How to Calculate Percent Increase and Decrease

In 9.2 you took a percent of a fixed number. Here the number changes and you want the new amount. A $30 %$ rise makes the new amount $130 %$ of the original, one multiply by $1.30$ . That number is the multiplier. A drop of $20 %$ leaves $80 %$ , a multiplier of $0.80$ .

Problem

A garden trail of

240

m was extended by

35 %

. Find

35 %

240

, then add it to

240

. How many meters of trail are there now?

Show a hint

The extra trail is $35 %$ of the ORIGINAL $240$ meters, not of some new total. Use part $=$ percent $\times$ whole, so the added length is $0.35 \times 240$ .
Once you know the added piece is $84$ meters, the new trail is just the old length plus that piece, $240 + 84$ . Adding the extra back on is what makes this an increase.

Show the full solution

35 %

240

0.35 \times 240 = 84

meters of new trail, and

240 + 84 = 324

. The trail is now

324

meters. Faster route, keeping the original and adding

35 %

more is

135 %

, so

240 \times 1.35 = 324

in one step.

Problem

A streaming channel had

480

subscribers and grew by

35 %

. The new count is

135 %

480

. How many subscribers does the channel have now?

Show a hint

The original count is $100 %$ of itself, and an increase piles more on top of that. A $35 %$ increase means the new amount is $100 % + 35 % = 135 %$ of the original.
Turn $135 %$ into the multiplier $1.35$ , then compute $480 \times 1.35$ in one step instead of adding the part separately.

Show the full solution

The new count is

100 % + 35 % = 135 %

480

, a multiplier of

1.35

480 \times 1.35 = 648

The channel now has

648

subscribers. Adding the part separately gives the same thing, since

0.35 \times 480 = 168

and

480 + 168 = 648

Problem

A coastal marsh had

350

wading birds at dawn. By midday,

12 %

had flown off. The survivors are

88 %

350

. How many birds remain at midday?

Show a hint

A $12 %$ decrease does not leave $12 %$ . It leaves the rest. Find the leftover percent by taking $100 % - 12 % = 88 %$ , and that fraction of the flock is what you want.
Turn $88 %$ into the multiplier $0.88$ and multiply the original count by it, $350 \times 0.88$ . If you prefer the long way, take $12 %$ of $350$ and subtract it from $350$ . Both roads land on the same number.

Show the full solution

12 %

drop leaves

100 % - 12 % = 88 %

, so the multiplier is

0.88

350 \times 0.88 = 308

At midday there are

308

birds. The long way agrees, since

0.12 \times 350 = 42

birds flew off and

350 - 42 = 308

Problem

A kayak rental costs

$ 90

a day. A

40 %

discount removes

40 %

and leaves

60 %

. What is the discounted price in dollars?

Show a hint

A $40 %$ discount removes $40 %$ of the price, so what is left is $100 % - 40 % = 60 %$ of it. That surviving $60 %$ is your multiplier.
The multiplier for a $40 %$ decrease is $1 - 0.40 = 0.60$ . Multiply the original price by it, $90 \times 0.60$ .

Show the full solution

40 %

discount leaves

100 % - 40 % = 60 %

of the price, so the multiplier is

0.60

90 \times 0.60 = 54

The discounted price is

54

dollars. The multiplier folds the subtracting into one step, since

0.40 \times 90 = 36

and

90 - 36 = 54

Both strips start from the same original, the full

100 %

bar. Adding

30 %

stacks a new block on the right, so the total is

100 % + 30 % = 130 %

, which is the multiplier

\times 1.30

. Cutting

20 %

peels a slice off the right, leaving

100 % - 20 % = 80 %

, the multiplier

\times 0.80

. An increase always lands above

1

and a decrease always lands below

1

, and either way you are scaling the original

100 %

Problem

A podcast went from

250

plays to

320

. The change is

70

plays. Divide by the original

250

, not the new

320

. What percent increase was this? Give the number only.

Show a hint

First find the change in plays, the new amount minus the old amount. Then remember a percent change is measured against the ORIGINAL, so the comparison you want is that change divided by $250$ , not divided by $320$ .
You have $70$ more plays out of the starting $250$ . Compute $\frac{70}{250}$ and turn that fraction into a percent by multiplying by $100$ .

Show the full solution

The change is

320 - 250 = 70

plays, and a percent change divides by the original.

\frac{70}{250} \times 100 = 28

The increase is

28

. Forward check, a

28 %

rise scales by

1.28

, and

250 \times 1.28 = 320

. Dividing by the new

320

answers a different question.

Problem

A library had

640

visitors on a sunny Saturday and

480

on a rainy one, a drop of

160

. Divide by the original

640

. What percent decrease was this? Give the number only.

Show a hint

First find the drop, then ask what fraction of the ORIGINAL it is. The drop is $640 - 480 = 160$ , and the original is $640$ , so the percent change is $\frac{160}{640}$ written as a percent.
Simplify $\frac{160}{640}$ . Both top and bottom share a factor of $160$ , so it becomes $\frac{1}{4}$ , and $\frac{1}{4} = 25 %$ . Resist the urge to divide by $480$ . The original count is the whole here.

Show the full solution

The drop is

640 - 480 = 160

visitors, and a percent change divides by the original

640

\frac{640 - 480}{640} = \frac{160}{640} = \frac{1}{4} = 0.25 = 25 %

So the answer is

25

. Dividing by

480

instead gives about

33 %

, which answers how much the count would have to grow to climb back to

640

Problem

A bamboo shoot grew from

14

cm to

35

cm, a change of

21

cm. Since

21 > 14

, the percent will sail past

100 %

. What percent increase was this? Give the number only.

Show a hint

The change is $35 - 14 = 21$ centimeters. For a percent change you always divide by the ORIGINAL amount, the $14$ centimeters it started at, not the $35$ it ended at.
Compute $\frac{21}{14}$ and turn it into a percent. Since $21$ is more than $14$ , this fraction is bigger than $1$ , so the percent lands above $100 %$ .

Show the full solution

The change is

35 - 14 = 21

centimeters, and a percent change divides by the original

14

\frac{35 - 14}{14} = \frac{21}{14} = \frac{3}{2} = 1.5

That is

150 %

, so the increase is

150

. A plain doubling to

28

would be a

100 %

increase, and this went past double, so the percent lands above

100

A podcast climbs from

250

plays to

320

. The gold slice is the change,

320 - 250 = 70

, and the percent change is that change measured against where we started,

\frac{70}{250} = 0.28 = 28 %

. Dividing by the new amount,

\frac{70}{320}

, is the trap, since the change is always compared to the original, never to where you landed.

Problem

After a

20 %

increase, a shelf holds 738 books. To recover last year's count, divide by

1.20

, not subtract

20 %

of 738. How many books were there before?

Show a hint

The tempting wrong path is $738 - 20 %$ of $738 = 590.4$ , which is not even a whole number of books. The $20 %$ growth was a slice of last year's count, not of this year's. So the right question is what number, when grown by $20 %$ , lands on $738$ .
A $20 %$ increase multiplies the original by $1.20$ , so $1.20 \times original = 738$ . To undo a multiplication, divide. Compute $738 \div 1.20$ .

Show the full solution

20 %

increase multiplies by

1.20

, so

1.20 \times original = 738

. Undo the multiplication by dividing,

738 \div 1.20 = 615

. Check it forward,

1.20 \times 615 = 738

. Last year the shelf held

615

books. Subtracting

20 %

738

gives

590.4

, too low, because the

20 %

was a slice of last year's smaller count.

Problem

A ferry pass now costs

$ 510

after a

15 %

fare cut. A

15 %

decrease multiplied the old price by

0.85

. What was the original season pass price in dollars?

Show a hint

A $15 %$ decrease keeps $100 % - 15 % = 85 %$ of the original, so it multiplies the old price by $0.85$ to give $510$ . The $510$ is the new amount, not the original.
Since $original \times 0.85 = 510$ , undo the multiplier by dividing. Compute $510 \div 0.85$ . Adding $15 %$ of $510$ back on is the wrong move, because that $15 %$ would be a slice of the smaller current price, not of the original.

Show the full solution

15 %

cut leaves

100 % - 15 % = 85 %

, so

original \times 0.85 = 510

. Divide by the multiplier to recover the old price.

original = \frac{510}{0.85} = 600

The season pass was

600

dollars. Adding

15 %

510

back gives

586.50

, short of the mark, since

15 %

of the smaller current price is smaller.

Problem

A bee colony of

200

grew by

30 %

one week, then shrank by

30 %

the next. Many guess it returns to

200

. It does not. How many workers are in the colony at the end?

Show a hint

Do the two weeks in order, one change at a time. A $30 %$ increase multiplies by $1.30$ , so the boosted colony is $200 \times 1.30$ . Whatever that comes to is the new starting size for week two.
For week two, a $30 %$ decrease multiplies by $0.70$ . The catch is that the $30 %$ is taken off the bigger boosted number, not off the original $200$ , so it removes more bees than the increase added.

Show the full solution

30 %

rise multiplies by

1.30

and a

30 %

fall by

0.70

. Week one gives

200 \times 1.30 = 260

workers, and week two gives

260 \times 0.70 = 182

. The colony ends at

182

workers. Chained,

1.30 \times 0.70 = 0.91

, under

1

, because the drop came off the larger

260

and removed more than the rise added.

Going up

20 %

then down

20 %

does not undo itself. The rise multiplies by

1.20

and the cut multiplies by

0.80

, and

1.20 \times 0.80 = 0.96

, so you land at

100 \times 0.96 = 96

, a hair below the dashed start line, not back at

100

. The reason is that the

20 %

cut came off the taller

120

, which is

24

, while the

20 %

rise had added only

20

. Same percent, different size to act on, so the trip down outweighs the trip up.

Every percent change in this lesson is a single multiplier. It is above $1$ for an increase and below $1$ for a decrease. To apply several changes in a row, multiply their multipliers together.

Problem

A recycling program collects

5, 000

kg in Q1 and expects each later quarter to grow by

10 %

. Each quarter multiplies by

1.10

. How many kilograms are expected in Q4 (three quarters later)?

Show a hint

A $10 %$ increase multiplies the amount by $1.10$ . One quarter later it is $5, 000 \times 1.10$ . Each new quarter multiplies the previous quarter's amount by $1.10$ again.
Three quarters later means three multipliers stacked, so compute $5, 000 \times 1.10 \times 1.10 \times 1.10$ . Take it one step at a time, $5, 000$ to $5, 500$ to $6, 050$ , then once more.

Show the full solution

Each quarter multiplies by

1 + \frac{10}{100} = 1.10

, and three quarters later means three of those multipliers.

5, 000 \times 1.1 0^{3} = 5, 000 \times 1.331 = 6, 655

Q4 is expected to bring

6655

kilograms. Step by step that is

5, 500

, then

6, 050

, then

6, 655

, with the jumps growing because each

10 %

comes off a bigger amount.

Problem

An art class had

400

students. Enrollment rose

50 %

for summer, then fell

40 %

from that summer peak. A

50 %

rise and a

40 %

fall do not cancel. What is the fall enrollment?

Show a hint

Do it in two steps, one change at a time. A $50 %$ increase multiplies by $1.50$ , so the summer peak is $400 \times 1.50$ . Whatever that comes to, the fall drop of $40 %$ is measured off that peak, not off the original $400$ .
A $40 %$ decrease multiplies by $1 - 0.40 = 0.60$ , so multiply the summer peak by $0.60$ . If you like, chain the multipliers first, $1.50 \times 0.60 = 0.90$ , which already tells you the fall figure lands at $90 %$ of the original, not back at $100 %$ .

Show the full solution

50 %

rise multiplies by

1.50

, so the summer peak is

400 \times 1.50 = 600

. A

40 %

fall multiplies by

0.60

, so the fall count is

600 \times 0.60 = 360

. Enrollment is

360

. The two changes do not cancel, since

1.50 \times 0.60 = 0.90

. The rise was figured on

400

but the fall on the larger

600

Practice these ideas

Practice

A bakery sold

80

loaves on Tuesday and

15 %

more on Wednesday. How many loaves did the bakery sell on Wednesday?

Show the solution

15 %

increase multiplies by

1 + \frac{15}{100} = 1.15

, so Wednesday's total is

80 \times 1.15 = 92.

The bakery sold

92

loaves. The long way matches, since

0.15 \times 80 = 12

and

80 + 12 = 92

Practice

A winter jacket priced at

$ 250

is marked

30 %

off. What is the sale price in dollars?

Show the solution

30 %

decrease keeps

70 %

of the price, so the multiplier is

1 - 0.30 = 0.70

. Apply it to the original price.

250 \times 0.70 = 175

The sale price is

175

dollars.

Practice

A stadium that held

600

fans expanded its seating by

45 %

. How many seats are there now?

Show the solution

45 %

increase makes the new seating

100 % + 45 % = 145 %

of the old, a multiplier of

1.45

600 \times 1.45 = 870

The stadium now holds

870

seats.

Practice

A bamboo shoot rose from

40

cm to

54

cm, a change of

14

cm. Divide by the original. What percent increase is this? Give the number only.

Show the solution

The change is

54 - 40 = 14

centimeters, and a percent change divides by the original

40

. That gives

\frac{14}{40} = 0.35

, or

35 %

, so the increase is

35

. Forward check,

40 \times 1.35 = 54

Practice

A reservoir dropped from

900

720

megalitres, a fall of

180

. What percent decrease is this? Give the number only.

Show the solution

The drop is

900 - 720 = 180

megaliters, and a percent change divides by the original

900

\frac{180}{900} = \frac{1}{5} = 0.20

That is a

20 %

decrease, so the answer is

20

. The multiplier agrees, since

\frac{720}{900} = 0.80

means the reservoir kept

80 %

of its water.

Practice

A startup grew from

25

customers to

75

, a jump of

50

. Since

50 > 25

, the percent lands above

100 %

. What percent increase is this? Give the number only.

Show the solution

The change is

75 - 25 = 50

customers, and a percent increase divides by the original

25

. That gives

\frac{50}{25} = 2

, which as a percent is

200 %

, so the answer is

200

. The new count is

3

times the old, and a multiplier of

3

1 + 2

, the original

100 %

plus a

200 %

increase on top.

Practice

After a

25 %

increase, a road bike sells for

$ 500

. The

25 %

was on the OLD price. What was the original price in dollars?

Show the solution

25 %

increase multiplies by

1 + \frac{25}{100} = 1.25

, so

1.25 \times original = 500

. Divide to undo it.

original = \frac{500}{1.25} = 400

The original price was

400

dollars. Subtracting

25 %

500

would give

375

, which is wrong because the

25 %

was taken on the smaller original.

Practice

After a

20 %

decrease, a sofa sells for

$ 480

. The multiplier was

0.80

. What was the original price in dollars?

Show the solution

20 %

decrease keeps

100 % - 20 % = 80 %

, so

original \times 0.80 = 480

. Divide by the multiplier instead of adding

20 %

back.

original = \frac{480}{0.80} = 600

The original price was

600

dollars. Forward check,

600 \times 0.80 = 480

Practice

A vlogger started with

200

subscribers, gained

40 %

, then lost

25 %

of that new total. How many subscribers does the channel have after both months?

Show the solution

40 %

gain multiplies by

1.40

and a

25 %

loss by

0.75

. Month one gives

200 \times 1.40 = 280

subscribers, and month two gives

280 \times 0.75 = 210

. The final count is

210

. It does not land back at

200

, because the

25 %

drop came off the larger

280

Practice

A koi pond holds

2, 000

liters and gains

20 %

each day. How many liters does the pond hold after three full days of growth?

Show the solution

20 %

increase multiplies by

1.20

, and three days means three of those multipliers.

2, 000 \times 1.2 0^{3} = 3, 456

The pond holds

3456

liters. Day by day that is

2, 400

, then

2, 880

, then

3, 456

Practice

A factory used

1, 500

kWh last month and cut usage by

8 %

this month. How many kWh did the factory use this month?

Show the solution

8 %

decrease leaves

100 % - 8 % = 92 %

of the original, so the multiplier is

0.92

. Multiply the original amount by it.

1, 500 \times 0.92 = 1, 380

So this month the factory used

1380

kilowatt-hours.

Practice

A scout troop grew from

120

156

members, a change of

36

. Divide by the original. What percent increase is this? Give the number only.

Show the solution

The change is

156 - 120 = 36

members, and a percent change divides by the original

120

\frac{36}{120} = 0.30

That is a

30 %

increase, so the answer is

30

. Forward check,

120 \times 1.30 = 156

Practice

A camping tent is marked down

40 %

and now sells for

$ 90

. The sale price is

60 %

of the original. What was the original price in dollars?

Show the solution

40 %

markdown leaves

1 - 0.40 = 0.60

of the price, so

0.60 \times original = 90

. Divide to undo the multiplier,

90 \div 0.60 = 150

. The original price was

150

dollars. Check it forward,

150 - 0.40 \times 150 = 150 - 60 = 90

Practice

A club had

$ 500

. A fundraiser raised it by

20 %

, then supply costs cut it by

20 %

. How many dollars are left in the fund?

Show the solution

The fundraiser multiplies by

1 + 0.20 = 1.20

and the supply costs by

1 - 0.20 = 0.80

500 \times 1.20 \times 0.80 = 600 \times 0.80 = 480

The fund has

480

dollars left. It lands below

500

because the

20 %

drop was taken from the larger

600

, so it removed more than was added.