and pharmacy technicians as well as us, in our everyday lives.

  21. Grade 6 Mathematics Module: Solving Problems Involving Different Types

    1. finding a missing term in a proportion (direct, inverse, and partitive); and 2. solving problems involving direct proportion, partitive proportion and inverse proportion in different contexts such as distance, rate, and time using appropriate strategies and tools. The module is divided into three lessons, namely: