Increase and decrease whole numbers mentally by benchmark percents

Ontario Grade 7 Mathematics

Number

Use familiar percent amounts to find a new whole-number value

A school club has 200 buttons. It receives 25% more buttons. How can you find the new total without writing a long calculation? A benchmark percent is a familiar percent, such as 10%, 25%, or 50%, that is easy to work with. First find that part of the number. Then add it to the starting number for an increase, or subtract it for a decrease. In this lesson, you will practise those steps with whole numbers.

What you will learn

Explain what a percent means.
Use benchmark percents to find part of a whole number mentally.
Increase or decrease a whole number by a benchmark percent.
Check that a new value makes sense.

Review: What does percent mean?

A whole number is a number with no fractional part, such as 40 or 250. A percent tells how many parts there are out of 100 equal parts. For example, 10% means 10 out of every 100.

A benchmark percent is a percent that is especially handy to find mentally. For example, 50% is half, 25% is one quarter, and 10% is one tenth. To find 10% of a whole number, divide it by 10. To find 50%, find half. To find 25%, find one quarter by dividing by 4.

You can use one benchmark to find another. For example, 20% is two groups of 10%. Also, 75% is 50% plus 25%. These connections let you work with more than one familiar amount.

10\%=\frac{1}{10},\quad 25\%=\frac{1}{4},\quad 50\%=\frac{1}{2}

Percent means parts out of 100.
10% is one tenth; 25% is one quarter; 50% is one half.
Find the benchmark part before changing the starting number.

Increase: add the benchmark part

To increase a number by a percent, find that percent of the starting number. Then add the part to the starting number. An increase means the new amount is greater than the starting amount.

For example, 10% of 300 is 30. Increasing 300 by 10% means adding 30 to 300, so the new amount is 330. The 30 is the increase, not the new total.

For a 25% increase, find one quarter of the starting number and add it. For a 50% increase, find half and add it. If a benchmark part is easy to find, the rest can often be done mentally.

\text{new amount}=\text{starting amount}+\text{percent amount}

Find the percent amount first.
Add that amount to the starting number.
The result of an increase should be larger than the starting number.

Decrease: subtract the benchmark part

To decrease a number by a percent, find that percent of the starting number. Then subtract the part from the starting number. A decrease means the new amount is less than the starting amount.

For example, 10% of 500 is 50. Decreasing 500 by 10% means taking away 50, leaving 450. The amount taken away is the decrease; 450 is the new amount.

Some decreases are easier to find by thinking about what remains. If you decrease a number by 25%, then 75% remains. If you decrease it by 50%, then half remains. Both ways describe the same result.

\text{new amount}=\text{starting amount}-\text{percent amount}

Find the percent amount, then subtract it.
Or find the percent that remains and calculate that part.
The result of a decrease should be smaller than the starting number.

A mental-math plan

Read the question and identify the starting number and the percent. Decide whether the amount increases or decreases. Then choose a benchmark connection that is easy to find.

Keep the percent amount separate from the new amount. For a 25% increase, for example, the quarter of the starting number is the increase. Add it to get the new amount.

Check the direction. An increase must give a larger number, and a decrease must give a smaller number. You can also estimate: a 10% change should be a much smaller part of the starting number than a 50% change.

Identify the starting number, percent, and direction.
Find the benchmark part mentally.
Add for an increase; subtract for a decrease.
Check whether the answer is larger or smaller as expected.

Useful benchmark percents

Percent	Easy way to find it
10%	Divide the number by 10
25%	Divide the number by 4
50%	Find half
75%	Find 50% and 25%, or find what remains after a 25% decrease

Worked example

Increase by 25%

A library shelf has 240 books. It receives 25% more books. How many books are on the shelf now?

Books before and after the increase

The bar values are the stated book counts. The new total includes the 25% increase.

Find the added amount
Since 25% is one quarter, find one quarter of 240. Dividing 240 into four equal groups gives 60 books. So 60 books are added.
$240\div4=60$
Add to the starting amount
An increase means adding the added amount to the original 240 books.
$240+60=300$
Check
The new total is greater than 240, as an increase should be.
$300>240$

Answer: There are 300 books on the shelf now.

Check: One quarter of 240 is 60, and 240 plus 60 is 300.

Worked example

Decrease by 10%

A community garden has 680 seedlings. After sharing some, it has 10% fewer seedlings. How many seedlings remain?

Seedlings before and after the decrease

The bar values are the stated seedling counts. The remaining amount is 90% of the starting amount.

Find the amount shared
Ten percent is one tenth. Divide 680 into 10 equal groups to find one tenth. This gives the number of seedlings shared.
$680\div10=68$
Subtract the decrease
The garden has 68 fewer seedlings, so subtract 68 from the starting number.
$680-68=612$
Check
The remaining number is less than 680, as a decrease should be.
$612<680$

Answer: 612 seedlings remain.

Check: Ten percent of 680 is 68, and subtracting 68 leaves 612.

Worked example

Decrease by 75%

A collection has 360 cards. It decreases by 75%. How many cards are left?

Cards before and after the decrease

The bar values are the stated card counts. The remaining amount is one quarter of the starting amount.

Find what remains
If 75% is taken away, the part left is 25%, because 100% is the whole collection. One quarter is a quick way to find 25%.
$100\%-75\%=25\%$
Find one quarter
Divide 360 into four equal groups. One group is the number of cards left.
$360\div4=90$
Check
Only one quarter remains, so 90 is less than the starting amount of 360.
$90<360$

Answer: 90 cards are left.

Check: A 75% decrease leaves 25%. One quarter of 360 is 90.

Common mistakes and how to avoid them

Finding the percent amount and calling it the new total.

Correction: The percent amount is only the part added or removed. Add it to or subtract it from the starting number.

Adding the percent amount for a decrease.

Correction: A decrease removes part of the starting amount, so subtract the percent amount.

Thinking that a 75% decrease leaves 75%.

Correction: A 75% decrease removes 75%, so 25% remains.

Giving an answer larger than the starting number for a decrease.

Correction: Check the direction. An increase gives a larger amount; a decrease gives a smaller amount.

Lesson summary

A percent tells how many parts there are out of 100.
Use benchmark percents such as 10%, 25%, and 50% to find a part mentally.
For an increase, add the percent amount to the starting number.
For a decrease, subtract the percent amount, or find the part that remains.
Check that the final amount is larger for an increase and smaller for a decrease.

Check your understanding

Question 1

A box has 160 markers. How many markers are there after a 25% increase?

40
120
200
185

Show answer and explanation

200

One quarter of 160 is 40. Add 40 to 160 to get 200.

Question 2

A school has 450 notebooks. What number remains after a 10% decrease?

45
405
495
440

Show answer and explanation

405

One tenth of 450 is 45. Subtract 45 from 450 to get 405.

Question 3

A set of 280 tiles decreases by 50%. How many tiles remain?

140
230
420
50

Show answer and explanation

140

A 50% decrease leaves half. Half of 280 is 140.

Key terms

Whole number: A number with no fractional part, such as 12 or 300.
Percent: A way to describe parts out of 100.
Benchmark percent: A familiar percent that is easy to find, such as 10%, 25%, or 50%.
Increase: A change that makes an amount larger.
Decrease: A change that makes an amount smaller.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 7 Mathematics, expectation B2.3. 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.

Official curriculum reference

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B2.3 · Increase and decrease whole numbers mentally by benchmark percents

Ontario Grade 7 Mathematics

Use familiar percent amounts to find a new whole-number value

What you will learn

Review: What does percent mean?

Increase: add the benchmark part

Decrease: subtract the benchmark part

A mental-math plan

Useful benchmark percents

Increase by 25%

Decrease by 10%

Decrease by 75%

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Grade 7 Math

About this lesson and its review