Math 7 · 7.RP.A.3

Percent increase and decrease

A rent goes up by 120 dollars a month. Is that a big rise or a small one? How do you measure a change so that it means the same thing whatever size the amount started at?

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Percent increase and decrease

Percent error The multiplier for a percent change Two percent changes one after the other

When you reach for it

Something has changed, and you know what it was before and what it is now. A flat’s rent went from 800 to 920 dollars a month. A climbing gym had 240 members and now has 317. A city park grew from 12 hectares to 15.

Subtracting tells you how much it changed, and sometimes that is all you need. But the amount alone cannot say whether the change was large. A rise of 120 dollars is a lot on a rent of 400 dollars and very little on a rent of 4{,}000. To judge a change, and to compare changes to amounts of different sizes, you measure it against where it started. That measurement, written as a percent, is the percent increase (or percent decrease, if the amount went down).

It is the question from finding what percent one number is of another, asked about a change: the change is the part, and the starting amount is the whole.

This lesson is taught over two days. Part 1 is percent increase. Part 2 is percent decrease, what happens when a change is undone, and the difference between percent and percentage points.

The procedure

Part 1: percent increase

Two numbers go in: the original amount, what the quantity was before, and the new amount, what it is after. Take the rent going from 800 to 920 dollars.

Find the amount of increase. The increase is how far the new amount is above the original:

920 - 800 = 120

So the rent went up by 120 dollars.

Write the increase as a percent of the original. Now ask what percent 120 is of 800. The increase is the part. The original is the whole, because the question being answered is “how much bigger is it than it was?”, and “it was” means the original amount:

\frac{120}{800} = \frac{15}{100} = 15\%

The rent went up by 15\%. Put together, the two steps are one formula:

\begin{aligned} \text{increase} &= \text{new} - \text{original} \\ \text{percent increase} &= \frac{\text{increase}}{\text{original}} \times 100\% \end{aligned}

Multiplying by 100\% changes nothing about the size, since 100\% = 1; it only writes the fraction as a number of hundredths.

The picture is a bar for the old rent with the rise stuck on the end. The old rent is the whole, so it is 100\%; the extra piece reaches 15\% further.

Figure 1The old rent of 800 dollars is the whole, 100\%. The rise of 120 dollars is the extra piece, and it reaches 15\% past the end of the whole.

Why the original, and not the new amount. A fraction means nothing until its whole is named, the same point that naming the whole makes for fractions. Here the whole is fixed by the question. Dividing 120 by the new rent, 920, gives about 13\%, which answers a different question: what part of today’s rent is the recent rise. Nobody asks that when they ask how much the rent went up.

A size check. Comparing the increase with the original tells you roughly where the answer sits before you divide. An increase smaller than the original gives a percent under 100\%. An increase equal to the original means the amount doubled, a 100\% increase. An increase bigger than the original gives more than 100\%, which is the territory of percents above one hundred.

The percent increase and the new amount as a percent. The new rent is the old rent plus 15\% of it, so it is 100\% + 15\% = 115\% of the old rent; in the figure, the whole bar reaches 115\%. These two numbers answer different questions and differ by exactly 100\%: the increase is 15\% of the original, the new amount is 115\% of it. Using that 115\% as a single multiplication to get from the old amount to the new one, or back again, is the subject of the multiplier for a percent change.

Part 2: percent decrease, and percentage points

Percent decrease. When the amount goes down, the change is how far the new amount is below the original, and it is still measured against the original:

\begin{aligned} \text{decrease} &= \text{original} - \text{new} \\ \text{percent decrease} &= \frac{\text{decrease}}{\text{original}} \times 100\% \end{aligned}

The subtraction is written the other way round so that the decrease comes out as a positive amount; the word “decrease” already says which way the change went. Some books write a single formula, \displaystyle \frac{\text{new} - \text{original}}{\text{original}} \times 100\%, and let the sign carry the direction: positive for an increase, negative for a decrease. The two say the same thing.

A decrease can never pass 100\%. A decrease of 100\% takes away the whole original, leaving 0, and a count, a price or an area cannot go below 0. An increase has no such ceiling; a newsletter can grow by 500\%.

The whole changes when the direction changes. Suppose a monthly bus pass goes from 60 to 75 dollars, and the next year back from 75 to 60. The change is 15 dollars both times. Going up, the original is 60, so the increase is \displaystyle \frac{15}{60} = 25\%. Coming down, the original is 75, so the decrease is \displaystyle \frac{15}{75} = 20\%. The same 15 dollars is a bigger share of the smaller amount.

Figure 2The same 15 dollars, measured against two different wholes. Going up, the whole is the four pieces of 60; coming down, it is the five pieces of 75.

So a 25\% increase is undone by a 20\% decrease, not by a 25\% one. Percent changes that come one after another do not add or cancel the way plain amounts do, which is the subject of two percent changes one after the other.

Percentage points. Sometimes the quantity that changed is itself a percent. Say parks cover 8\% of a city’s land, and new parks raise that to 10\%. There are two honest ways to describe the change, and they give different numbers.

Subtract the two percents. 10\% - 8\% = 2\% of the city’s land. To keep this apart from a percent change, it is called a change of 2 percentage points.
Find the percent increase. The change is 2 and the original is 8, so the percent increase is \displaystyle \frac{2}{8} = 25\%. The parkland itself grew by a quarter of what it was.

Both are correct; they measure different things. “Rose by 2 percentage points” compares the two percents by subtraction. “Rose by 25\%” compares them by measuring the change against the original. Writing “rose by 2\%” is unclear, because a reader cannot tell which was meant.

Worked examples

Examples 1 to 5 go with Part 1. Examples 6 to 8 go with Part 2.

Example 1

A rent going up

The monthly rent on a flat rises from 1{,}100 dollars to 1{,}232 dollars. Find the percent increase.

Solution

The amount of increase is the new rent minus the old rent:

1{,}232 - 1{,}100 = 132 \text{ dollars}

The rent went up from 1{,}100 dollars, so 1{,}100 is the original and the whole. The question is what percent 132 is of 1{,}100. Dividing top and bottom by 11 makes the bottom 100:

\frac{132}{1{,}100} = \frac{12}{100} = 12\%

The rent rose by 12\%.

Check by building 12\% of 1{,}100 from easy pieces: 10\% of 1{,}100 is 110, and 1\% is 11, so 2\% is 22. Together 110 + 22 = 132, the increase. Since the new rent is the old one plus 12\% of it, it is 112\% of the old rent, and 1.12 \times 1{,}100 = 1{,}232.

Example 2

A fare in dollars and cents

A single train ticket into town goes from 3.20 dollars to 3.60 dollars. Find the percent increase.

Solution

The increase is

3.60 - 3.20 = 0.40 \text{ dollars}

and the original fare is 3.20 dollars, so the percent increase is \displaystyle \frac{0.40}{3.20}. Decimals in a fraction are easier to see in whole cents: 0.40 dollars is 40 cents and 3.20 dollars is 320 cents, and measuring both in the same unit does not change their ratio.

\frac{40}{320} = \frac{1}{8} = 0.125 = 12.5\%

The fare went up by 12.5\%. Dividing 0.40 by 3.20 on a calculator gives the same 0.125.

Example 3

A percent that does not come out exactly

A climbing gym had 240 members at the start of the year and 317 at the end. Find the percent increase, to the nearest tenth of a percent.

Solution

The increase is 317 - 240 = 77 members, and the original is 240:

\frac{77}{240} = 0.320833\ldots

The 3 repeats for ever, so the division never ends. Written as a percent this is 32.0833\ldots\%. To round to the nearest tenth of a percent, look at the hundredths digit, 8, which is 5 or more, so the tenths digit goes up:

\frac{77}{240} \approx 32.1\%

The membership grew by about 32.1\%.

Check by estimating. A third of 240 is 80, and 77 is a little less than 80, so the increase should be a little less than a third, 33\tfrac{1}{3}\%. It is.

Example 4

More than doubling

A newsletter had 350 readers when it started and 980 readers a year later. Find the percent increase. Then find the new number of readers as a percent of the original number.

Solution

The increase is 980 - 350 = 630 readers. Measured against the original 350:

\frac{630}{350} = \frac{9}{5} = 1.8 = 180\%

The increase, 630, is bigger than the original, 350, so the percent was always going to be over 100\%. The readership grew by 180\%.

The second question asks for the whole new amount, not the change, measured against the original:

\frac{980}{350} = \frac{14}{5} = 2.8 = 280\%

There are 280\% as many readers as at the start. The two answers differ by exactly 100\%, the original readers themselves: 280\% = 100\% + 180\%. “Grew by 180\%” and “is 280\% of what it was” describe the same newsletter.

Example 5

A bigger amount but a smaller percent

Over ten years, Alder Park grew from 12 hectares to 15 hectares, and Riverside Park grew from 40 hectares to 46 hectares. Which park gained more land? Which grew by the larger percent?

Solution

Find each increase and measure it against that park’s own original area.

Alder Park gained 15 - 12 = 3 hectares:

\frac{3}{12} = \frac{1}{4} = 25\%

Riverside Park gained 46 - 40 = 6 hectares:

\frac{6}{40} = \frac{15}{100} = 15\%

Riverside gained more land, twice as much. Alder grew by the larger percent, because its 3 hectares were measured against a much smaller starting park. The two questions have different answers, and which one matters depends on what is being asked: a walker who wants more room cares about hectares, and someone asking which park changed more for its size cares about the percent.

Example 6

A streamer losing followers

A streamer had 2{,}500 followers at the start of the month and 2{,}150 at the end. Find the percent decrease.

Solution

The number went down, so the decrease is the original minus the new:

2{,}500 - 2{,}150 = 350 \text{ followers}

It is measured against the original, 2{,}500:

\frac{350}{2{,}500} = \frac{14}{100} = 14\%

The streamer lost 14\% of their followers. Check: 14\% of 2{,}500 is 0.14 \times 2{,}500 = 350, the decrease.

Example 7

Down and then back up

A gym had 160 members in the autumn. Over the winter the number fell to 120, and by the spring it was back at 160. Find the percent decrease and the percent increase. Are they the same?

Solution

Both changes are 40 members. What differs is the original each one is measured against.

The fall starts at 160:

\frac{40}{160} = \frac{1}{4} = 25\%

The rise starts at 120:

\frac{40}{120} = \frac{1}{3} = 33\tfrac{1}{3}\%

The membership fell by 25\% and then had to rise by 33\tfrac{1}{3}\% to get back where it began. This is the bus pass of Part 2 run the other way: the 40 members are one of four equal pieces of 160, but one of only three equal pieces of 120. Measured against the smaller amount, the climb back is the bigger percent.

Example 8

Percentage points on the way down

Rent takes up 30\% of a family’s monthly income. After a pay rise, with the rent unchanged, it takes up 24\%. One report says the share fell by 6 percentage points; another says it fell by 20\%. Which is right?

Solution

Both are, because they measure the change in two different ways.

Subtracting the two percents gives the change in percentage points:

30\% - 24\% = 6 \text{ percentage points}

Measuring that change against the original share gives the percent decrease:

\frac{30 - 24}{30} = \frac{6}{30} = 20\%

The share of income spent on rent fell by 6 percentage points, which is a 20\% decrease in the share. If the share later went back up from 24\% to 30\%, it would rise by the same 6 percentage points but by \displaystyle \frac{6}{24} = 25\%, since that change starts from 24.

Boundaries

The original must not be 0. The percent change divides by the original amount, and division by zero is not defined. A newsletter that goes from 0 readers to 40 has grown by 40 readers, but by no percent at all.
The amounts are sizes, so they are positive. Rent, fares, members and areas cannot be negative, and the formula is built for amounts like these. Put a temperature through it, from -4 to 2 degrees, and it gives \displaystyle \frac{2 - (-4)}{-4} = -150\%: a rise reported with a minus sign. For quantities that can be negative, give the change in units, 6 degrees, instead.
A decrease stops at 100\%; an increase does not stop. Losing the whole original leaves nothing, which is a 100\% decrease, and an amount of things cannot drop below that. An increase can be any size.
The original is the whole, so the same change gives two percents. A rise followed by an equal fall in amount is a larger percent going up than coming down. Percent changes in a row neither add nor cancel; see Two percent changes one after the other.
When the quantities are percents, name the comparison. A change from 8\% to 10\% is 2 percentage points or a 25\% increase, and “2\%” alone is ambiguous.
Change over time, not distance from a true value. Percent increase compares a quantity with its own earlier value. Comparing a measurement with the true value it should have had uses the same division but is a different question, Percent error.

Where this goes next

The next step is to stop thinking of a change as “add the increase” and see it as one multiplication: a 15\% increase turns every amount into 115\% of itself. That single number is the multiplier for a percent change, and with it you can find the new amount from the original, or the original from the new amount, in one step. Chaining multipliers explains the bus pass in Part 2, the gym in Example 7 and every other run of changes one after the other.

The same measurement, “change against the original”, sits under markup, where a shop’s added amount is measured against what an item cost it, and under Percent error, where the gap between a measurement and the true value is measured against the true value. In each case the work is the work of this lesson; what changes is which amount counts as the original.

Exercises

Part 1: finding the percent increase

Find the percent increase from the first number to the second.

From 40 to 50.
From 25 to 32.
From 75 to 120.
From 8 to 20.
From 1.20 to 1.35.

Part 1: in context

The rent on a flat goes from 1{,}250 dollars a month to 1{,}325 dollars a month. Find the percent increase.
A bus fare rises from 2.25 dollars to 2.43 dollars. Find the percent increase.
A climbing gym grows from 180 members to 243 members. Find the percent increase.
A newsletter grows from 1{,}600 readers to 3{,}600 readers. (a) Find the percent increase. (b) Find the new number of readers as a percent of the original number.
A city park is enlarged from 36 hectares to 41 hectares. Find the percent increase, to the nearest tenth of a percent.
Over the same month, Ingrid’s channel grew from 400 followers to 520, and Leopold’s channel grew from 1{,}500 followers to 1{,}800. Who gained more followers? Whose channel grew by the larger percent?
Two flats each have their rent raised by 60 dollars a month. One rent was 600 dollars and the other was 1{,}200 dollars. Find each percent increase.

Part 2: finding the percent decrease

Find the percent decrease from the first number to the second.

From 50 to 41.
From 240 to 180.
From 12.50 to 10.

Part 2: in context

A gym’s membership falls from 320 to 296. Find the percent decrease.
A newsletter’s readers fall from 2{,}400 to 2{,}016. Find the percent decrease.
Say whether each change is an increase or a decrease, and find the percent. (a) A rent goes from 950 dollars to 1{,}045 dollars a month. (b) A channel goes from 6{,}000 followers to 5{,}100 followers.
A club’s membership rises from 250 to 350 and then falls back to 250. (a) Find the percent increase. (b) Find the percent decrease, to the nearest tenth of a percent.

Part 2: percentage points

Parks cover 5\% of a town’s land, and then 6\%. (a) By how many percentage points did the share rise? (b) By what percent did the share increase?
The share of a gym’s members who hold a yearly pass rises from 40\% to 50\%. (a) By how many percentage points did it rise? (b) By what percent did it increase?

Reasoning

Explain why a percent decrease in the number of members of a club can never be more than 100\%, while a percent increase can be any size.
A train fare goes from 2 dollars to 3 dollars and later back to 2 dollars. Cosima says the fare went up by 50\% and then down by 50\%. Find the two percents, and explain which part of her statement is right.
A city doubles the area of one of its parks. What is the percent increase? Explain why the answer is not 200\%, and what 200\% does describe.
A new channel starts with 0 followers and has 90 after a week. Explain why the percent increase cannot be found, and say how the growth could be described instead.

Answers

25\%. The increase is 50 - 40 = 10, and \displaystyle \frac{10}{40} = \frac{1}{4} = 25\%.
28\%. The increase is 32 - 25 = 7, and \displaystyle \frac{7}{25} = \frac{28}{100} = 28\%.
60\%. The increase is 120 - 75 = 45, and \displaystyle \frac{45}{75} = \frac{3}{5} = 60\%.
150\%. The increase is 20 - 8 = 12, and \displaystyle \frac{12}{8} = 1.5 = 150\%. The increase is bigger than the original, so the percent is over 100\%.
12.5\%. The increase is 1.35 - 1.20 = 0.15, and \displaystyle \frac{0.15}{1.20} = \frac{15}{120} = \frac{1}{8} = 12.5\%.
6\%. The increase is 1{,}325 - 1{,}250 = 75 dollars, and \displaystyle \frac{75}{1{,}250} = \frac{6}{100} = 6\%.
8\%. The increase is 2.43 - 2.25 = 0.18 dollars, which is 18 cents of 225 cents, and \displaystyle \frac{18}{225} = \frac{8}{100} = 8\%.
35\%. The increase is 243 - 180 = 63 members, and \displaystyle \frac{63}{180} = \frac{7}{20} = 35\%.
(a) 125\%. The increase is 3{,}600 - 1{,}600 = 2{,}000 readers, and \displaystyle \frac{2{,}000}{1{,}600} = \frac{5}{4} = 125\%. (b) 225\%, since \displaystyle \frac{3{,}600}{1{,}600} = \frac{9}{4} = 225\%, which is 100\% + 125\%.
About 13.9\%. The increase is 41 - 36 = 5 hectares, and \displaystyle \frac{5}{36} = 0.13888\ldots, which is 13.888\ldots\% \approx 13.9\%.
Leopold gained more followers, 300 against Ingrid’s 120. Ingrid’s channel grew by the larger percent: \displaystyle \frac{120}{400} = 30\%, against \displaystyle \frac{300}{1{,}500} = 20\% for Leopold.
10\% and 5\%. The same 60 dollars is \displaystyle \frac{60}{600} = 10\% of the smaller rent and \displaystyle \frac{60}{1{,}200} = 5\% of the larger one.
18\%. The decrease is 50 - 41 = 9, and \displaystyle \frac{9}{50} = \frac{18}{100} = 18\%.
25\%. The decrease is 240 - 180 = 60, and \displaystyle \frac{60}{240} = \frac{1}{4} = 25\%.
20\%. The decrease is 12.50 - 10 = 2.50, and \displaystyle \frac{2.50}{12.50} = \frac{1}{5} = 20\%.
7.5\%. The decrease is 320 - 296 = 24 members, and \displaystyle \frac{24}{320} = 0.075 = 7.5\%.
16\%. The decrease is 2{,}400 - 2{,}016 = 384 readers, and \displaystyle \frac{384}{2{,}400} = 0.16 = 16\%.
(a) An increase of 10\%: the rent went up by 1{,}045 - 950 = 95 dollars, and \displaystyle \frac{95}{950} = \frac{1}{10} = 10\%. (b) A decrease of 15\%: the channel lost 6{,}000 - 5{,}100 = 900 followers, and \displaystyle \frac{900}{6{,}000} = \frac{15}{100} = 15\%.
(a) 40\%, since the change is 100 members and \displaystyle \frac{100}{250} = \frac{2}{5} = 40\%. (b) About 28.6\%, since the same 100 is now measured against 350: \displaystyle \frac{100}{350} = \frac{2}{7} = 0.285714\ldots \approx 28.6\%.
(a) 1 percentage point, since 6\% - 5\% = 1\% of the land. (b) 20\%, since the change of 1 measured against the original 5 is \displaystyle \frac{1}{5} = 20\%.
(a) 10 percentage points, since 50\% - 40\% = 10\%. (b) 25\%, since \displaystyle \frac{10}{40} = \frac{1}{4} = 25\%.
A percent decrease is the decrease divided by the original number of members. The club cannot lose more members than it had, so the decrease is at most the original, and the fraction is at most \displaystyle \frac{\text{original}}{\text{original}} = 1 = 100\%, which is losing every member. An increase has no such limit: the club can gain any number of members, so the increase can be any number of times the original, and the percent can be any size.
The fare went up by \displaystyle \frac{1}{2} = 50\% and down by \displaystyle \frac{1}{3} = 33\tfrac{1}{3}\%. Cosima is right about the rise, since the 1 dollar change is measured against 2. She is wrong about the fall: going down, the original is 3, and 1 dollar is a third of 3, not a half. Both changes are the same amount, 1 dollar, but measured against different wholes.
100\%. If the old area is A, the new area is 2A, so the increase is 2A - A = A, and \displaystyle \frac{A}{A} = 1 = 100\%. The figure 200\% describes the new area as a percent of the old one, \displaystyle \frac{2A}{A} = 200\%, not the change.
The percent increase is the increase divided by the original, here \displaystyle \frac{90}{0}, and dividing by zero is not defined. There is no starting amount to measure the change against. The growth can still be given as an amount: the channel gained 90 followers in a week.