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B2.1 · Use operation properties with numbers, ratios, rates, and percents

Learn to use operation properties with numbers, ratios, rates, and percents through clear examples and targeted practice.

Ontario Grade 6 Mathematics

Number

Ontario Grade 6 Mathematics — B2.1

A calculation can often be changed in a helpful way without changing its answer. This can make work with numbers, ratios, rates, and percents easier. Addition combines amounts. Multiplication finds equal groups. Division shares or compares amounts. A ratio compares two amounts. A rate compares amounts with different units, such as kilometres and hours. A percent means a number of parts out of 100. In this lesson, you will use familiar operation properties to work with each of these.

What you will learn

Use operation properties to make number calculations easier.
Use multiplication and division to scale ratios and rates.
Use familiar fraction and multiplication facts to find percents of amounts.
Explain why a calculation works and check whether an answer makes sense.

1. Use operation properties with numbers

An operation is an action with numbers, such as adding, subtracting, multiplying, or dividing. An operation property is a rule that tells us when changing the order or grouping of a calculation keeps its value the same.

For addition, changing the order does not change the total. For example, adding 8 and 12 gives the same total as adding 12 and 8. This is the commutative property of addition. Commutative means the numbers can switch places.

For multiplication, changing the order also keeps the same product. A product is the answer to a multiplication question. Three groups of 5 have the same total as 5 groups of 3. This is the commutative property of multiplication.

When adding or multiplying three or more numbers, we can also choose which numbers to work with first. This is the associative property. Associative means the numbers can be grouped in a different way. We can use this property with addition and multiplication, but not to regroup subtraction or division.

The distributive property connects multiplication with addition and subtraction. It lets us multiply each part of a sum or difference, then combine the results. For example, 6 groups of 19 can be found by finding 6 groups of 20 and taking away 6 groups of 1. This works because 19 is 1 less than 20.

a+b=b+a,\quad ab=ba,\quad a(b+c)=ab+ac

Addition and multiplication can be reordered.
Addition and multiplication can be regrouped.
Multiplication can be applied to each part of a sum or difference.
Do not treat subtraction or division as if they can always be reordered or regrouped.

2. Use multiplication and division with ratios and rates

A ratio compares two amounts. Imagine a mix with 2 cups of oats for every 3 cups of raisins. The ratio is 2 to 3. The order matters: the first number describes oats, and the second describes raisins.

If we multiply both parts of a ratio by the same number, the comparison stays the same. For example, doubling both amounts gives 4 cups of oats for every 6 cups of raisins. The amounts are larger, but the relationship has not changed. Dividing both parts by the same number also keeps the comparison when both divide evenly.

A rate compares amounts with different units. For example, 12 kilometres in 3 hours is a rate. To find the distance for 1 hour, divide both amounts by 3. The units help us remember what each number means.

Keep the order and units clear. If the question asks for a rate per hour, the hour amount should be 1. Do not change only one part of a ratio or rate when scaling it, because that changes the comparison.

2:3=4:6

Multiply or divide both parts by the same number to make an equivalent ratio or rate.
Keep the quantities in the same order.
Use units to describe and check a rate.

3. Use operation properties with percents

Percent means “out of 100.” For example, 25% means 25 parts out of 100 equal parts. Some percents match familiar fractions: 50% is one half, 25% is one quarter, and 10% is one tenth.

To find a percent of an amount, use a matching fraction or multiplication fact. For example, 25% of an amount is one quarter of it. Dividing the amount by 4 finds one quarter because four equal parts make the whole.

You can also split a percent into easier parts. For example, 15% is 10% plus 5%. Find each part of the amount, then add the results. This uses the distributive property: finding each part and adding them gives the same result as finding the combined percent.

Check the size of your answer. A percent less than 100% is a part smaller than the whole. If you find 25% of a number, the result should be one quarter of that number, not the full amount.

25\%=\frac{25}{100}=\frac14

Connect familiar percents to familiar fractions.
Use division to find equal parts.
Split a percent into easier parts when helpful, then add the results.
Check that the result is a sensible part of the whole.

4. Choose a method and check your work

There is often more than one correct way to calculate. Choose a method that uses facts you know. You might switch factors to put an easier number first, group factors to make a friendly product, or split a number into parts.

For ratios and rates, multiplying or dividing every part by the same number is often helpful. For percents, you might use a familiar fraction or split the percent into easier parts.

After calculating, ask what the numbers represent. A ratio has two related parts. A rate has units. A percent is a part of a whole. These meanings help you catch mistakes.

Try these practice tasks: find 7 groups of 18 by using a nearby multiple of 10; simplify the ratio 6 to 9 by dividing both parts by the same number; then find 10% and 5% of 80 and combine them to find 15% of 80.

Choose a property that makes the arithmetic easier.
Keep the meaning of each number and unit in view.
Use a familiar fact or an estimate to check the answer.

Worked example

Use the distributive property to multiply

Find the total number of counters in 7 equal groups of 18. Use a nearby friendly number.

Choose a nearby number
Eighteen is 2 less than 20. Finding 7 groups of 20 is easy, so use it as a starting point.
$18=20-2$
Find the two totals
Seven groups of 20 make 140. Seven groups of 2 make 14. Take away 14 because we used 20 instead of 18 in each group.
$7\times18=(7\times20)-(7\times2)=140-14$
State the result
There are 126 counters in all. This is reasonable because 18 is close to 20, and 7 groups of 20 would be 140.
$140-14=126$

Answer: 126 counters

Check: Since 18 is 2 less than 20, seven groups of 18 must be 14 less than 140. The result, 126, matches.

Worked example

Make an equivalent ratio

A mix uses 3 cups of oats for every 5 cups of fruit. How many cups of each ingredient are needed to make a batch that is 3 times as large?

Keep the two amounts together
The ratio is 3 cups of oats to 5 cups of fruit. To make a batch 3 times as large, multiply both parts by 3. Changing both parts by the same factor keeps the comparison the same.
$3:5\rightarrow(3\times3):(5\times3)$
Calculate each amount
There are 9 cups of oats and 15 cups of fruit. Both amounts are three times their starting amounts, so the ratio is preserved.
$9:15$

Answer: 9 cups of oats and 15 cups of fruit

Check: The new amounts are both three times the original amounts. Dividing 9 and 15 by 3 returns the original ratio, 3 to 5.

Worked example

Find a percent by splitting it

Find 15% of 80. Treat 15% as 10% plus 5%.

Find 10%
Ten percent means one tenth. Find one tenth of 80 by dividing 80 by 10. 10% of 80=80\div10=8
Find 5%
Five percent is half of 10%. Half of 8 is 4, so 5% of 80 is 4. 5% of 80=8\div2=4
Combine the parts
Fifteen percent is 10% plus 5%. Add the two parts to find 15% of 80.
$8+4=12$

Answer: 15% of 80 is 12.

Check: Since 10% of 80 is 8, 15% should be a little more than 8. The result, 12, is also less than 80, as expected.

Common mistakes and how to avoid them

Changing the order of numbers in subtraction or division because order works for addition and multiplication.

Correction: Reordering is safe for addition and multiplication, but not generally for subtraction or division. Keep the original order for those operations.

Multiplying only one part of a ratio to make a larger batch.

Correction: Multiply both parts of the ratio by the same number. This keeps the comparison unchanged.

Finding 15% by adding 10 and 5 to the amount.

Correction: The 10 and 5 describe percent parts, not amounts. Find 10% and 5% of the whole, then add those two results.

Lesson summary

Addition and multiplication can be reordered or regrouped without changing the result.
The distributive property lets multiplication act on each part of a sum or difference.
Multiply or divide both parts of a ratio or rate by the same number to keep an equivalent comparison.
Use familiar fractions or split a percent into easier parts.
Check the units, comparison, and size of the answer.

Check your understanding

Question 1

Which calculation correctly finds 4 groups of 29 using the distributive property?

4 × 30 − 4 × 1
4 × 30 + 4 × 1
4 × 29 − 1
30 − 4 × 1

Show answer and explanation

4 × 30 − 4 × 1

Since 29 is 30 minus 1, four groups of 29 equal four groups of 30 minus four groups of 1.

Question 2

A drink mix has a ratio of 2 scoops of powder to 7 cups of water. Which ratio describes twice as much mix?

4 to 14
4 to 7
2 to 14
7 to 4

Show answer and explanation

4 to 14

Both parts must be multiplied by 2. The equivalent ratio is 4 scoops to 14 cups.

Question 3

What is 25% of 36?

Show answer and explanation

Twenty-five percent is one quarter. One quarter of 36 is 36 divided by 4, which is 9.

Key terms

Operation: An action with numbers, such as adding, multiplying, or dividing.
Operation property: A rule that helps explain how numbers can change in a calculation without changing its value.
Product: The answer to a multiplication question.
Commutative property: The property that lets numbers switch order in addition or multiplication without changing the answer.
Associative property: The property that lets numbers be grouped differently in addition or multiplication without changing the answer.
Distributive property: The property that lets multiplication be applied to each part of a sum or difference.
Ratio: A comparison of two amounts.
Rate: A comparison of amounts with different units.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 6 Mathematics, expectation B2.1. It is a study resource, not an official curriculum publication.

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