How to solve percent problems
TEAS 7 outline: Solve real-world percentage problems
Also covers HESI A2 Math (Percentages)
- Every percent problem uses part = percent × whole.
- Change the percent to a decimal first: 7.5% = 0.075.
- Percent change: divide the change by the original amount, not the new one.
- 20% off means you pay 80%: sale price = 0.80 × original.
- Back-to-back discounts multiply: 25% then 10% off is 32.5% off.
Key facts
- Percent means per hundred; to convert, move the decimal point two places left (7.5% = 0.075, 125% = 1.25).
- Part = percent × whole, where "of" means multiply and "is" means equals.
- Whole = part ÷ percent, and percent = part ÷ whole × 100.
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- Percent change = (new − original) ÷ original × 100; always divide by the starting amount.
- Sale price = original × (1 − discount rate); a total with tax or tip = price × (1 + rate).
- Two percent changes in a row multiply: 25% off and then 10% off leaves 0.75 × 0.90 = 0.675 of the price, a 32.5% total discount.
- Mental shortcuts: 10% moves the decimal one place left, 5% is half of 10%, and 1% moves it two places left.
Quick check1 question · 2 more at the end
A clinic has 240 patients scheduled this week, and 35% of them are new patients. How many new patients are scheduled?
Correct answer: B.
Find the part with part = percent × whole. Write 35% as 0.35, then 0.35 × 240 = 84. Mental check: 10% of 240 is 24, so 30% is 72 and 5% is 12; 72 + 12 = 84.
A This finds the 65% who are not new patients (240 − 84 = 156) instead of the new patients.
B Correct: 0.35 × 240 = 84.
C This writes 35% as 0.035 instead of 0.35, so the answer is 10 times too small.
D This divides 240 by 35 instead of multiplying 240 by 0.35.
Full lesson
Each part starts with its main idea. Open a part when you want the details.
The short answer: Write the percent as a decimal and use part = percent × whole. To find the part, multiply. To find the whole, divide the part by the percent. To find the percent, divide the part by the whole and multiply by 100. For percent change, divide the amount of change by the original amount.
1 What is the one equation behind every percent problem?Almost every percent problem is part = percent × whole. 1 min
Almost every percent problem is part = percent × whole. A percent means "per hundred," so change it to a decimal by moving the decimal point two places left: 30% = 0.30 and 7.5% = 0.075.
Translate the sentence word by word: "of" means ×, "is" means =, and "what" is the unknown.
| The question sounds like… | Do this |
|---|---|
| What is 30% of 240? (find the part) | 0.30 × 240 = 72 |
| 72 is 30% of what? (find the whole) | 72 ÷ 0.30 = 240 |
| 72 is what percent of 240? (find the percent) | 72 ÷ 240 = 0.30 = 30% |
| By what percent did it change? | change ÷ original × 100 |
2 How do you find a part, like tax or a tip?Change the percent to a decimal and multiply it by the whole. 1 min
Change the percent to a decimal and multiply it by the whole. Example: a $64 pair of clinic shoes has 7.5% sales tax. What is the total cost?
- Convert: 7.5% = 0.075.
- Tax: 0.075 × $64 = $4.80.
- Total: $64 + $4.80 = $68.80. (Or in one step: 1.075 × $64.)
- Mental check: 10% of $64 is $6.40, 5% is $3.20, and 2.5% is $1.60. Then 5% + 2.5% = $3.20 + $1.60 = $4.80.
Classic wrong turn: writing 7.5% as 0.75, which gives $48 of tax on a $64 item. Tax that big should look wrong right away.
3 How do you find the whole when you know a part?Divide the part by the percent written as a decimal. 1 min
Divide the part by the percent written as a decimal. Example: a patient has had 360 mL of fluid, which is 30% of the day's goal. What is the full goal?
- Translate: 360 is 30% of what? → 360 = 0.30 × G.
- Divide both sides by 0.30: G = 360 ÷ 0.30 = 1,200 mL.
- Check: 30% of 1,200 is 0.30 × 1,200 = 360.
Classic wrong turn: multiplying 360 × 0.30 = 108. Sense check: if 360 mL is only part of the goal, the whole goal must be bigger than 360.
4 How do you find a percent increase or decrease?Divide the amount of change by the original amount, not the new one. 1 min
Divide the amount of change by the original amount, not the new one. Example: a medication dose is lowered from 40 mg to 30 mg. By what percent was it decreased?
- Change: 40 − 30 = 10 mg.
- Divide by the original: 10 ÷ 40 = 0.25.
- Convert: 0.25 = 25% decrease.
- Check: 25% of 40 is 10, and 40 − 10 = 30.
Classic wrong turn: dividing by the new amount, 10 ÷ 30 ≈ 33.3%. The starting amount is always the base.
5 How do discounts work, including two in a row?A discount leaves you paying the remaining percent of the price. 1 min
A discount leaves you paying the remaining percent of the price. With 20% off, you pay 80%, so sale price = 0.80 × original.
| Situation | Set-up |
|---|---|
| $80 shoes, 25% off | 0.75 × 80 = $60 |
| Then 10% more off the sale price | 0.90 × 60 = $54 (total 32.5% off, not 35%) |
| Sale price $48 after 20% off; original? | 48 ÷ 0.80 = $60 |
The TEAS has a basic on-screen calculator. Type percents as decimals (0.075, not 7.5) and use mental shortcuts for round numbers like 10% or 25%.
Common mistakes
The traps students fall into most. Read these before you practice.
Watch out: 7.5% is 0.075, not 0.75.
Watch out: Divide by the original amount for percent change: a drop from 40 to 30 is 25%, not 33.3%.
Watch out: To undo a discount, divide by the percent that is left: a $48 price after 20% off came from 48 ÷ 0.80 = $60, not 48 × 1.20 = $57.60.
Watch out: Back-to-back percents do not add: 25% off and then 10% off is 32.5% off, not 35%.
Watch out: Check which number the question wants: the tax or the total, the discount or the sale price.
How to remember it
The part over the whole always equals the percent over 100. A short hook is "is over of equals percent over 100." For "72 is what percent of 240?", write 72/240 = P/100, so P = 30.
Self-check:
- Find 15% of $72 using the 10% + 5% shortcut.
- A price went from $50 to $60. Find the percent increase (divide by $50).
- Undo a 25% discount on a $45 sale price, then check your answer.
Show answers
- $10.80.
- 10% of $72: move the decimal one place left, $7.20.
- 5% is half of that: $3.60.
- Add: $7.20 + $3.60 = $10.80.
- 20% increase.
- Change: $60 − $50 = $10.
- Divide by the original: 10 ÷ 50 = 0.20.
- Convert: 0.20 = 20%.
- $60.
- 25% off means you paid 75% of the original: 45 = 0.75 × original.
- Divide by the percent that is left: 45 ÷ 0.75 = $60.
- Check: 0.75 × 60 = 45.