Practice Set · Pre-Algebra · Young Fermats · Class 10

Pre-Algebra Class 10: Percent — Three Faces (Part/Whole/Percent), Tip, Tax, Discount

A one-page percent cheat sheet from Class 10 of the SOMATH Young Fermats — Pre-Algebra course, followed by 25 practice questions with hidden step-by-step answer explanations. Free to use at home, in class, or as a warm-up before enrolling.

· By the SOMATH team · 226 W 79th St, UWS · (646) 668-6151

Short answer up top. Every percent problem is really one equation with one missing box: part / whole = percent / 100. Depending on which number is missing, the same equation answers all three “faces” of percent — find the part (“what is 15% of 80?”), find the whole (“15 is 20% of what number?”), or find the percent (“what percent of 80 is 15?”). Once you can set that up cleanly, tip, sales tax, discount, sale price, markup, and percent change are all just short recipes on top of it. Class 10 is the direct sequel to Class 9 — Proportional Reasoning and is one of the highest-leverage classes for the SHSAT and the Digital SAT. Book a free 30-minute evaluation at /evaluation or call (646) 668-6151, and any 5th- or 6th-grader can join Young Fermats — Pre-Algebra on the next Monday at our 226 W 79th Street classroom on the Upper West Side.

What percent actually means

The word percent literally means “per hundred” — from the Latin per centum. Writing 15% is just a shorter way of writing the fraction 15/100 or the decimal 0.15. That is the whole idea. Percent is not a new kind of number — it is a third way to write a ratio, on top of fractions and decimals.

Definition. A percent is a ratio with a denominator of 100. 15% = 15/100 = 0.15. Percent is the third face of every ratio: as a fraction (15/100), as a decimal (0.15), and as a percent (15%). All three name the same amount.

Because percent is just a ratio, it plugs directly into the proportion machinery from Class 9. Every percent question is really the proportion:

part / whole = percent / 100

That single equation is the entire class. Everything below is just filling in what you know and solving for what you don’t.

The percent-fraction-decimal conversion table

Fluent percent thinking means moving between the three forms without stopping to think. Memorize these until they are automatic — every percent problem starts with a conversion.

PercentFraction (simplified)Decimal
1%1/1000.01
5%1/200.05
10%1/100.10
20%1/50.20
25%1/40.25
50%1/20.50
75%3/40.75
100%11.00
125%5/41.25
200%22.00

PERCENT → DECIMAL

move the decimal point 2 places LEFT, drop the % sign

15% → 0.15    8.25% → 0.0825    250% → 2.5

DECIMAL → PERCENT

move the decimal point 2 places RIGHT, add the % sign

0.6 → 60%    0.075 → 7.5%    1.4 → 140%

Why the “two places” rule works. To turn a percent into a decimal you divide by 100. Dividing by 100 moves the decimal point two places to the left. That is it — the rule is not a trick, it is just what dividing by 100 does. And multiplying by 100 (going the other way) moves the decimal point two places to the right.

The three faces of percent

The single equation part / whole = percent / 100 answers all three shapes of percent question. You will be given two of the three values — find the missing one. The name of the game is label the three boxes, then solve.

FACE 1 — FIND THE PART

given the whole and the percent

15% of 80 = ? → 12

FACE 2 — FIND THE WHOLE

given the part and the percent

15 is 20% of ? → 75

FACE 3 — FIND THE PERCENT

given the part and the whole

15 is what % of 80? → 18.75%

SHORTCUT (all three faces)

translate: “of” = ×, “is” = =, “what” = x, “percent” = ÷100

“15% of 80” = 0.15 × 80 = 12

Face 1 — Find the part (“what is 15% of 80?”)

Setup. Whole = 80, percent = 15, part = x (unknown). Proportion: x/80 = 15/100.

  1. Cross-multiply: 100x = 80 × 15 = 1200.
  2. Divide both sides by 100: x = 12.

Decimal shortcut. 15% × 80 = 0.15 × 80 = 12. Same answer, one line.

Face 2 — Find the whole (“15 is 20% of what number?”)

Setup. Part = 15, percent = 20, whole = x (unknown). Proportion: 15/x = 20/100.

  1. Simplify the right side: 20/100 = 1/5. So 15/x = 1/5.
  2. Cross-multiply: 15 × 5 = 1 × x → x = 75.

Decimal shortcut. 15 = 0.20 × x → x = 15 / 0.20 = 75. Same answer.

Face 3 — Find the percent (“15 is what percent of 80?”)

Setup. Part = 15, whole = 80, percent = p (unknown). Proportion: 15/80 = p/100.

  1. Cross-multiply: 15 × 100 = 80 × p → 1500 = 80p.
  2. Divide both sides by 80: p = 18.75. Attach the percent sign: 18.75%.

Decimal shortcut. 15 ÷ 80 = 0.1875 = 18.75%. Same answer.

The one rule for identifying the three faces. The whole is almost always the number that comes right after the word of. The part is the smaller piece being compared to it. The percent is the value with a % sign attached. Label the three roles before you write anything down, and the setup is unambiguous.

Tip — percent ADDED to a bill

A tip is a percent OF the bill, ADDED to the bill. Two things you may be asked to find: (1) the tip amount, or (2) the total the customer pays.

TIP AMOUNT

tip = bill × (tip% ÷ 100)

20% tip on $60 → $12

TOTAL PAID

total = bill × (1 + tip%÷100)

$60 × 1.20 → $72

Full example — 18% tip on a $45 bill.

  1. Tip amount: 45 × 0.18 = $8.10.
  2. Total paid: 45 + 8.10 = $53.10, or in one step 45 × 1.18 = $53.10.

Mental-math tip trick. 10% of any number is just that number with the decimal shifted one place to the left. So 10% of $45 is $4.50. 20% = double = $9.00. 15% = 10% + half of 10% = $4.50 + $2.25 = $6.75. This is the fastest way to tip in your head at a restaurant.

Sales tax — percent ADDED to a price

Sales tax works exactly like a tip — a percent OF the pre-tax price, ADDED to it — only the percent is set by the state and city, not by the customer.

NYC sales tax (as of 2026). The combined sales tax on most goods in Manhattan is 8.875% (4% NY State + 4.5% New York City + 0.375% MCTD). That is the rate we use in our examples and word problems for practical, right-outside-the-classroom relevance.

Full example — sales tax on a $40 book in NYC.

  1. Tax amount: 40 × 0.08875 = $3.55.
  2. Total paid: 40 + 3.55 = $43.55, or in one step 40 × 1.08875 = $43.55.

Discount — percent SUBTRACTED from a price

A discount is a percent OFF the original price. Two things you may be asked to find: (1) the discount amount (the dollars you save), or (2) the sale price (the final price you pay).

DISCOUNT AMOUNT

discount = original × (discount% ÷ 100)

25% off $120 → save $30

SALE PRICE

sale price = original × (1 − discount%÷100)

$120 × 0.75 → pay $90

Full example — 30% off a $80 jacket.

  1. Discount amount: 80 × 0.30 = $24. You save $24.
  2. Sale price: 80 − 24 = $56, or in one step 80 × 0.70 = $56.

The “paying the complement” shortcut. If something is 30% off, you PAY the other 70%. If something is 15% off, you pay 85%. If something is 60% off, you pay 40%. Multiplying the original by that complement percent gives you the sale price in ONE step — no subtraction needed. This shortcut is the single most useful percent move on the Digital SAT.

Tip / tax / discount at a glance. Tip is a percent ADDED — multiply by (1 + p/100). Tax is a percent ADDED — multiply by (1 + p/100). Discount is a percent SUBTRACTED — multiply by (1 − p/100). Same recipe, one sign difference.

Percent change — how much did it go up or down?

Percent change measures how a quantity moved relative to its ORIGINAL value. Positive = percent increase, negative = percent decrease.

percent change = (new − original) / original × 100

Full example — a book’s price went from $20 to $25.

  1. Change: 25 − 20 = 5. Divided by original: 5 / 20 = 0.25.
  2. As a percent: 0.25 × 100 = 25% increase.

Full example — a shirt on sale went from $40 to $30.

  1. Change: 30 − 40 = −10. Divided by the original: −10 / 40 = −0.25.
  2. As a percent: −0.25 × 100 = 25% decrease.

The most common percent-change mistake. Dividing by the wrong number. The denominator is ALWAYS the ORIGINAL value (the “starting” number, the “before” number, the “old” price). Never divide by the new value. If a price went from $40 to $30, that is a 25% decrease (10 divided by 40), not a 33% decrease (10 divided by 30). Getting this right earns SHSAT and SAT points that everyone else loses.

25 practice questions with hidden step-by-step answers

Try each one before revealing the answer. Every explanation shows the setup, the arithmetic, and a “why this works” note so the pattern sticks.

PART 1 — Percent, fraction, decimal fluency (Q1–Q5)
Q1
Write 32% as a fraction (do not simplify) and as a decimal.
Show answer & explanation
Fraction: 32/100. Decimal: 0.32.
Why. Percent means “per 100,” so 32% is literally 32/100. To convert 32% to a decimal, move the point two places left: 32.0 → 0.32. Both forms name the same amount.
Q2
Write 7.5% as a decimal.
Show answer & explanation
0.075.
Why. Move the decimal point two places to the left: 7.5 → 0.75 → 0.075. Careful with the zero — you have to shift TWO places, not one, so a single-digit percent like 7.5% picks up a leading zero.
Q3
Write 0.6 as a percent.
Show answer & explanation
60%.
Why. To convert a decimal to a percent, move the decimal point two places to the RIGHT and add the percent sign: 0.6 → 6.0 → 60. So 0.6 = 60%.
Q4
Write 3/4 as a percent.
Show answer & explanation
75%.
Why. Convert to a decimal first: 3 ÷ 4 = 0.75. Then decimal → percent: 0.75 = 75%. Or use the proportion 3/4 = p/100. Cross-multiply: 300 = 4p, so p = 75. Either method gives 75%.
Q5
Write 125% as a decimal and as a fraction (mixed number if needed).
Show answer & explanation
Decimal: 1.25. Fraction: 125/100 = 5/4 = 1 1/4.
Why. 125% just means 125 per 100, which is more than one whole. Move the decimal two places left: 125 → 12.5 → 1.25. As a fraction, 125/100 simplifies (divide top and bottom by 25) to 5/4, which is the mixed number 1 1/4. Percents can absolutely be greater than 100%, and they name amounts bigger than the whole.
PART 2 — Face 1: find the PART (Q6–Q10)
Q6
What is 25% of 80?
Show answer & explanation
20.
Why. Decimal shortcut: 25% = 0.25, so 0.25 × 80 = 20. Or use proportion: part/80 = 25/100 → part × 100 = 80 × 25 → 100 · part = 2000 → part = 20. Mental check: 25% is one quarter, and one quarter of 80 is 20.
Q7
What is 15% of 60?
Show answer & explanation
9.
Why. 15% = 0.15, so 0.15 × 60 = 9. Or use the mental-math trick: 10% of 60 is 6, and 5% is half of that (3), so 15% = 6 + 3 = 9. Same answer.
Q8
What is 8% of 250?
Show answer & explanation
20.
Why. 8% = 0.08, so 0.08 × 250 = 20. Mental check: 10% of 250 is 25, so 8% should be slightly less — 20 fits. Proportion form: part/250 = 8/100 → 100 · part = 2000 → part = 20.
Q9
What is 120% of 40?
Show answer & explanation
48.
Why. 120% = 1.20, so 1.20 × 40 = 48. When the percent is bigger than 100, the answer is bigger than the whole — 48 > 40. That is the sign check for a percent above 100: the answer must exceed the original number.
Q10
A class has 24 students. 75% of them ride the subway to school. How many students ride the subway?
Show answer & explanation
18 students.
Why. Whole = 24, percent = 75, find the part. 0.75 × 24 = 18. Or: 75% is three quarters, and three quarters of 24 is 3 × (24/4) = 3 × 6 = 18. Same answer.
PART 3 — Face 2: find the WHOLE (Q11–Q15)
Q11
12 is 25% of what number?
Show answer & explanation
48.
Why. Part = 12, percent = 25, whole = x. Proportion: 12/x = 25/100. Cross-multiply: 12 × 100 = 25x → 1200 = 25x → x = 48. Decimal shortcut: 12 = 0.25x → x = 12 ÷ 0.25 = 48. Sanity check: 25% of 48 is 12. ✓
Q12
18 is 20% of what number?
Show answer & explanation
90.
Why. 18 = 0.20 × x → x = 18 ÷ 0.20 = 90. Or scale-factor thinking: 20% is 1/5 of the whole, so the whole is 5 × 18 = 90. Sanity check: 20% of 90 is 18. ✓
Q13
7 is 35% of what number?
Show answer & explanation
20.
Why. 7 = 0.35 × x → x = 7 ÷ 0.35 = 20. Sanity check: 35% of 20 = 0.35 × 20 = 7. ✓ When the percent is a “messy” number like 35, division does the work.
Q14
150 is 120% of what number?
Show answer & explanation
125.
Why. 150 = 1.20 × x → x = 150 ÷ 1.20 = 125. When a percent above 100% is applied to a number, the answer is bigger than that number — so the original whole (125) must be smaller than the “part” (150). Sanity check: 120% of 125 = 1.20 × 125 = 150. ✓
Q15
In a survey, 45 students said they walk to school. That is 30% of the students surveyed. How many students were surveyed in total?
Show answer & explanation
150 students.
Why. Part = 45, percent = 30, find the whole. 45 = 0.30 × x → x = 45 ÷ 0.30 = 150. Sanity check: 30% of 150 = 45. ✓ Common mistake: multiplying 45 × 0.30 instead of dividing — that would give 13.5, which is smaller than the part, which is impossible.
PART 4 — Face 3: find the PERCENT (Q16–Q19)
Q16
What percent of 40 is 10?
Show answer & explanation
25%.
Why. Part = 10, whole = 40, find the percent. 10/40 = 0.25 = 25%. Or proportion form: 10/40 = p/100 → 1000 = 40p → p = 25.
Q17
9 is what percent of 12?
Show answer & explanation
75%.
Why. Part = 9, whole = 12. 9 ÷ 12 = 0.75 = 75%. Fraction check: 9/12 = 3/4 = 75%. Everything matches.
Q18
A student answered 34 out of 40 questions correctly on a math quiz. What percent did she get right?
Show answer & explanation
85%.
Why. Part = 34, whole = 40. 34 ÷ 40 = 0.85 = 85%. Sanity check: 10% of 40 is 4, so 85% is 40 − (15% × 40) = 40 − 6 = 34. ✓ This is the exact setup for every test-score percent problem.
Q19
A basketball player made 9 out of 20 free throws in a game. What percent did she make? What percent did she miss?
Show answer & explanation
Made 45%. Missed 55%.
Why. Made percent: 9 ÷ 20 = 0.45 = 45%. Missed percent: 20 − 9 = 11 misses, and 11 ÷ 20 = 0.55 = 55%. Sanity check: 45% + 55% = 100%. ✓ The two percents must always sum to 100% because they cover every free throw.
PART 5 — Tip, tax, discount, percent change (Q20–Q25)
Q20
A family’s restaurant bill is $60. They want to leave a 20% tip. What is the tip amount and what is the total they pay?
Show answer & explanation
Tip = $12. Total = $72.
Why. Tip amount = 60 × 0.20 = $12. Total = bill + tip = 60 + 12 = $72, or in one step 60 × 1.20 = $72. Mental-math check: 10% of $60 is $6, so 20% is $12. ✓
Q21
A meal costs $45 before tip. Leaving an 18% tip, what is the total the customer pays?
Show answer & explanation
$53.10.
Why. Tip amount = 45 × 0.18 = $8.10. Total = 45 + 8.10 = $53.10, or in one step 45 × 1.18 = $53.10. Mental-math shortcut: 10% is $4.50, 20% is $9, so 18% is a hair under $9, meaning the total is a hair under $54 — $53.10 fits perfectly.
Q22
A pair of sneakers costs $80 at a store in Manhattan. NYC combined sales tax is 8.875%. How much sales tax is added, and what is the total?
Show answer & explanation
Tax = $7.10. Total = $87.10.
Why. Tax = 80 × 0.08875 = $7.10 exactly. Total = 80 + 7.10 = $87.10, or in one step 80 × 1.08875 = $87.10. Sanity check: 10% of $80 would be $8, so a tax rate just under 10% (8.875%) should give a bit less than $8 — $7.10 is in the right zone.
Q23
A $120 jacket is marked 25% off. What is the sale price?
Show answer & explanation
$90.
Why. Fast way — use the “paying the complement” shortcut: 25% off means paying 75%. Sale price = 120 × 0.75 = $90. Long way: discount = 120 × 0.25 = $30, and 120 − 30 = $90. Same answer.
Q24
A book’s price went from $20 to $25. What is the percent increase?
Show answer & explanation
25% increase.
Why. Change = 25 − 20 = 5. Divide by the ORIGINAL price (20, not 25): 5 ÷ 20 = 0.25 = 25% increase. The most common mistake is dividing by 25 instead of 20; percent change ALWAYS uses the original value as the denominator.
Q25
A restaurant bill totals $54.60 after a 20% tip. What was the original bill (before the tip)?
Show answer & explanation
$45.50.
Why. The $54.60 total is 120% of the original bill (100% bill + 20% tip). So 54.60 = 1.20 × x, which gives x = 54.60 ÷ 1.20 = $45.50. Sanity check: 20% tip on $45.50 = 0.20 × 45.50 = $9.10, and 45.50 + 9.10 = $54.60. ✓ This is the reverse of a tip problem, and it is exactly the “find the whole” face of percent.

About the Young Fermats Pre-Algebra course

Class 10 is one of 48 rolling classes in our Young Fermats Pre-Algebra program for grade 5–6 students (ages 10–12). The full arc runs from integers on the number line through systems of linear equations and probability — the direct on-ramp to Algebra 1 in 7th or 8th grade. Classes are small-group (max 6 students), 120 minutes per week, and students can start any Monday because the syllabus is rolling.

Every new family starts with a free 30-minute evaluation in our 226 W 79th Street classroom plus a written diagnostic delivered within 48 hours. The diagnostic maps your child’s current level against the readiness signals for Pre-Algebra and Algebra 1 — and gives you a clear next step whether you enroll or not.

Ready to see the whole 48-class arc?

Class 10 is a taste — Young Fermats runs 48 rolling classes, students can join any Monday, and the syllabus takes them from integers through systems of equations. Small-group (max 6), 120 minutes a week, at 226 W 79th Street. First class is free. Cancel any time with 15 days’ notice.

See the course →   Book a free evaluation   or call (646) 668-6151

Related posts