Solve problems within and between measurement systems

Ontario Grade 9 Mathematics

Geometry and Measurement

Using units within and between measurement systems

A recipe may list a length in centimetres, while a tool is marked in inches. To solve a problem like this, you need to know what each unit measures and how the units are related. A measurement system is a set of units used to describe measurements. The metric system uses units such as millimetres, centimetres, metres, and kilometres. The imperial system uses units such as inches, feet, yards, and miles. In a real problem, read the units carefully and use a conversion relationship that matches them. Convert units within one system when useful, and convert between systems when the problem requires it.

What you will learn

Identify the unit and measurement system that fit a problem.
Convert measurements within a system using the relationship between units.
Use a given conversion relationship to change from one measurement system to another.
Choose a reasonable answer and include the correct unit.

1. Start with the quantity and unit

A quantity is what is being measured, such as length, mass, or capacity. A unit tells how that quantity is measured. For example,

4

metres is a length, and metres are the unit. Before calculating, identify both the quantity and the unit in every measurement.

Check that the units describe the same kind of quantity before comparing or combining measurements. For example, metres and feet both measure length, so they can be compared after conversion. A length and a mass cannot be added because they describe different quantities.

Review two useful number skills. Multiplying by a number greater than

1

makes a positive measurement larger. Dividing by a number greater than

1

makes it smaller. These patterns help you check whether a conversion makes sense.

Write the unit beside each measurement.
Only compare or combine measurements of the same kind.
Estimate whether a converted value should be larger or smaller.

2. Convert within one system

Within a system means changing from one unit in that system to another. In the metric system, units of length are related by powers of

10

. For example,

1

metre is

100

centimetres. To change metres to centimetres, multiply by

100

. To change centimetres to metres, divide by

100

.

A conversion factor is a number relationship between two units. For metric length, examples include

1

kilometre equals

1000

metres and

1

metre equals

100

centimetres. The units tell you which direction to convert. A smaller unit takes more units to describe the same length, so the number becomes larger.

Imperial units also have relationships within that system. For example,

1

foot is

12

inches. If a measurement is given in feet and the answer needs inches, multiply the number of feet by

12

. If the answer needs feet, divide inches by

12

. Keep the unit with the result so it is clear what the number means.

1\text{ m}=100\text{ cm}

Use a relationship that connects the starting unit to the target unit.
Moving to a smaller unit usually gives a larger number.
Moving to a larger unit usually gives a smaller number.

3. Convert between systems

Between systems means changing from a metric unit to an imperial unit, or the other way around. These units are not related by powers of

10

, so use a suitable conversion relationship. A problem may provide the relationship. If it does, use that value rather than guessing.

For example,

1

inch is exactly

2.54

centimetres. This relationship can convert inches to centimetres or centimetres to inches. To convert inches to centimetres, multiply by

2.54

. To convert centimetres to inches, divide by

2.54

.

Other useful length relationships include

1

mile being about

1.609

kilometres and

1

yard being about

0.9144

metres. The word “about” matters: a rounded conversion gives an approximate answer. Use the level of precision that suits the situation, and state when your answer is approximate.

For a multi-step problem, first convert to a unit that is easy to use. For instance, change mixed metric units into one metric unit before converting systems. This avoids combining unlike unit labels by mistake.

Use the stated or appropriate conversion relationship.
Multiply or divide according to the direction of conversion.
Label an answer approximate when the conversion relationship is approximate.

4. Check whether the answer fits the situation

After calculating, check the unit and size of the answer. If you changed metres to centimetres, the number should become larger. If you changed centimetres to metres, the number should become smaller. These checks can reveal a reversed operation.

Some situations need a practical decision as well as a conversion. If you need to buy a whole number of one-yard pieces, a requirement of

2.6

yards means buying

3

yards, not

2

. Rounding down would leave too little material. In other settings, such as reporting a distance, a decimal value may be suitable.

Do not round too early. Keep enough digits during the calculation, then round the final value only as the situation requires. Always include a unit in the final answer.

Check that the answer's unit matches the question.
Use the context to decide how to round.
Do not round down when that would leave too little of a needed material.

Useful length relationships

Relationship	System change
1 m = 100 cm	Within metric
1 ft = 12 in	Within imperial
1 in = 2.54 cm	Between systems
1 yd ≈ 0.9144 m	Between systems
1 mi ≈ 1.609 km	Between systems

Worked example

Buying fabric for a project

A project needs

2.4

metres of fabric. The store sells fabric by the whole yard. How many yards should you buy? Use

1

yard

\approx 0.9144

metres.

Rewrite in centimetres
First change the metric measurement to centimetres. Since $1$ metre is $100$ centimetres, multiply by $100$ . This is a within-system conversion, and it gives a whole-number metric amount that is easy to picture.
$2.4\text{ m}\times100=240\text{ cm}$
Convert metres to yards
Use the provided relationship in metres, so convert the original $2.4$ metres directly. Divide by $0.9144$ because each yard is about $0.9144$ metres. The result is a little more than $2.6$ yards.
$2.4\div0.9144\approx2.625$
Choose an amount to buy
The store sells only whole yards. Two yards would be less than the amount needed, so round up to the next whole yard. Buying three yards gives enough fabric.
$3\text{ yd}$

Answer: Buy

3

yards of fabric.

Check: Two yards is about

2\times0.9144=1.8288

metres, which is less than

2.4

metres. Three yards is about

3\times0.9144=2.7432

metres, so it is enough.

Common mistakes and how to avoid them

Multiplying when converting centimetres to metres.

Correction: There are

100

centimetres in one metre, so divide the number of centimetres by

100

to find metres.

Leaving the unit off the answer.

Correction: Write the unit requested in the problem. A number alone does not say what was measured.

Rounding a purchase amount down to a whole unit.

Correction: Check whether rounding down provides enough material. If it does not, use the next whole unit.

Using a conversion relationship for a different quantity.

Correction: Match the relationship to the unit and quantity in the question. A length relationship cannot convert a mass.

Lesson summary

Identify what is measured and record each unit.
Use a conversion relationship to change units within or between systems.
Choose multiplication or division based on the starting and target units.
Check the size, unit, and practical meaning of the result.

Check your understanding

Question 1

A ribbon is

350

centimetres long. How many metres is this?

$3.5$ m
$35$ m
$0.35$ m
$35000$ m

Show answer and explanation

3.5

m

There are

100

centimetres in a metre, so divide

350

by

100

. The result is

3.5

metres.

Question 2

A measurement is

8

inches. Using

1

inch equals

2.54

centimetres, which value is closest in centimetres?

$3.15$ cm
$10.54$ cm
$20.32$ cm
$254$ cm

Show answer and explanation

20.32

cm

Multiply because each inch is

2.54

centimetres. The calculation is

8\times2.54=20.32

, so the measurement is

20.32

centimetres.

Key terms

Measurement system: A set of units used to measure quantities, such as the metric or imperial system.
Quantity: What is being measured, such as length, mass, or capacity.
Unit: A named amount used to measure a quantity, such as a metre or an inch.
Conversion factor: A number relationship used to change a measurement from one unit to another.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 9 Mathematics (MTH1W), expectation E1.3. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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E1.3 · Solve problems within and between measurement systems