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 $\boxed{3.6}$ 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?

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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 $\boxed{94.40}$ 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

A $35\%$ 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 $\boxed{243}$ 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

A $25\%$ discount leaves $75\%$, so that step is $\times 0.75$. A $6\%$ tax adds on top, so that step is $\times 1.06$. $$240 \times 0.75 = 180, \qquad 180 \times 1.06 = 190.80$$ The final price paid is $\boxed{190.80}$ 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 $12\%$ as $0.12$ and multiply by the sale. $$0.12 \times 1{,}450 = 174$$ Devon earns $\boxed{174}$ dollars. Split it to check, $10\%$ of $1{,}450$ is $145$ and $2\%$ is $29$, 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 $510 = 0.085 \times \text{sale}$.
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 $510 = 0.085 \times \text{sale}$. Divide the part by the rate. $$\text{sale} = \frac{510}{0.085} = 6000$$ The telescope sold for $\boxed{6000}$ 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 $\tfrac{208}{320} = 0.65 = 65\%$ and against is $\tfrac{112}{320} = 0.35 = 35\%$. 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. $$\frac{208}{320}\times 100 = 0.65 \times 100 = 65$$ So $\boxed{65}$ 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

A $40$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?

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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 $25\%$.

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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 $\boxed{20}$ 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?

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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 $9 \div 0.15$.

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The 9 members are the part and $15\%$ is the rate, so $0.15 \times (\text{group size}) = 9$. Divide the part by the rate. $$9 \div 0.15 = 60$$ So $\boxed{60}$ 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 $400\times0.75\times0.80\times0.50 = 120$ 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 $50\%$. $$720 \times 0.75 = 540, \qquad 540 \times 0.60 = 324, \qquad 324 \times 0.50 = 162$$ After Wednesday, $\boxed{162}$ 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

A $60\%$ 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 $\boxed{8}$ of their remaining games. Checking, $7 + 8 = 15$ wins out of 25 is $\tfrac{15}{25} = \tfrac{3}{5} = 60\%$, 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

A $25\%$ discount is $\times 0.75$ and an $8\%$ tax is $\times 1.08$, so $\text{listed} \times 0.75 \times 1.08 = 162$. Since $0.75 \times 1.08 = 0.81$, divide to undo it. $$\text{listed} = \frac{162}{0.81} = 200$$ The original listed price was $\boxed{200}$ dollars. Running it forward confirms the chain, $200 \times 0.75 = 150$ and $150 \times 1.08 = 162$.

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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 $\boxed{3}$ 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 $\boxed{159}$ 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

A $15\%$ tip means paying $115\%$ of the bill, a multiplier of $1.15$. $$40 \times 1.15 = 46$$ Naomi pays $\boxed{46}$ 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 $5\%$ as $0.05$ and multiply by the job. $$0.05 \times 2400 = 120$$ Mara earns $\boxed{120}$ 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 $0.09 \times \text{price} = 630$. Divide the part by the rate. $$\text{price} = \frac{630}{0.09} = 7000$$ The kayak sold for $\boxed{7000}$ 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

A $40\%$ 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 $\boxed{126}$ 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

A $15\%$ discount leaves $100\% - 15\% = 85\%$ of the price, a multiplier of $0.85$. $$220 \times 0.85 = 187$$ Devang pays $\boxed{187}$ 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

A $10\%$ discount leaves $90\%$, so multiply by $0.90$. Then $8\%$ tax on the reduced price multiplies by $1.08$. $$250 \times 0.90 = 225, \qquad 225 \times 1.08 = 243$$ The final price paid is $\boxed{243}$ 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. $$\frac{21}{84}\times 100 = \frac{1}{4}\times 100 = 25$$ So $\boxed{25}$ 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. $$\frac{60}{200} = 0.30$$ so $\boxed{30}$ 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 $0.20 \times \text{total} = 12$. Divide the part by the rate. $$\text{total} = \frac{12}{0.20} = 60$$ So $\boxed{60}$ 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. $$600 \times 0.80 = 480, \qquad 480 \times 0.75 = 360$$ The crew has $\boxed{360}$ 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

An $80$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 $\boxed{20}$ 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

A $50\%$ season needs $\tfrac{1}{2} \times 28 = 14$ wins in all. They have 9, so they still need $14 - 9 = 5$. The team must win $\boxed{5}$ 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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