Metric Math Conversion Problems
Related Pages Introduction to the Metric System Metric Worksheets Measurement Games
In these lessons, we will look into some methods that can be used for metric conversion problems. The first method uses the metric table and can also be regarded as a shortcut method. The second method uses the metric staircase and the third method uses the unit fraction method.
The following diagrams show some metric conversions for lengths and weights. Scroll down the page for examples and solutions.
Convert using a Metric Table or Shortcut Method
The metric system uses prefixes to denote multiple of 10.
The following table shows some common prefixes of metric units and how to convert between them.
Each prefix differs by a multiple of 10 from the next prefix. When converting between the different units of measure, we look at the number of “jumps” between the prefixes of the two units and then multiply or divide by the powers of 10 accordingly (moving to the right would mean to multiply and moving to the left would mean to divide).
Example: Convert 3 km to m
Solution: There are 3 “jumps” to the right from kilometer to meter.
So, we multiply by 10 3 = 1000
3 km = 3 × 1,000 = 3,000 m
Example: Convert 20 mm to cm
Solution: There is 1 “jump” to the left from millimeter to centimeter
So, we divide by 10.
20 mm = 20 ÷ 10 = 2 cm
How to convert to different metric units of measure for length, capacity, and mass? The metric system is widely used because it is based on the powers of 10. This allows for easy conversion between smaller and larger units. The basic unit of length is the meter (m). It is just over 1 yard. The basic unit of weight is the gram (g). One gram is the mass of 1 cubic centimeter of water. The basic unit of capacity or volume is liter (l). A liter equals 1000 cubic centimeters. Examples: Complete each conversion
- 6.3km = ____ m
- 62,300cm = ___ hm
- 37,200g = ___ kg
- 1.73hg = ___ dg
- 93.7L = ___ mL
- 62,300cL = ___ kL
How to perform metric conversion using the table or shortcut method?
Use the mnemonic “King Henry Drinks Ucky Dark Chocolate Milk”
- 0.521 mm to m
- 9501 g to kg
- 40.6 L to mL
- 1457 dm to hm
- 53.45 dag to mg .4 kL to mL
How to convert metric units of weight without using a calculator? The presentation focuses on conversion between kg, g, and mg. Imagine a 1 kilogram(kg) tablet which is made up of a 1000 smaller gram(g) tablets which is in turn made up of 1000 even smaller milligram (mg) tablets. Big to small: multiply Small to big: divide
How to multiply and divide by 1000 in conversion of metric units? This tutorial explains answers to exercises in converting metric units of weight. The exercises involve multiplying and dividing without a calculator.
Metric Staircase or Ladder Method
How to use the metric staircase to convert units within the metric system? Another method similar to the table method is to use the metric staircase. It is also called the ladder method.
Use the mnemonic “Kangaroo Helps Dingo Because Dingo Can’t Multiply”
How to use the metric staircase to convert between metric measurements? When you go up the stairs to the left, move the decimal to the left. When you go down the stairs to the right, move the decimal to the right.
Convert using the Unit Fraction Method
In this method, we multiply by a unit fraction in order to perform the metric conversion.
Common Metric Length Conversions Using Unit Fractions
Common Metric Weight Conversions Using Unit Fractions
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5.6: Unit Conversion- Metric System
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- David Arnold
- College of the Redwoods
The metric system of units is the standard system of units preferred by scientists. It is based on the base ten number system and its decimal format is more friendly to users of this system. There is a common set of prefixes adopted by the metric system to indicate a power of ten to apply to the base unit.
Metric System Preferences
This is a list of standard prefixes for the metric system and their meanings.
\[ \begin{array}{c c} \text{deka = 10} & \text{deci = 1/10} \\ \text{hecto = 100} & \text{centi = 1/100} \\ \text{kilo = 1000} & \text{milli = 1/1000} \end{array}\nonumber \]
Thus, for example, a decameter is 10 meters, a hectoliter is 100 liters, and a kilogram is 1000 grams. Similarly, a decimeter is 1/10 of a meter, a centiliter is 1/100 of a liter, and a milligram is 1/1000 of a gram.
Units of Length
The standard measure of length in the metric system is the meter.
Historically, the meter was defined by the French Academy of Sciences as the length between two marks on a platinum-iridium bar, which was designed to represent 110,000,000 of the distance from the equator to the north pole through Paris. In 1983, it was redefined by the International Bureau of Weights and Measures (BIPM) as the distance travelled by light in free space in 1299,792,458 of a second. (Wikipedia)
We can apply the standard prefixes to get the following result.
Metric Units of Length
These units of length are used in the metric system.
We can use these facts to build conversion factors as we did in Section 6.3. For example, because
1 km = 1000 m,
we can divide both sides by 1000 m to produce the conversion factor
\[1 = \frac{1 \text{ km}}{1000 \text{ m}}.\nonumber \]
This conversion factor can help change meters into kilometers.
Before using this conversion factor in an example, we repeat here the rules for multiplying and dividing by powers of ten. We will be making heavy use of these rules in this section.
Multiplying and Dividing by Powers of Ten
- Multiplying a decimal number by 10 n will move the decimal point n places to the right. For example, 3.2567 · 10 2 = 3.2567 · 100 = 325.67.
- Dividing a decimal number by 10 n will move the decimal point n places to the left. For example, 3.2567/10 2 = 3.2567/100 = 0.032567.
And now the example.
Change 2,326 meters to kilometers.
Multiply by the conversion factor 1 km/1000m.
\[ \begin{aligned} 2326 \text{ m} = 2326 \text{ m} \cdot \frac{1 \text{ km}}{1000 \text{ m}} ~ & \textcolor{red}{ \text{ Apply conversion factor.}} \\ = 2326 \cancel{ \text{ m}} \cdot \frac{1 \text{ km}}{1000 \cancel{ \text{ m}}} ~ & \textcolor{red}{ \text{ Cancel common units.}} \\ = \frac{2326 \cdot 1}{1000} \text{ km} ~ & \textcolor{red}{ \text{ Multiply fractions.}} \\ = 2.326 \text{ km } & \textcolor{red}{ \text{ Simplify.}} \end{aligned}\nonumber \]
In the last step, note that dividing by 1000 moves the decimal point three places to the left. Thus, 2326 meters is equal to 2.326 kilometers.
Alternate Solution
A second solution depends upon the fact that multiplying or dividing by a power of ten will move the decimal point right or left a number of places equal to the number of zeros present in the multiplier or divisor. Thus, as we saw above, dividing by 1000 moved the decimal point 3 places to the left. Suppose that we arrange the metric units of length in order, from largest to smallest, as shown below.
Note that we must move 3 places left to move from the meters (m) abbreviation to the kilometers (km) abbreviation. In like manner, if we write 2,326 meters as 2,326.0meters, then we can convert to kilometers by moving the decimal 3 places to the left.
Change 1,156 meters to kilometers
1.156 kilometers
Change 537 centimeters to meters.
We know that
\[1 \text{ cm } = \frac{1}{100} \text{ m,}\nonumber \]
or multiplying both sides of this result by 100,
\[100 \text{ cm } = 1 \text{ m.}\nonumber \]
Dividing both sides of this last result by 100 cm, we obtain the conversion factor 1 m/100 cm.
\[\begin{aligned} 537 \text{ cm } = 537 \text{ cm } \cdot \frac{1 \text{ m}}{100 \text{ cm}} ~ & \textcolor{red}{ \text{ Apply conversion factor.}} \\ = 537 ~ \cancel{ \text{cm}} \cdot \frac{1 \text{ m}}{100 ~ \cancel{ \text{ cm}}} ~ & \textcolor{red}{ \text{ Cancel common units.}} \\ = \frac{537 \cdot 1}{100} \text{ m} ~ & \textcolor{red}{ \text{ Multiply fractions.}} \\ = 5.37 \text{ m} \end{aligned}\nonumber \]
In the last step, note that dividing by 100 moves the decimal point two places to the left. Alternately, we can set up our ordered list of units.
Note that we must move 2 places left to move from the centimeters (cm) abbreviation to the meters (m) abbreviation. In like manner, if we write 537 centimeters as 537.0 centimeters, then we can convert to meters by moving the decimal 2 places to the left.
Sometimes more than one conversion factor is needed.
Change 276 centimeters to meters.
2.76 meters
Change 10.2 dekameters to centimeters.
We have two facts:
- 1 dam=10 m, which yields the conversion factor 10 m/1 dam.
- 1 cm=(1/100) m or 100 cm=1 m, which yields the conversion factor 100 cm/1 m.
\[\begin{aligned} 10.2 \text{ dam } = 10.2 \text{ dam } \cdot \frac{10 \text{ m}}{1 \text{ dam}} \cdot \frac{100 \text{ cm}}{1 \text{ m}} ~ & \textcolor{red}{ \text{ Apply conversion factor.}} \\ = 10.2 ~ \cancel{ \text{dam}} \cdot \frac{10 ~ \cancel{ \text{ m}}}{1 ~ \cancel{ \text{ dam}}} \cdot \frac{100 \text{ cm}}{1 ~ \cancel{ \text{ m}}} ~ & \textcolor{red}{ \text{ Cancel common units.}} \\ = \frac{10.2 \cdot 10 \cdot 100}{1} \text{ cm} ~ & \textcolor{red}{ \text{ Multiply fractions.}} \\ = 10,200 \text{ cm} \end{aligned}\nonumber \]
In the last step, note that multiplying by 10, then by 100, moves the decimal point three places to the right.
Alternately, we can set up our ordered list of units.
Note that we must move 3 places right to move from the dekameters (dam) abbreviation to the centimeters (cm) abbreviation. In like manner, we can convert 10.2 dekameters to centimeters by moving the decimal 3 places to the right.
Change 13.5 dekameters to centimeters
13,500 centimeters
Units of Mass
The fundamental unit of mass in the metric system is called a gram. Originally, it was defined to be equal to one cubic centimeter of water measured at the temperature of melting ice. Now it is simply defined as 1/1000 of a kilogram, which is defined by a physical prototype preserved by the International Bureau of Weights and Measures (Wikipedia). The mass of an object is not the same as an object’s weight, but rather a resistance to motion when an external force is applied.
The same metric system prefixes apply.
Metric Units of Mass
These units of mass are used in the metric system.
Convert 0.025 dekagrams to milligrams.
We’ll use two conversion factors:
- 1 dag=10 g, which yields the conversion factor 10 g/1 dag.
- 1 mg=(1/1000) g, which yields the conversion factor 1000 mg/1 g.
\[\begin{aligned} 0.025 \text{ dag } = 0.025 \text{ dag } \cdot \frac{10 \text{ g}}{1 \text{ dag}} \cdot \frac{1000 \text{ mg}}{1 \text{ g}} ~ & \textcolor{red}{ \text{ Apply conversion factor.}} \\ = 0.025 ~ \cancel{ \text{dag}} \cdot \frac{10 ~ \cancel{ \text{ g}}}{1 ~ \cancel{ \text{ dag}}} \cdot \frac{1000 \text{ mg}}{1 ~ \cancel{ \text{ g}}} ~ & \textcolor{red}{ \text{ Cancel common units.}} \\ = \frac{0.025 \cdot 10 \cdot 1000}{1} \text{ mg} ~ & \textcolor{red}{ \text{ Multiply fractions.}} \\ = 250 \text{ mg} \end{aligned}\nonumber \]
Note that we must move 4 places right to move from the dekagrams (dag) abbreviation to the milligrams (mg) abbreviation. In like manner, we can convert 0.025 dekagrams to milligrams by moving the decimal 4 places to the right.
Convert 0.05 dekagrams to milligrams.
500 milligrams
Units of Volume
The fundamental unit of volume in the metric system is called a litre. Originally, one litre was defined as the volume of one kilogram of water measured at 4 ◦ C at 760 millimeters of mercury (Wikipedia). Currently, 1 litre is defined as 1 cubic decimeter (imagine a cube with each edge 1/10 of a meter).
Metric Units of Volume
These units of volume are used in the metric system.
Convert 11,725 millilitres to dekalitres.
- 1 daL=10 L, which yields the conversion factor 1 daL/10 L.
- 1 mL=(1/1000) L, which yields the conversion factor 1 L/1000 mL.
\[ \begin{aligned} 11, 725 \text{ mL } = 11, 725 \text{ mL } \cdot \frac{1 \text{ L}}{1000 \text{ mL}} \cdot \frac{1 \text{ daL}}{10 \text{ L}} ~ & \textcolor{red}{ \text{ Apply conversion factors.}} \\ = 11, 725 ~ \cancel{\text{mL}} \cdot \frac{1 ~ \cancel{\text{L}}}{1000 ~ \cancel{\text{mL}}} \cdot \frac{1 \text{ daL}}{10 ~ \cancel{\text{L}}} ~ & \textcolor{red}{ \text{ Cancel common units.}} \\ = \frac{11, 725 \cdot 1 \cdot 1}{1000 \cdot 10} \text{ daL } & \textcolor{red}{ \text{ Multiply fractions.}} \\ = 1.1725 \text{ daL} \end{aligned}\nonumber \]
Note that we must move 4 places left to move from the millitres (mL) abbreviation to the dekalitres (daL) abbreviation. In like manner, we can convert 11,725 millilitres to dekalitres by moving the decimal 4 places to the left.
Convert 5,763 millilitres to dekalitres.
0.5763 dekalitres
1. What is the meaning of the metric system prefix centi?
2. What is the meaning of the metric system prefix deka?
3. What is the meaning of the metric system prefix hecto?
4. What is the meaning of the metric system prefix kilo?
5. What is the meaning of the metric system prefix deci?
6. What is the meaning of the metric system prefix milli?
7. What is the meaning of the metric system abbreviation mg?
8. What is the meaning of the metric system abbreviation g?
9. What is the meaning of the metric system abbreviation m?
10. What is the meaning of the metric system abbreviation km?
11. What is the meaning of the metric system abbreviation kL?
12. What is the meaning of the metric system abbreviation daL?
13. What is the meaning of the metric system abbreviation hm?
14. What is the meaning of the metric system abbreviation dm?
15. What is the meaning of the metric system abbreviation dam?
16. What is the meaning of the metric system abbreviation cm?
17. What is the meaning of the metric system abbreviation dL?
18. What is the meaning of the metric system abbreviation L?
19. What is the meaning of the metric system abbreviation hg?
20. What is the meaning of the metric system abbreviation kg?
21. What is the meaning of the metric system abbreviation dg?
22. What is the meaning of the metric system abbreviation dag?
23. What is the meaning of the metric system abbreviation hL?
24. What is the meaning of the metric system abbreviation cL?
25. Change 5,490 millimeters to meters.
26. Change 8,528 millimeters to meters.
27. Change 64 meters to millimeters.
28. Change 65 meters to millimeters.
29. Change 4,571 millimeters to meters.
30. Change 8,209 millimeters to meters.
31. Change 15 meters to centimeters.
32. Change 12 meters to centimeters.
33. Change 569 centimeters to meters.
34. Change 380 centimeters to meters.
35. Change 79 meters to centimeters.
36. Change 60 meters to centimeters.
37. Change 7.6 kilometers to meters.
38. Change 4.9 kilometers to meters.
39. Change 861 centimeters to meters.
40. Change 427 centimeters to meters.
41. Change 4,826 meters to kilometers.
42. Change 1,929 meters to kilometers.
43. Change 4,724 meters to kilometers.
44. Change 1,629 meters to kilometers.
45. Change 6.5 kilometers to meters.
46. Change 7.9 kilometers to meters.
47. Change 17 meters to millimeters.
48. Change 53 meters to millimeters.
49. Change 512 milligrams to centigrams.
50. Change 516 milligrams to centigrams.
51. Change 541 milligrams to centigrams.
52. Change 223 milligrams to centigrams.
53. Change 70 grams to centigrams.
54. Change 76 grams to centigrams.
55. Change 53 centigrams to milligrams.
56. Change 30 centigrams to milligrams.
57. Change 83 kilograms to grams.
58. Change 70 kilograms to grams.
59. Change 8,196 grams to kilograms.
60. Change 6,693 grams to kilograms.
61. Change 564 centigrams to grams.
62. Change 884 centigrams to grams.
63. Change 38 grams to centigrams.
64. Change 88 grams to centigrams.
65. Change 77 centigrams to milligrams.
66. Change 61 centigrams to milligrams.
67. Change 5,337 grams to kilograms.
68. Change 4,002 grams to kilograms.
69. Change 15 kilograms to grams.
70. Change 45 kilograms to grams.
71. Change 833 centigrams to grams.
72. Change 247 centigrams to grams.
73. Change 619,560 centilitres to kilolitres.
74. Change 678,962 centilitres to kilolitres.
75. Change 15.2 litres to millilitres.
76. Change 9.7 litres to millilitres.
77. Change 10,850 centilitres to litres.
78. Change 15,198 centilitres to litres.
79. Change 10.7 litres to millilitres.
80. Change 17.3 litres to millilitres.
81. Change 15,665 millilitres to litres.
82. Change 12,157 millilitres to litres.
83. Change 6.3 kilolitres to centilitres.
84. Change 8.3 kilolitres to centilitres.
85. Change 4.5 kilolitres to centilitres.
86. Change 6.2 kilolitres to centilitres.
87. Change 10.6 litres to centilitres.
88. Change 16.6 litres to centilitres.
89. Change 14,383 centilitres to litres.
90. Change 11,557 centilitres to litres.
91. Change 9.9 litres to centilitres.
92. Change 19.5 litres to centilitres.
93. Change 407,331 centilitres to kilolitres.
94. Change 827,348 centilitres to kilolitres.
95. Change 14,968 millilitres to litres.
96. Change 18,439 millilitres to litres.
7. milligram
11. kilolitre or kiloliter
13. hectometer
15. dekameter
17. decilitre or deciliter
19. hectogram
21. decigram
23. hectolitre or hectoliter
25. 5.49 meters
27. 64,000 millimeters
29. 4.571 meters
31. 1,500 centimeters
33. 5.69 meters
35. 7,900 centimeters
37. 7,600 meters
39. 8.61 meters
41. 4.826 kilometers
43. 4.724 kilometers
45. 6,500 meters
47. 17,000 millimeters
49. 51.2 centigrams
51. 54.1 centigrams
53. 7,000 centigrams
55. 530 milligrams
57. 83,000 grams
59. 8.196 kilograms
61. 5.64 grams
63. 3,800 centigrams
65. 770 milligrams
67. 5.337 kilograms
69. 15,000 grams
71. 8.33 grams
73. 6.1956 kilolitres
75. 15,200 millilitres
77. 108.5 litres
79. 10,700 millilitres
81. 15.665 litres
83. 630,000 centilitres
85. 450,000 centilitres
87. 1,060 centilitres
89. 143.83 litres
91. 990 centilitres
93. 4.07331 kilolitres
95. 14.968 litres
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Converting Metric Units
Here we will learn about converting metric units, including metric units of length, metric units of mass and metric units of capacity (volume).
There are also converting metric units 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 converting metric units?
Converting metric units is being able to convert between different metric units of length, mass or volume.
To do this we need to know what the metric units are and their conversion factors.
We can use prefixes to make these metric units bigger and smaller.
The main ones used are kilo, centi and milli:
Metric units of length
The SI unit (international system of units) of length is the metre (m) .
For length we mostly use kilometres (km) , metres (m) , centimetres (cm) and millimetres (mm) .
Covert 7 \ m to centimetres
So, 7 \ m=700 \ cm
Metric units of mass
The metric system for mass is based around grams (g) .
For mass we mostly use tonnes (t) , kilograms (kg) , and grams (g) .
Covert 4 \ kg to centimto grams
So, 4 \ kg=4000 \ g
Metric units of capacity (volume)
The SI unit The metric system for capacity is based on litres (l) .
For volume we mostly use litres (l) , centilites (cl) , and millilitres (ml) .
We also need to be aware that:
Covert 6000 \ ml to litres
So, 6000 \ ml=6 \ l
How to convert metric units
In order to convert metric units:
- Find the unit conversion.
- Multiply or divide.
Write the answer.
Converting metric units worksheet
Get your free converting metric units worksheet of 20+ questions and answers. Includes reasoning and applied questions.
Related lessons on units of measurement
Converting metric units is part of our series of lessons to support revision on units of measurement . You may find it helpful to start with the main units of measurement 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:
- Units of measurement
- Conversion of units
- Metric units of measurement
- Converting units of area and volume
- Converting units of time
Converting metric units examples
Example 1: converting length.
Convert 3.2 \ m to cm
2 Multiply or divide.
As we are going from larger units to smaller units we multiply.
3 Write the answer.
Example 2: converting length
Convert 780 \ mm to cm
Find the unit conversion .
Multiply or divide .
As we are going from smaller units to larger units we divide.
780 \div 10=78
Example 3: converting mass
Convert 12.5 \ kg to g
12.5 \times 1000=12 \ 500
Example 4: converting mass
Convert 3800 \ g to kg
3800\div 1000=3.8
Example 5: converting capacity
Convert 7.1 \ l to ml
7.1 \times 1000=7100
Example 6: converting capacity
Convert 750 \ ml to l
750 \div 1000=0.75
Example 7: converting capacity
Convert 6.8 \ l to cm^3
6.8 \times 1000=6800
Common misconceptions
- Multiply or divide?
If you are going from larger units to smaller units – multiply If you are going from smaller units to larger units – divide
Practice converting metric units questions
1. Convert: 390 \ cm to m
Converting centimetres to metres we divide by 100 .
2. Convert: 57 \ cm to mm
Converting centimetres to millimetres we multiply by 10 .
3. Convert: 81 \ 000 \ g to kg
Converting grams to kilograms we divide by 1000 .
4. Convert: 630 \ kg to tonnes
Converting kilograms to tonnes we divide by 1000 .
5. Convert: 4.8 \ l to ml
Converting litres to millilitres we multiply by 1000 .
6. Convert: 560 \ ml to l
Converting millilitres to litres we divide by 1000 .
Converting metric units GCSE questions
1. Write 376 \ cm in metres.
2. Here is a signpost
How far apart are the Town Centre and the Park?
Give your answer in kilometres.
Converting distance to km
Adding the two distances
3. A bottle contains 1.75 litres of lemonade.
A can contains 330 \ ml of lemonade.
Phil has 2 bottles of lemonade and 6 cans of lemonade.
How much lemonade does he have in total?
Give your answer in millilitres.
Converting capacity to millilitres
For finding the total
4. 4 tins of soup have a mass of 1.8 kg.
3 tins of soup and 2 packets of soup have a mass of 1420 \ g.
Find the mass of 7 tins of soup and 5 packets of soup.
Give your answer in kilograms.
Finding the mass of one tin
Converting mass to kg
Learning checklist
You have now learned how to:
- Convert metric units of length
- Convert metric units of mass
- Convert metric units of capacity (volume)
The next lessons are
- Conversion graphs
- Scale maths
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Metric Units: Reasoning and Problem Solving
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2.6: Problem Solving and Unit Conversions
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Learning Objectives
- To convert a value reported in one unit to a corresponding value in a different unit using conversion factors.
During your studies of chemistry (and physics also), you will note that mathematical equations are used in many different applications. Many of these equations have a number of different variables with which you will need to work. You should also note that these equations will often require you to use measurements with their units. Algebra skills become very important here!
Converting Between Units with Conversion Factors
A conversion factor is a factor used to convert one unit of measurement into another. A simple conversion factor can convert meters into centimeters, or a more complex one can convert miles per hour into meters per second. Since most calculations require measurements to be in certain units, you will find many uses for conversion factors. Always remember that a conversion factor has to represent a fact; this fact can either be simple or more complex. For instance, you already know that 12 eggs equal 1 dozen. A more complex fact is that the speed of light is \(1.86 \times 10^5\) miles/\(\text{sec}\). Either one of these can be used as a conversion factor depending on what type of calculation you are working with (Table \(\PageIndex{1}\)).
*Pounds and ounces are technically units of force, not mass, but this fact is often ignored by the non-scientific community.
Of course, there are other ratios which are not listed in Table \(\PageIndex{1}\). They may include:
- Ratios embedded in the text of the problem (using words such as per or in each , or using symbols such as / or %).
- Conversions in the metric system, as covered earlier in this chapter.
- Common knowledge ratios (such as 60 seconds \(=\) 1 minute).
If you learned the SI units and prefixes described, then you know that 1 cm is 1/100th of a meter.
\[ 1\; \rm{cm} = \dfrac{1}{100} \; \rm{m} = 10^{-2}\rm{m} \nonumber \]
\[100\; \rm{cm} = 1\; \rm{m} \nonumber \]
Suppose we divide both sides of the equation by \(1 \text{m}\) (both the number and the unit):
\[\mathrm{\dfrac{100\:cm}{1\:m}=\dfrac{1\:m}{1\:m}} \nonumber \]
As long as we perform the same operation on both sides of the equals sign, the expression remains an equality. Look at the right side of the equation; it now has the same quantity in the numerator (the top) as it has in the denominator (the bottom). Any fraction that has the same quantity in the numerator and the denominator has a value of 1:
\[ \dfrac{ \text{100 cm}}{\text{1 m}} = \dfrac{ \text{1000 mm}}{\text{1 m}}= \dfrac{ 1\times 10^6 \mu \text{m}}{\text{1 m}}= 1 \nonumber \]
We know that 100 cm is 1 m, so we have the same quantity on the top and the bottom of our fraction, although it is expressed in different units.
Performing Dimensional Analysis
Dimensional analysis is amongst the most valuable tools that physical scientists use. Simply put, it is the conversion between an amount in one unit to the corresponding amount in a desired unit using various conversion factors. This is valuable because certain measurements are more accurate or easier to find than others. The use of units in a calculation to ensure that we obtain the final proper units is called dimensional analysis .
Here is a simple example. How many centimeters are there in 3.55 m? Perhaps you can determine the answer in your head. If there are 100 cm in every meter, then 3.55 m equals 355 cm. To solve the problem more formally with a conversion factor, we first write the quantity we are given, 3.55 m. Then we multiply this quantity by a conversion factor, which is the same as multiplying it by 1. We can write 1 as \(\mathrm{\dfrac{100\:cm}{1\:m}}\) and multiply:
\[ 3.55 \; \rm{m} \times \dfrac{100 \; \rm{cm}}{1\; \rm{m}} \nonumber \]
The 3.55 m can be thought of as a fraction with a 1 in the denominator. Because m, the abbreviation for meters, occurs in both the numerator and the denominator of our expression, they cancel out:
\[\dfrac{3.55 \; \cancel{\rm{m}}}{ 1} \times \dfrac{100 \; \rm{cm}}{1 \; \cancel{\rm{m}}} \nonumber \]
The final step is to perform the calculation that remains once the units have been canceled:
\[ \dfrac{3.55}{1} \times \dfrac{100 \; \rm{cm}}{1} = 355 \; \rm{cm} \nonumber \]
In the final answer, we omit the 1 in the denominator. Thus, by a more formal procedure, we find that 3.55 m equals 355 cm. A generalized description of this process is as follows:
quantity (in old units) × conversion factor = quantity (in new units)
You may be wondering why we use a seemingly complicated procedure for a straightforward conversion. In later studies, the conversion problems you encounter will not always be so simple . If you master the technique of applying conversion factors, you will be able to solve a large variety of problems.
In the previous example, we used the fraction \(\dfrac{100 \; \rm{cm}}{1 \; \rm{m}}\) as a conversion factor. Does the conversion factor \(\dfrac{1 \; \rm m}{100 \; \rm{cm}}\) also equal 1? Yes, it does; it has the same quantity in the numerator as in the denominator (except that they are expressed in different units). Why did we not use that conversion factor? If we had used the second conversion factor, the original unit would not have canceled, and the result would have been meaningless. Here is what we would have gotten:
\[ 3.55 \; \rm{m} \times \dfrac{1\; \rm{m}}{100 \; \rm{cm}} = 0.0355 \dfrac{\rm{m}^2}{\rm{cm}} \nonumber \]
For the answer to be meaningful, we have to construct the conversion factor in a form that causes the original unit to cancel out . Figure \(\PageIndex{1}\) shows a concept map for constructing a proper conversion.
General Steps in Performing Dimensional Analysis
- Identify the " given " information in the problem. Look for a number with units to start this problem with.
- What is the problem asking you to " find "? In other words, what unit will your answer have?
- Use ratios and conversion factors to cancel out the units that aren't part of your answer, and leave you with units that are part of your answer.
- When your units cancel out correctly, you are ready to do the math . You are multiplying fractions, so you multiply the top numbers and divide by the bottom numbers in the fractions.
Significant Figures in Conversions
How do conversion factors affect the determination of significant figures?
- Numbers in conversion factors based on prefix changes, such as kilograms to grams, are not considered in the determination of significant figures in a calculation because the numbers in such conversion factors are exact.
- Exact numbers are defined or counted numbers, not measured numbers, and can be considered as having an infinite number of significant figures. (In other words, 1 kg is exactly 1,000 g, by the definition of kilo-.)
- Counted numbers are also exact. If there are 16 students in a classroom, the number 16 is exact.
- In contrast, conversion factors that come from measurements (such as density, as we will see shortly) or that are approximations have a limited number of significant figures and should be considered in determining the significant figures of the final answer.
Example \(\PageIndex{1}\)
Exercise \(\pageindex{1}\).
Perform each conversion.
- 101,000 ns to seconds
- 32.08 kg to grams
- 1.53 grams to cg
- Conversion factors are used to convert one unit of measurement into another.
- Dimensional analysis (unit conversions) involves the use of conversion factors that will cancel unwanted units and produce the appropriate units.
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Conversion: metric/imperial units; line graphs - Problem-Solving Investigation - Year 6
Subject: Mathematics
Age range: 7-11
Resource type: Worksheet/Activity
Last updated
20 February 2019
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This in-depth maths investigation is an open-ended problem solving activity for Year 6 children. It can be used to support teaching towards the objectives: conversion of units: metric and common imperial; conversion line graphs.
In-depth Investigation: Weights in a Line Children use systematic working to calculate the number of possibilities of making weights using just two set weights; then look for patterns using line graphs.
This investigation will develop maths meta-skills, support open-ended questioning and logical reasoning, and enable children to learn to think mathematically and articulate mathematical ideas.
This problem-solving investigation is part of our Year 6 Measures and Data block. Each Hamilton maths block contains a complete set of planning and resources to teach a term’s worth of objectives for one of the National Curriculum for England’s maths areas.
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Scientific Inquiry & Reasoning Skills - General Mathematical Concepts and Techniques
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It’s important for you to know that questions on the natural, behavioral, and social sciences sections will ask you to use certain mathematical concepts and techniques. As the descriptions of the scientific inquiry and reasoning skills suggest, some questions will ask you to analyze and manipulate scientific data to show that you can
Recognize and interpret linear, semilog, and log-log scales and calculate slopes from data found in figures, graphs, and tables
Demonstrate a general understanding of significant digits and the use of reasonable numerical estimates in performing measurements and calculations
Use metric units, including converting units within the metric system and between metric and English units (conversion factors will be provided when needed), and dimensional analysis (using units to balance equations)
Perform arithmetic calculations involving the following: probability, proportion, ratio, percentage, and square-root estimations
Demonstrate a general understanding (Algebra II-level) of exponentials and logarithms (natural and base 10), scientific notation, and solving simultaneous equations
Demonstrate a general understanding of the following trigonometric concepts: definitions of basic (sine, cosine, tangent) and inverse (sin‒1, cos‒1, tan‒1) functions; sin and cos values of 0°, 90°, and 180°; relationships between the lengths of sides of right triangles containing angles of 30°, 45°, and 60°
Demonstrate a general understanding of vector addition and subtraction and the right-hand rule (knowledge of dot and cross products is not required)
Note also that an understanding of calculus is not required, and a periodic table will be provided during the exam.
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Convert Units of Length – Reasoning and Problem Solving
Convert Units of Length - Reasoning and Problem Solving
This worksheet includes a range of reasoning and problem solving questions for pupils to practise the main skill of converting metric units of length.
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Convert Units of Length Reasoning and Problem Solving worksheet Answer sheet
National Curriculum Objectives:
- (5M5) Convert between different units of metric measure (for example, kilometre and metre; centimetre and metre; centimetre and millimetre; gram and kilogram; litre and millilitre)
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Differentiation: Questions 1, 4 and 7 (Reasoning) Developing Convert and add the given metric measurements to determine whether a statement is correct. Using multiples of 5 with up to 1 decimal place. Expected Convert and add the given metric measurements to determine whether a statement is correct. Using any number with up to 3 decimal places.
Reasoning and Problem Solving Metric Measures Developing 1a. Various answers, for example: 1km, 880m. Each is around the same distance and both are plausible distances for children to walk. 2a. Various answers, for example:
Mathematics Year 5: (5M4) Solve problems involving converting between units of time Mathematics Year 5: (5M5) Convert between different units of metric measure (for example, kilometre and metre; centimetre and metre; centimetre and millimetre; gram and kilogram; litre and millilitre) Mathematics Year 5: (5M6) Understand and use approximate equiv...
The metric system is widely used because it is based on the powers of 10. This allows for easy conversion between smaller and larger units. The basic unit of length is the meter (m). It is just over 1 yard. The basic unit of weight is the gram (g). One gram is the mass of 1 cubic centimeter of water.
There are 56 NRICH Mathematical resources connected to Length/distance, you may find related items under Measuring and calculating with units. Broad Topics > Measuring and calculating with units > Length/distance
Unit 1 Module 1: Place value, rounding, and algorithms for addition and subtraction Unit 2 Module 2: Unit conversions and problem solving with metric measurement Unit 3 Module 3: Multi-digit multiplication and division Unit 4 Module 4: Angle measure and plane figures Unit 5 Module 5: Fraction equivalence, ordering, and operations
Reasoning and Problem Solving Calculate with Metric Measures Developing 1a. 150ml 2a. 984cm > 0.8m; 889cm > 0.4m; 988cm > 0.4m 3a. Marcus is incorrect, because 8 x 70kg is greater than 500kg (560kg). Expected 4a. 5 jugs 5a. 0.7kg > 500g; 0.7kg > 005g; 0.5kg > 007g 6a.
Unit reasoning helps us make sense of measurements by converting between measurement units and making reasonable estimates. For the test, we need to know common measurement units for length, time, volume, and mass, as well as how common units are related in both metric and U.S. customary units.
Use the factor label method and unit fractions to convert from meters to kilometers. 18,000 1 ⋅ 1 kilometer 1,000 = 18,000 kilometers 1,000. 18,000 kilometers 1,000 = 18 kilometers. Cancel, multiply, and solve. Answer.
Start Lesson In this lesson, we will investigate and model problems and problem solving activities where we must convert between metric units.
The fundamental unit of mass in the metric system is called a gram. Originally, it was defined to be equal to one cubic centimeter of water measured at the temperature of melting ice. Now it is simply defined as 1/1000 of a kilogram, which is defined by a physical prototype preserved by the International Bureau of Weights and Measures (Wikipedia).
Metric Units Year 5 Reasoning and Problem Solving with answers. National Curriculum Objectives. Mathematics Year 5: ... Reasoning and Problem Solving Questions 1, 4 and 7 (Problem Solving) Developing Use the given clues to calculate the amount of space used/needed (multiples of 10 only).
Convert units multi-step word problems (metric) Google Classroom. You might need: Calculator. Abigail's mom said she will be home from work in 65 minutes. It is 4: 27 p.m. , and Abigail has dance practice at 6: 00 p.m. How much time will Abigail have between the time that her mom gets home from work and the beginning of dance practice?
Example 2: converting length. Convert 780 \ mm 780 mm to cm cm. Find the unit conversion. Show step. 1 \ cm=10 \ mm 1 cm = 10 mm. Multiply or divide. Show step. As we are going from smaller units to larger units we divide. 780 \div 10=78 780 ÷ 10 = 78.
In this lesson, we will investigate and model problems and problem solving activities where we must convert between metric units. Grid View List View. Presentation. Video. Unsigned Video Signed Video. Intro Quiz. Project In Class. ... Converting between metric units of length; Solving problems involving converting between metric units of length;
Metric Units: Reasoning and Problem Solving. Resource. Maths. Year 5. Download. White Rose. Everyone can do maths: Everyone can! White Rose is used in most UK Primary schools and is the most trusted primary maths provider in the UK. Everything White Rose offers is created by maths experts and directed at teachers using a mastery approach to ...
If there are 100 cm in every meter, then 3.55 m equals 355 cm. To solve the problem more formally with a conversion factor, we first write the quantity we are given, 3.55 m. Then we multiply this quantity by a conversion factor, which is the same as multiplying it by 1. We can write 1 as 100cm 1m and multiply:
Reasoning and Problem Solving Step 4: Imperial Units Teaching Note: The conversions used in this resource are: 1 inch = 2.54cm, 1 pint = 568ml and 1kg = 2.2lbs. National Curriculum Objectives: Mathematics Year 5: (5M6) Understand and use approximate equivalences between metric units and common imperial units such as inches, pounds and pints
This in-depth maths investigation is an open-ended problem solving activity for Year 6 children. It can be used to support teaching towards the objectives: conversion of units: metric and common imperial; conversion line graphs. In-depth Investigation: Weights in a Line
This teaching pack is stuffed full of useful things for your year 5 class. It has been written to support the White Rose Maths small step 3: 'Convert units of length'. Included in the pack is a PowerPoint that takes the children through fluency, reasoning and problem-solving activities. Then, the children can use the activity sheets, which have similar questions and contexts, to tackle the ...
Course: 5th grade > Unit 14. Lesson 2: Converting metric units. Converting units: metric distance. Converting units: centimeters to meters. Convert units (metrics) Metric units of mass review (g and kg) Metric units of length review (mm, cm, m, & km) Metric units of volume review (L and mL) U.S. customary and metric units.
Use metric units, including converting units within the metric system and between metric and English units (conversion factors will be provided when needed), and dimensional analysis (using units to balance equations) ... Skill 2: Scientific Reasoning and Problem-solving; Skill 3: Reasoning about the Design and Execution of Research; Skill 4 ...
This worksheet includes a range of reasoning and problem solving questions for pupils to practise the main skill of converting metric units of length.
The human-like automatic deductive reasoning has always been one of the most challenging open problems in the interdiscipline of mathematics and artificial intelligence. This paper is the third in a series of our works. We built a neural-symbolic system, called FGeoDRL, to automatically perform human-like geometric deductive reasoning. The neural part is an AI agent based on reinforcement ...