How to Solve Percent Word Problems

This lesson applies percent-as-fraction, percent of a number, and percent change to word problems. The routine is the same every time. Read the problem and find what whole each percent is taken of, translate the words into a multiplication or a division, solve, then check that the size and direction of your answer make sense.

Problem

A hoodie costs

$ 48

and the local sales tax is

7.5 %

. How many dollars is the sales tax alone (not the total)?

Show a hint

The tax is just $7.5 %$ OF the price, exactly the kind of "percent of a number" you did in 9.2. The whole is the 48 dollar price and the rate is $7.5 %$ , so the tax is $7.5 %$ of 48. Be careful, the question wants the tax by itself, not the 48 plus the tax.
Turn the percent into a decimal before multiplying. Since $7.5 % = 0.075$ , the tax equals $0.075 \times 48$ .

Show the full solution

Write

7.5 %

as the decimal

0.075

and multiply by the price.

0.075 \times 48 = 3.6

The tax alone is

3.6

dollars. The question asks for the tax by itself, not the total, so there is nothing to add on at the end.

Problem

Four friends have an

$ 80

dinner bill and agree on an

18 %

tip. What is the total amount they pay, food plus tip, in dollars?

Show a hint

You are paying for the whole meal plus a little extra, so think in one multiplier. You owe all $100 %$ of the food and another $18 %$ on top, which is $118 %$ of the food. Turn $118 %$ into a decimal multiplier and you only have to do one multiplication.
Multiply the food, 80 dollars, by $1.18$ . If you would rather build it in two pieces, find the tip with $0.18 \times 80$ first, then add that to 80.

Show the full solution

Paying the whole bill plus an

18 %

tip is

118 %

of the food, a multiplier of

1.18

80 \times 1.18 = 94.40

They pay

94.40

dollars. Building the tip separately gives the same total, since

0.18 \times 80 = 14.40

and

80 + 14.40 = 94.40

Problem

Mira bought a kettle for

$ 180

and marks it up

35 %

of cost. What is the selling price of the kettle in dollars?

Show a hint

A markup works exactly like a tax. The selling price is the whole cost plus another $35 %$ of that same cost, so it is $100 % + 35 % = 135 %$ of the cost. Turn that percent into a single multiplier.
$135 %$ as a decimal is $1.35$ , so the selling price is $180 \times 1.35$ . Multiply that out.

Show the full solution

35 %

markup keeps the full cost and adds

35 %

on top, so the selling price is

135 %

of cost, a multiplier of

1.35

180 \times 1.35 = 243

The selling price is

243

dollars. The markup alone is

0.35 \times 180 = 63

, and

180 + 63 = 243

, the same answer the slow way.

Stacking a discount and a tax is just chaining two multipliers, one that pulls below

1

and one that pushes above

1

. Here

20 %

off is

\times 0.80

and

5 %

tax is

\times 1.05

, so 120 dollars becomes

120 \times 0.80 \times 1.05 = 100.80

dollars. Because

0.80 \times 1.05 = 1.05 \times 0.80

, the net effect is the same product whichever order you run the gates in, and the price still ends at 100.80 dollars.

Problem

A longboard is tagged at

$ 240

. The shop takes

25 %

off, then the register adds

6 %

tax on the reduced price. What is the final price paid in dollars?

Show a hint

Handle the two moves one at a time, and notice each one is just a multiplier. Taking $25 %$ off keeps the $75 %$ that survives, so that step is $\times 0.75$ . Adding $6 %$ tax keeps the whole price and piles $6 %$ on top, so that step is $\times 1.06$ .
Chain the two multipliers in a row rather than doing each as a separate add or subtract. The final price is $240 \times 0.75 \times 1.06$ . Work left to right, $240 \times 0.75$ first, then multiply that result by $1.06$ .

Show the full solution

25 %

discount leaves

75 %

, so that step is

\times 0.75

. A

6 %

tax adds on top, so that step is

\times 1.06

240 \times 0.75 = 180, 180 \times 1.06 = 190.80

The final price paid is

190.80

dollars. The tax rides on the reduced 180, not the original 240. Order does not matter though, since

240 \times 1.06 \times 0.75

lands on the same 190.80.

Problem

Devon earns a

12 %

commission and sold

$ 1, 450

in glassware on Sunday. How many dollars does Devon earn in commission that day?

Show a hint

Commission is a percent OF the sale, so this is a plain "percent of a number" problem. The whole is the $1, 450$ dollar sale and the rate is $12 %$ , so translate the words into $0.12 \times 1, 450$ .
Turn $12 %$ into the decimal $0.12$ , then multiply by $1, 450$ . A quick way is to find $10 %$ of the sale, which is $145$ , then add another $2 %$ , which is $29$ , and combine them.

Show the full solution

A commission is a percent of the sale. Write

12 %

0.12

and multiply by the sale.

0.12 \times 1, 450 = 174

Devon earns

174

dollars. Split it to check,

10 %

1, 450

145

and

2 %

29

, and

145 + 29 = 174

Problem

Naomi earns

8.5 %

commission and received

$ 510

on one telescope sale. What was the sale price of that telescope in dollars?

Show a hint

Name the unknown the sale price and translate the sentence straight across. Her commission is $8.5 %$ of the sale price, and that commission equals $510$ . So $510$ is the part, $8.5 %$ is the rate, and the sale price is the whole you are hunting for. As an equation that reads $510 = 0.085 \times sale$ .
You are finding the whole from a part, exactly like in 9.2, so undo the multiplication by dividing. Convert $8.5 %$ to the decimal $0.085$ , then compute $510 \div 0.085$ . The answer should come out much bigger than $510$ , since the commission is only a thin slice of the full price.

Show the full solution

The commission is

8.5 %

of the sale price, so

510 = 0.085 \times sale

. Divide the part by the rate.

sale = \frac{510}{0.085} = 6000

The telescope sold for

6000

dollars. When you know the part and want the whole, you divide, and dividing by a small rate makes the whole far bigger than the part.

A two-way split divides one whole into two groups. Here

320

votes split into

208

in favor and

112

against, and the gold slice fills its share of the bar. Each group's percent is its own count over the total of everyone, so in favor is

\frac{208}{320} = 0.65 = 65 %

and against is

\frac{112}{320} = 0.35 = 35 %

. The two shares always add to the whole, since

65 % + 35 % = 100 %

Problem

Maple Ridge Middle School cast

320

ballots on a schedule plan. Of those,

208

voted in favor. What percent of the votes were in favor of the plan?

Show a hint

Decide what is the part and what is the whole. The question asks about the votes in favor, so the part is the 208 in favor and the whole is all 320 votes that were cast. To find what percent one number is of another, set up the part over the whole and turn it into a percent.
Compute $\frac{208}{320} \times 100$ . Dividing 208 by 320 gives $0.65$ , and multiplying by 100 turns that decimal into a percent.

Show the full solution

The part is the 208 votes in favor and the whole is all 320 ballots cast.

\frac{208}{320} \times 100 = 0.65 \times 100 = 65

65

percent voted in favor. The against share is

\frac{112}{320} \times 100 = 35

, and

65 + 35 = 100

, which is the check worth running on any two-way split.

Problem

40

L barrel of mango cooler is

25 %

juice. The owner stirs in

10

extra liters of plain water, adding no juice. What percent of the new barrel is real mango juice?

Show a hint

The amount of mango juice never changes, only water is added. First find how many liters of juice are in the barrel, that is $25 %$ of 40, and then find the new total number of liters after the 10 liters of water go in.
The new concentration is the juice amount divided by the new total, turned into a percent. Compute $\frac{10}{50} \times 100$ . Since you only added water, expect a number below $25 %$ .

Show the full solution

The juice is

0.25 \times 40 = 10

liters, and pouring in water leaves that 10 liters untouched. The new total is

40 + 10 = 50

liters, so the juice share is

\frac{10}{50} \times 100 = 20.

The barrel is

20

percent juice. Adding only water can only weaken the drink, so a number below the original

25 %

is the right direction.

Problem

At a beekeeping challenge,

15 %

of members did not find the queen in time, and that came to exactly

9

members. How many people took the challenge in all?

Show a hint

The 9 who missed the queen are not the whole club, they are the slice that failed. Translate the sentence into math, those 9 people are $15 %$ of the full group, so $0.15$ times the group size equals 9. The group size is the whole you are hunting for, the base.
You know the part is 9 and its rate is $15 %$ , and you want the base. Finding a whole from a part means you divide, so compute $9 \div 0.15$ .

Show the full solution

The 9 members are the part and

15 %

is the rate, so

0.15 \times (group size) = 9

. Divide the part by the rate.

9 \div 0.15 = 60

60

people took the challenge. Check it forward,

0.15 \times 60 = 9

. Only

15 %

of the group failed, so the whole group has to be much larger than 9.

Each rush sells a percent of what is left, not of the original 400. Keeping

0.75

, then

0.80

, then

0.50

of each remainder gives

400 \times 0.75 \times 0.80 \times 0.50 = 120

muffins.

Problem

A warehouse has

720

boxes. Monday ships

25 %

. Tuesday ships

40 %

of what remains. Wednesday ships

50 %

of what is left then. How many boxes remain after Wednesday's shipment?

Show a hint

Each percent is taken of what is LEFT at that moment, not of the starting 720. So instead of tracking what ships, track the fraction that stays. If $25 %$ ships on Monday, then $75 %$ stays. If $40 %$ of the rest ships on Tuesday, then $60 %$ of the rest stays. Turn each step into the leftover multiplier.
Multiply the leftover fractions one after another onto 720. That is $720 \times 0.75$ for Monday, then times $0.60$ for Tuesday, then times $0.50$ for Wednesday. The question wants the boxes that REMAIN after Wednesday, not the boxes that were shipped, so the final product is your answer.

Show the full solution

Track what stays instead of what leaves. Monday leaves

75 %

, Tuesday leaves

60 %

, Wednesday leaves

50 %

720 \times 0.75 = 540, 540 \times 0.60 = 324, 324 \times 0.50 = 162

After Wednesday,

162

boxes remain. Each percent is taken of the current pile, not of the original 720, so the leftover fractions multiply in a chain.

Problem

The Maple Hollow Otters play a

25

-game season. After

13

games they have

7

wins. The coach targets a

60 %

win rate for the whole season. How many of the remaining games must they win?

Show a hint

First find how many wins the goal needs across the whole season. The target is $60 %$ of all 25 games, so translate "60 percent of 25" into multiplication to get the total number of wins the Otters need by the end.
They already have 7 of those wins banked. Subtract to see how many more wins they still need, and notice that those extra wins all have to come out of the games that remain.

Show the full solution

60 %

win rate over 25 games is

0.60 \times 25 = 15

wins. They already have 7, so they need

15 - 7 = 8

more, and there are

25 - 13 = 12

games left to get them. They must win

8

of their remaining games. Checking,

7 + 8 = 15

wins out of 25 is

\frac{15}{25} = \frac{3}{5} = 60 %

, the target exactly.

Problem

A keyboard is

25 %

off, then

8 %

tax is added to the reduced price. Devansh pays

$ 162

. What was the original listed price in dollars?

Show a hint

Run the story forward first to find the chain. Taking 25 percent off multiplies the listed price by $0.75$ , and adding 8 percent tax multiplies by $1.08$ . So the listed price times $0.75$ times $1.08$ equals the 162 he paid. Combine the two multipliers into one number before you do anything else.
Since $0.75 \times 1.08 = 0.81$ , the whole deal is just listed price times $0.81 = 162$ . You are given the result and want the starting number, so undo the multiply by dividing. Compute $162 \div 0.81$ .

Show the full solution

25 %

discount is

\times 0.75

and an

8 %

tax is

\times 1.08

, so

listed \times 0.75 \times 1.08 = 162

. Since

0.75 \times 1.08 = 0.81

, divide to undo it.

listed = \frac{162}{0.81} = 200

The original listed price was

200

dollars. Running it forward confirms the chain,

200 \times 0.75 = 150

and

150 \times 1.08 = 162

Practice these ideas

Practice

A kettle sells for

$ 60

and a

5 %

sales tax is added. How many dollars is the sales tax alone?

Show the solution

Write

5 %

as the decimal

0.05

and multiply by the 60 dollar price.

0.05 \times 60 = 3

The sales tax alone is

3

dollars. The total with tax would be 63 dollars, but the question asked only for the tax, so stop at 3.

Practice

A ukulele is listed at

$ 150

before

6 %

sales tax. What is the total amount you pay including the tax?

Show the solution

Adding

6 %

tax means multiplying by

1 + 0.06 = 1.06

150 \times 1.06 = 159

You pay

159

dollars. The tax on its own is

0.06 \times 150 = 9

, and

150 + 9 = 159

, the same total.

Practice

Naomi's grooming bill is

$ 40

and she adds a

15 %

tip. How many dollars does Naomi pay in all?

Show the solution

15 %

tip means paying

115 %

of the bill, a multiplier of

1.15

40 \times 1.15 = 46

Naomi pays

46

dollars in all. The tip by itself is

0.15 \times 40 = 6

dollars, and

40 + 6 = 46

Practice

Mara earns

5 %

commission and closed a

$ 2400

solar job. How much does she earn in commission, in dollars?

Show the solution

Commission is a percent of the sale. Write

5 %

0.05

and multiply by the job.

0.05 \times 2400 = 120

Mara earns

120

dollars in commission. Ten percent of 2400 is 240, and

5 %

is half of that, which is 120.

Practice

Priya earns

9 %

commission and received

$ 630

on one kayak sale. What was the sale price of that kayak in dollars?

Show the solution

The

9 %

commission on the sale price came to 630 dollars, so

0.09 \times price = 630

. Divide the part by the rate.

price = \frac{630}{0.09} = 7000

The kayak sold for

7000

dollars. Check it forward,

0.09 \times 7000 = 630

, and the price should come out far larger than the commission.

Practice

A garden center pays

$ 90

for a planter and marks it up

40 %

. What is the shelf price in dollars?

Show the solution

40 %

markup makes the shelf price

140 %

of what the store paid, a multiplier of

1.40

90 \times 1.40 = 126

The shelf price is

126

dollars. The markup alone is

0.40 \times 90 = 36

, and

90 + 36 = 126

Practice

A peacoat is priced at

$ 220

and marked

15 %

off. What is the discounted price Devang pays in dollars?

Show the solution

15 %

discount leaves

100 % - 15 % = 85 %

of the price, a multiplier of

0.85

220 \times 0.85 = 187

Devang pays

187

dollars. The discount itself is

0.15 \times 220 = 33

dollars, and

220 - 33 = 187

Practice

A desk lamp is priced at

$ 250

. The shop takes

10 %

off, then rings up

8 %

tax on the reduced price. What is the final price paid in dollars?

Show the solution

10 %

discount leaves

90 %

, so multiply by

0.90

. Then

8 %

tax on the reduced price multiplies by

1.08

250 \times 0.90 = 225, 225 \times 1.08 = 243

The final price paid is

243

dollars. The tax is charged on the reduced 225, not the original 250, which is why the chain reads

250 \times 0.90 \times 1.08

Practice

A lending library has

84

books, and

21

were checked out on the weekend. What percent of the library's books were checked out?

Show the solution

The part is the 21 books checked out and the whole is all 84 books.

\frac{21}{84} \times 100 = \frac{1}{4} \times 100 = 25

25

percent of the books were checked out. A quarter of 84 is 21, which matches the count that left the shelves.

Practice

A robotics club of

200

members voted on practice time. Exactly

140

voted for mornings. What percent voted for evenings? Give the number only.

Show the solution

With only two choices, the evening voters are the

200 - 140 = 60

members left over. Compare that count to the whole club.

\frac{60}{200} = 0.30

30

percent voted for evenings. Divide by the total membership, never by the 140 on the other side. The morning share is

70 %

, and the two add to

100 %

Practice

Exactly

20 %

of people who sat a certification exam did not pass, and that group numbered

12

. How many people took the exam in all?

Show the solution

The 12 who did not pass are

20 %

of the whole group, so

0.20 \times total = 12

. Divide the part by the rate.

total = \frac{12}{0.20} = 60

60

people took the exam. Check it forward,

0.20 \times 60 = 12

, and the other

80 %

is 48 passers, with

48 + 12 = 60

Practice

A crew prints

600

flyers. They hand out

20 %

to passersby. Then rain ruins

25 %

of the flyers still in the box. How many flyers does the crew have left?

Show the solution

Handing out

20 %

leaves

80 %

in the box, and the rain ruining

25 %

leaves

75 %

of what was still there.

600 \times 0.80 = 480, 480 \times 0.75 = 360

The crew has

360

flyers left. That

25 %

comes off the 480 in the box, not off the original 600, so the leftover fractions multiply rather than the losses adding up.

Practice

80

L barrel of polishing fluid is

25 %

solvent. An apprentice stirs in

20

more liters of plain water. What percent of the new mixture is solvent? Give the number only.

Show the solution

The solvent is

0.25 \times 80 = 20

liters, and plain water adds none of it. The total grows to

80 + 20 = 100

liters, so the solvent share is

\frac{20}{100} \times 100 = 20.

The new mixture is

20

percent solvent. Compare the solvent to the whole new mixture, and expect a number below

25 %

since only water went in.

Practice

The Marlowe Middle quiz bowl team plays a

28

-game season. After

16

games they have

9

wins. They want a

50 %

win rate for the whole season. How many of their remaining games must they win?

Show the solution

50 %

season needs

\frac{1}{2} \times 28 = 14

wins in all. They have 9, so they still need

14 - 9 = 5

. The team must win

5

of its remaining games. There are

28 - 16 = 12

games left, so 5 is well within reach, and

9 + 5 = 14

out of 28 is exactly

50 %

Percents in Word Problems

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