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Percentage Word Problems with Solutions

Grade: 7-8 | Topic: Arithmetic

What You Will Learn​

After working through this page you will be able to translate real-life percentage scenarios into math equations, solve multi-step percentage word problems involving discounts, taxes, tips, test scores, and mixtures, and check your answers using estimation. The key is learning to identify the "part," the "whole," and the "percentage" in each problem.

Theory​

Setting up percentage word problems​

Every percentage word problem boils down to one relationship:

Part=Percentage100×Whole\text{Part} = \frac{\text{Percentage}}{100} \times \text{Whole}

Your job is to figure out which of the three values -- Part, Percentage, or Whole -- is missing, and then solve for it.

Signal words to look for​

Word problems contain clues that tell you what is being asked:

Phrase in the problemWhat it means
"What is _% of ...?"Find the Part
"What percentage of ...?"Find the Percentage
"... is _% of what number?"Find the Whole
"How much more / less ..."Find the change, then possibly percentage change
"After a _% discount / increase ..."Apply a percentage change to find the new value

The four-step strategy​

  1. Read the problem carefully and identify the three quantities (Part, Percentage, Whole). Underline numbers and key phrases.
  2. Set up an equation using Part=p100×Whole\text{Part} = \frac{p}{100} \times \text{Whole}.
  3. Solve for the unknown quantity.
  4. Check by substituting your answer back in. Does the sentence make sense with your number?

Multi-step problems​

Some problems require more than one percentage calculation. For example, you might need to calculate a discount first, then add tax on the discounted price. The key is to handle each step separately and in the correct order. Never apply multiple percentages to the original value at the same time unless the problem specifically says to.

Worked Examples​

Example 1: Finding the part -- test scores (easy)​

Problem: Lily answered 85% of 40 questions correctly on her math test. How many questions did she get right?

Step 1: Identify the values. Percentage = 85%, Whole = 40, Part = ?

Step 2: Apply the formula. Part=85100×40=0.85×40=34\text{Part} = \frac{85}{100} \times 40 = 0.85 \times 40 = 34

Step 3: Check -- 34 out of 40 is 3440=0.85=85%\frac{34}{40} = 0.85 = 85\%. Correct.

Answer: Lily got 34 questions right.

Example 2: Finding the percentage -- savings goal (easy)​

Problem: Marcus wants to save $600 for a new bike. So far he has saved $450. What percentage of his goal has he reached?

Step 1: Identify the values. Part = $450, Whole = $600, Percentage = ?

Step 2: Apply the formula rearranged for the percentage. p=450600×100=0.75×100=75%p = \frac{450}{600} \times 100 = 0.75 \times 100 = 75\%

Answer: Marcus has reached 75% of his goal.

Example 3: Finding the whole -- class attendance (medium)​

Problem: On a rainy day, only 72% of students came to school. If 252 students were present, how many students are enrolled in the school?

Step 1: Identify the values. Part = 252, Percentage = 72%, Whole = ?

Step 2: Rearrange the formula to solve for the whole. Whole=Part×100Percentage=252×10072=2520072=350\text{Whole} = \frac{\text{Part} \times 100}{\text{Percentage}} = \frac{252 \times 100}{72} = \frac{25200}{72} = 350

Step 3: Verify -- 72%72\% of 350 is 0.72×350=2520.72 \times 350 = 252. Correct.

Answer: The school has 350 students enrolled.

Example 4: Discount then tax -- shopping (medium)​

Problem: A jacket originally costs $80. The store offers a 25% discount. After the discount, 8% sales tax is added. What is the final price?

Step 1: Calculate the discount. Discount=25100×80=0.25×80=20\text{Discount} = \frac{25}{100} \times 80 = 0.25 \times 80 = 20

Step 2: Subtract the discount to get the sale price. 80−20=6080 - 20 = 60

Step 3: Calculate the tax on the discounted price (not the original). Tax=8100×60=0.08×60=4.80\text{Tax} = \frac{8}{100} \times 60 = 0.08 \times 60 = 4.80

Step 4: Add the tax. 60+4.80=64.8060 + 4.80 = 64.80

Answer: The final price is $64.80.

Example 5: Multi-step problem -- commission and bonus (challenging)​

Problem: A salesperson earns a 6% commission on all sales. In March she sold $15,000 worth of products. She also receives a $200 bonus if her commission exceeds $800. What was her total income from this job in March?

Step 1: Calculate the commission. 6100×15,000=0.06×15,000=900\frac{6}{100} \times 15{,}000 = 0.06 \times 15{,}000 = 900

Step 2: Check if the commission exceeds $800. 900>800900 > 800, so she qualifies for the bonus.

Step 3: Calculate total income. 900+200=1,100900 + 200 = 1{,}100

Answer: Her total income in March was $1,100.

Common Mistakes​

Mistake 1: Applying tax to the original price instead of the discounted price

❌ Jacket costs $80. After 25% discount and 8% tax: 80−20=6080 - 20 = 60, then tax = 0.08×80=6.400.08 \times 80 = 6.40, total = $66.40.

✅ Tax is calculated on the discounted price: 0.08×60=4.800.08 \times 60 = 4.80. Total = $64.80.

Why this matters: In real life, sales tax applies to the amount you actually pay, not the original sticker price. Always apply each percentage step to the result of the previous step.

Mistake 2: Confusing "what percent of" with "percent change"

❌ "30 is what percent of 50?" Student calculates the percent difference: 50−3050×100=40%\frac{50 - 30}{50} \times 100 = 40\%.

✅ "30 is what percent of 50?" means 3050×100=60%\frac{30}{50} \times 100 = 60\%.

Why this matters: "What percent of" asks you to compare a part to a whole. "Percent change" or "percent difference" asks how much a value changed relative to the original. Read the question carefully to determine which calculation is needed.

Mistake 3: Not re-reading the question to check what is actually being asked

❌ Problem asks "How much did the price decrease?" Student calculates the new price ($60) instead of the decrease amount ($20).

✅ Always re-read the question after solving to make sure your answer matches what was asked -- the decrease, the new price, or the percentage.

Why this matters: Many students solve the problem correctly but then report the wrong quantity. The final step should always be checking that your answer addresses the exact question.

Practice Problems​

Try these on your own before checking the answers:

  1. A store sells 240 items in one day. If 15% of them were returned, how many items were returned?
  2. Out of 80 apples, 12 are bruised. What percentage are bruised?
  3. A charity has raised $4,500, which is 60% of its target. What is the fundraising target?
  4. A computer costs $1,200. It is on sale for 30% off, and there is a 7% sales tax on the discounted price. What is the final cost?
  5. A farmer plants 500 seeds. 92% of them sprout. Of the sprouts, 80% survive to harvest. How many plants are harvested?
Click to see answers
  1. 0.15×240=360.15 \times 240 = 36 items returned.
  2. 1280×100=15%\frac{12}{80} \times 100 = 15\%
  3. Target=4500×10060=7,500\text{Target} = \frac{4500 \times 100}{60} = 7{,}500, so the target is $7,500
  4. Discount = 0.30×1200=3600.30 \times 1200 = 360. Sale price = 1200−360=8401200 - 360 = 840. Tax = 0.07×840=58.800.07 \times 840 = 58.80. Final = 840 + 58.80 = \898.80$
  5. Sprouts = 0.92×500=4600.92 \times 500 = 460. Harvested = 0.80×460=3680.80 \times 460 = 368 plants.

Summary​

  • Every percentage word problem involves three quantities: Part, Percentage, and Whole. Identify which is missing and solve for it.
  • Use signal words ("of," "out of," "what percent") to set up the correct equation.
  • For multi-step problems, apply each percentage to the result of the previous step, not to the original value.
  • Always re-read the question to make sure your answer matches what was asked.

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