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Fractions, Decimals & Percents — Three names for the same number.

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About this topic

Fractions, decimals, and percents are three different ways of writing the same kind of number — a piece of a whole. A kid who only thinks in one of these three is always translating; a kid who thinks in all three picks the easiest form for the question.

For example, all these are the same number:

  • ½ = 0.5 = 50%
  • ¾ = 0.75 = 75%
  • 1/5 = 0.2 = 20%
  • 1/8 = 0.125 = 12.5%

Memorize these conversions. They're the single most common building blocks in AMC 8 problems #1–#15.

This lesson teaches nine ideas: (1) move between the three forms fast, (2) treat percents as multipliers, (3) understand why compound percents multiply (not add), (4) compare fractions without computing them, (5) simplify complex fractions, (6) spot telescoping products, (7) work backward through fraction stories, (8) handle weighted averages, (9) distinguish 'percent OF' from 'percent MORE/LESS than'.

CHAPTER 1

Three forms, one number

THEORY

A fraction like 3/5 means “3 out of 5 equal parts.” A decimal like 0.6 means “6 tenths.” A percent like 60% means “60 per hundred.”

They’re all the same number. Three costumes for one value. The kid who thinks in all three picks the easiest costume for each problem.

Same number, three costumesFRACTION3/5DECIMAL0.6PERCENT60%F ↔ Ddivide / 0.xF ↔ Pscale to /100D ↔ P×100 / ÷100All three vertices = the same number. Move freely between them.

The six conversions, spelled out

DirectionRuleWorked example
Fraction → DecimalDivide top by bottom3 ÷ 5 = 0.6
Decimal → FractionRead place value, simplify0.6 = 6/10 = 3/5
Decimal → PercentMultiply by 100 (move dot 2 right)0.6 → 60%
Percent → DecimalDivide by 100 (move dot 2 left)60% → 0.60
Fraction → PercentScale denom to 100, OR divide × 1003/5 = 60/100 = 60%
Percent → FractionPut over 100 then simplify60% = 60/100 = 3/5
The AMC almost never gives you a number in the form you’d naturally compute with. A one-second translation at the start often turns a hard-looking problem into an easy one.

Conversions to KNOW COLD

FractionDecimalPercent
1/20.550%
1/3≈ 0.33333⅓%
2/3≈ 0.66766⅔%
1/40.2525%
3/40.7575%
1/50.220%
2/50.440%
3/50.660%
4/50.880%
1/80.12512.5%
3/80.37537.5%
5/80.62562.5%
7/80.87587.5%
THE TRICK

When the denominator divides 100 cleanly (4, 5, 10, 20, 25, 50, 100), scale to 100 instead of long-dividing.

Example. Convert 2/25 to a percent. Multiply top and bottom by 4: 2/25 = 8/100 = 0.08 = 8%. Done in 2 seconds. The slow way (long division: 2 ÷ 25) takes 30 seconds.

When the denominator is 2 or 10, you can read the decimal directly. When it's 8 or 16, scale to 1000.

WORKED EXAMPLE
PROBLEM · 1987 #2

2 ⁄ 25 =

A) .008 B) .08 C) .8 D) 1.25 E) 12.5

Convert 2/25 to a decimal.

Scale the denominator to 100: multiply top and bottom by 4.

2/25 = 8/100 = 0.08.

The slow approach is to do 2 ÷ 25 by long division. The fast approach: recognize that 25 × 4 = 100, and just multiply both top and bottom by 4. Now the decimal is automatic.

Answer: B — .08.
RULE OF THUMB

Memorize the common conversions. When dividing by 4, 5, 20, 25, 50, scale the denominator to 100. When dividing by 2 or 10, read directly. Long division is the last resort.

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1989 · #3 Which of the following numbers is the largest?

Which of the following numbers is the largest?

Show answer
Answer: A — .99.
Show hints
Hint 1 of 2
Compare the digits right after the decimal point.
Still stuck? Show hint 2 →
Hint 2 of 2
.99 has a 9 in the hundredths place; the others have 0 there.
Show solution
Approach: compare place by place
  1. All start with .9; in the hundredths place .99 has a 9 while the rest have 0.
  2. So .99 is largest.
1997 · #3 Which of the following numbers is the largest?

Which of the following numbers is the largest?

Show answer
Answer: B — 0.979.
Show hints
Hint 1 of 2
Line the decimals up and compare them place by place.
Still stuck? Show hint 2 →
Hint 2 of 2
0.97, 0.979, and 0.9709 stay tied until the thousandths digit.
Show solution
Approach: compare place by place
  1. All have 9 tenths; A, B, C also have 7 hundredths (D and E have 0), so only those three can be largest.
  2. In the thousandths place 0.979 has a 9 while 0.97 and 0.9709 have 0, so 0.979 is largest.
1998 · #5 Which of the following numbers is largest?

Which of the following numbers is largest?

Show answer
Answer: B — 9.1234̄ (B).
Show hints
Hint 1 of 2
They all agree through 9.1234 — compare the digits that come after.
Still stuck? Show hint 2 →
Hint 2 of 2
A repeating 4 keeps adding 4s, beating a number that simply stops.
Show solution
Approach: compare digit by digit past the common start
  1. All five begin 9.1234. The next digit is 4 for both A and B, but smaller for C, D, and E.
  2. A stops at 9.12344 while B continues 9.123444…, so B is the largest.
2018 · #10 The harmonic mean of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic...

The harmonic mean of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic mean of 1, 2, and 4?

Show answer
Answer: C — 12/7.
Show hints
Hint 1 of 2
Three steps: take reciprocals, average, take reciprocal again.
Still stuck? Show hint 2 →
Hint 2 of 2
Reciprocals: 1, 1/2, 1/4. Sum = 7/4. Average = 7/12. Final reciprocal: 12/7.
Show solution
Approach: follow the definition
  1. Reciprocals: 1, 1/2, 1/4. Sum: 1 + 1/2 + 1/4 = 7/4.
  2. Average of 3 reciprocals: (7/4)/3 = 7/12.
  3. Harmonic mean = reciprocal: 12/7.
2022 · #8 What is the value of13 · 24 · 35 · … · 1820 · 1921 · 2022 ?

What is the value of

13 · 24 · 35 · … · 1820 · 1921 · 2022 ?
Show answer
Answer: B — 1/231.
Show hints
Hint 1 of 2
Every number 3 through 20 appears once as a numerator and once as a denominator. What cancels?
Still stuck? Show hint 2 →
Hint 2 of 2
After cancellation: only 1 · 2 (from numerators) and 21 · 22 (from denominators) survive.
Show solution
Approach: telescoping cancellation
  1. Numerators run 1, 2, 3, …, 20. Denominators run 3, 4, 5, …, 22. Every number from 3 to 20 appears in both lists and cancels.
  2. Left over: 1 · 221 · 22 = 2462 = 1231.
2023 · #10 Harold made a plum pie to take on a picnic. He was able to eat only 14 of the pie, and he left the rest for his friends. A moose came by...

Harold made a plum pie to take on a picnic. He was able to eat only 14 of the pie, and he left the rest for his friends. A moose came by and ate 13 of what Harold left behind. After that, a porcupine ate 13 of what the moose left behind. How much of the original pie still remained after the porcupine left?

Show answer
Answer: D — 1/3.
Show hints
Hint 1 of 2
Each eater leaves behind a fraction of what they found. Multiply those leftovers together.
Still stuck? Show hint 2 →
Hint 2 of 2
Harold leaves 3/4, moose leaves 2/3, porcupine leaves 2/3. Multiply: 3/4 × 2/3 × 2/3.
Show solution
Approach: multiply the 'leftover' fractions
  1. Each step's leftover is (1 − what they ate). Harold leaves 34. Moose leaves 23 of that. Porcupine leaves 23 of that.
  2. 34 × 23 × 23 = 1236 = 13.
Another way — twelve slices (MAA):
  1. Cut the pie into 12 equal slices. Harold eats 3, leaving 9. Moose eats 13 of 9 = 3, leaving 6. Porcupine eats 13 of 6 = 2, leaving 4.
  2. 4 of 12 = 1/3.
1989 · #2 210 + 4100 + 61000 =

210 + 4100 + 61000 =

Show answer
Answer: D — .246.
Show hints
Hint 1 of 2
Each fraction is a decimal in its own place: tenths, hundredths, thousandths.
Still stuck? Show hint 2 →
Hint 2 of 2
Just write the digits in order.
Show solution
Approach: place each digit in its decimal slot
  1. 2/10 = 0.2, 4/100 = 0.04, 6/1000 = 0.006.
  2. Adding gives .246.
CHAPTER 2

Percent is a multiplier

THEORY

The single biggest mental shift on this topic: stop thinking of percent as a noun and start thinking of it as a multiplier.

PERCENT AS MULTIPLIER

  • A is p% of B means A = (p/100) × B.
  • Increase B by p% means new B = (1 + p/100) × B.
  • Decrease B by p% means new B = (1 − p/100) × B.

So +25% becomes ×1.25 and −25% becomes ×0.75. A 50% discount is ×0.5. A 10% raise is ×1.10. Once you do this conversion, percent arithmetic stops being arithmetic and becomes plain multiplication.

Concrete example. A $180 coat at 50% off:

$180 × 0.5 = $90 — done. No 'find half of 180 and subtract' two-step.

Another. A $40 lunch with 8% sales tax:

$40 × 1.08 = $43.20 — one multiplication.

THE TRICK

When percent off is followed by another percent off (or a tax added), the multipliers compose — just multiply them together.

50% off then 20% off: ×0.5 × ×0.8 = ×0.4. The combined effect is ×0.4 = 60% off, not 70% off. (The trap: 50 + 20 = 70.)

WORKED EXAMPLE
PROBLEM · 1986 #15

Sale prices at the Ajax Outlet Store are 50% below original prices. On Saturdays an additional discount of 20% off the sale price is given. What is the Saturday price of a coat whose original price is $180?

A) $54 B) $72 C) $90 D) $108 E) $110

Original price $180. The 50%-off sale: $180 × 0.5 = $90. The Saturday additional 20%: $90 × 0.8 = $72.

Or combined in one line: $180 × 0.5 × 0.8 = $180 × 0.4 = $72.

The second discount is taken off the sale price, not the original. That's why it's ×0.8 of $90, not ×0.8 of $180. AMC loves this. Read carefully: 'X% off the sale price' means stacked multipliers.

Answer: B — $72.
RULE OF THUMB

Every percent change is a multiplier. Sequential changes multiply. Never add or subtract percents directly when they're applied at different times.

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1992 · #4 During the softball season, Judy had 35 hits. Among her hits were 1 home run, 1 triple, and 5 doubles. The rest of her hits were...

During the softball season, Judy had 35 hits. Among her hits were 1 home run, 1 triple, and 5 doubles. The rest of her hits were singles. What percent of her hits were singles?

Show answer
Answer: E — 80%.
Show hints
Hint 1 of 2
Count the non-single hits first and subtract.
Still stuck? Show hint 2 →
Hint 2 of 2
Then write the singles as a fraction of 35.
Show solution
Approach: singles over total
  1. Non-singles: 1 + 1 + 5 = 7, so singles = 35 − 7 = 28.
  2. 28/35 = 80%.
1992 · #11 (figure problem)
ajhsme-1992-11
Show answer
Answer: B — 24%.
Show hints
Hint 1 of 2
Read every bar's frequency and add for the total.
Still stuck? Show hint 2 →
Hint 2 of 2
Blue's frequency over the total gives the percent.
Show solution
Approach: blue over the total of all bars
  1. The frequencies are 50, 60, 40, 60, 40, summing to 250.
  2. Blue is 60, so 60/250 = 24%.
2000 · #10 Ara and Shea were once the same height. Since then Shea has grown 20% while Ara has grown half as many inches as Shea. Shea is now 60...

Ara and Shea were once the same height. Since then Shea has grown 20% while Ara has grown half as many inches as Shea. Shea is now 60 inches tall. How tall, in inches, is Ara now?

Show answer
Answer: E — 55 inches.
Show hints
Hint 1 of 2
Find the common starting height from Shea's 20% growth.
Still stuck? Show hint 2 →
Hint 2 of 2
Ara grew half as many inches as Shea — match inches, not percents.
Show solution
Approach: back out the starting height, then add Ara's inches
  1. Shea grew 20% to reach 60, so the start was 60 ÷ 1.2 = 50 inches, meaning Shea grew 10 inches.
  2. Ara grew half that, 5 inches, ending at 50 + 5 = 55 inches.
2001 · #10 A collector offers to buy state quarters for 2000% of their face value. At that rate, how much will Bryden get for his four state quarters?

A collector offers to buy state quarters for 2000% of their face value. At that rate, how much will Bryden get for his four state quarters?

Show answer
Answer: A — $20.
Show hints
Hint 1 of 2
Four quarters is $1 of face value.
Still stuck? Show hint 2 →
Hint 2 of 2
2000% means 2000 ÷ 100 = 20 times.
Show solution
Approach: percent as a multiplier
  1. Four quarters have a face value of $1, and 2000% is 2000 ÷ 100 = 20 times that.
  2. So Bryden gets 20 × $1 = $20.
2002 · #7 The students in Mrs. Sawyer's class each chose one of five kinds of candy in a taste test. The bar graph shows their preferences. What...

The students in Mrs. Sawyer's class each chose one of five kinds of candy in a taste test. The bar graph shows their preferences. What percent of her class chose candy E?

SWEET TOOTH012345678ABCDEKinds of candyNumber of students
Show answer
Answer: E — 20%.
Show hints
Hint 1 of 2
First find how many students there are in all.
Still stuck? Show hint 2 →
Hint 2 of 2
With 25 students total, each one is 4% of the class.
Show solution
Approach: part over whole, then turn into a percent
  1. The class total is 6 + 8 + 4 + 2 + 5 = 25 students.
  2. Candy E was chosen by 5 of them: 5/25 = 1/5 = 20%.
2003 · #11 Business is a little slow at Lou's Fine Shoes, so Lou decides to have a sale. On Friday, Lou increases all of Thursday's prices by 10%....

Business is a little slow at Lou's Fine Shoes, so Lou decides to have a sale. On Friday, Lou increases all of Thursday's prices by 10%. Over the weekend, Lou advertises the sale: "Ten percent off the listed price. Sale starts Monday." How much does a pair of shoes cost on Monday that cost $40 on Thursday?

Show answer
Answer: B — $39.60.
Show hints
Hint 1 of 2
A 10% increase followed by a 10% decrease is not a wash — the cut is taken off a bigger number.
Still stuck? Show hint 2 →
Hint 2 of 2
Multiply the two factors: ×1.1 then ×0.9.
Show solution
Approach: chain the two percent changes
  1. Friday: 40 × 1.1 = 44. Monday: 44 × 0.9 = 39.60.
  2. Equivalently 40 × 1.1 × 0.9 = 40 × 0.99 = 39.60 — a hair under the original.
CHAPTER 3

Compound percent — when +25% and −20% return to start

THEORY

This chapter handles the most famous AMC percent trap: going up by p% then down by p% does NOT return to the original.

Pick a number. Say $100. Raise by 25%: $125. Now drop $125 by 25%: $125 × 0.75 = $93.75. You lost $6.25.

Why? Because the 25% down is taken from the raised price ($125), not the original ($100). A bigger base means a bigger discount in dollars.

COMPOUND PERCENT

+p% then +q% is ×(1+p/100)(1+q/100). Multiply the multipliers.

The result is never just +(p+q)%, except for the trivial case p=0 or q=0.

Two important special cases:

  • +25%, then −20%: ×1.25 × ×0.8 = ×1.00. Returns to start. Because 0.8 = 1/1.25.
  • +10% four times: ×(1.1)⁴ = ×1.4641 — about 46.4%, NOT 40%.

The trap is reasoning additively. Always multiply.

THE TRICK

If a question says 'percent up p, percent down q, what's net?', compute (1 + p/100)(1 − q/100) directly. Don't add or subtract.

WORKED EXAMPLE
PROBLEM · 1988 #22

Tom's Hat Shoppe increased all original prices by 25%. Now the shoppe is having a sale where all prices are 20% off these increased prices. Which statement best describes the sale price of an item?

A) The sale price is 5% higher than the original price. B) The sale price is higher than the original price, but by less than 5%. C) The sale price is higher than the original price, but by more than 5%. D) The sale price is lower than the original price. E) The sale price is the same as the original price.

+25% is ×1.25. −20% off the new price is ×0.8. Combined multiplier:

1.25 × 0.8 = 1.00.

The sale price equals the original. The 25% up and 20% down exactly cancel because 0.8 = 1/1.25.

The pair (+25%, −20%) is one of the AMC's favorite traps because the numbers look different (one is bigger). Once you see ×1.25 and ×0.8 as reciprocals, the answer is instant.

Answer: E — The sale price is the same as the original price.
RULE OF THUMB

+1/n turns into a ×(1 + 1/n). Its undo is ÷(1 + 1/n), which equals ×(1 − 1/(n+1)). So +1/n undoes with −1/(n+1). +25%=+1/4 undoes with −1/5 = −20%. Memorize this pattern.

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1996 · #18 Ana's monthly salary was $2000 in May. In June she received a 20% raise. In July she received a 20% pay cut. After the two changes in...

Ana's monthly salary was $2000 in May. In June she received a 20% raise. In July she received a 20% pay cut. After the two changes in June and July, Ana's monthly salary was

Show answer
Answer: A — 1920 dollars.
Show hints
Hint 1 of 2
A 20% raise then a 20% cut do NOT cancel out.
Still stuck? Show hint 2 →
Hint 2 of 2
Multiply by 1.2, then by 0.8.
Show solution
Approach: apply both percent changes in turn
  1. After the raise: 2000 × 1.2 = 2400. After the cut: 2400 × 0.8 = 1920.
  2. So her salary is $1920.
1990 · #9 (figure problem)
ajhsme-1990-09
Show answer
Answer: D — 33⅓%.
Show hints
Hint 1 of 2
A C is a score from 75 to 84 — count how many of the 15 scores fall there.
Still stuck? Show hint 2 →
Hint 2 of 2
Then divide by 15.
Show solution
Approach: count C scores, then take the percent
  1. Scores in 75–84 are 77, 75, 84, 78, 80 — that's 5 of the 15.
  2. 5/15 = 33⅓%.
1991 · #18 (figure problem)
ajhsme-1991-18
Show answer
Answer: C — 30%.
Show hints
Hint 1 of 2
Even with no scale, each X stands for the same number of employees, so count X's.
Still stuck? Show hint 2 →
Hint 2 of 2
Add the X's for 5 years or more and divide by the total number of X's.
Show solution
Approach: count symbols, then take the fraction
  1. Each X represents the same number of employees, so the percent is just (X's for 5+ years) ÷ (total X's).
  2. Counting the columns, that fraction comes out to 30%.
1997 · #16 Penni buys $100 of stock in each of three companies: AA, BB, and CC. After one year AA is up 20%, BB is down 25%, and CC is unchanged....

Penni buys $100 of stock in each of three companies: AA, BB, and CC. After one year AA is up 20%, BB is down 25%, and CC is unchanged. In the second year AA drops 20% from its new value, BB rises 25% from its new value, and CC is unchanged. If A, B, C are the final values, which ordering is correct?

Show answer
Answer: E — B < A < C.
Show hints
Hint 1 of 2
Track each $100 through both years.
Still stuck? Show hint 2 →
Hint 2 of 2
A 20% rise followed by a 20% fall does not return to the start.
Show solution
Approach: follow each $100 year by year
  1. AA: 100 → 120 → 96. BB: 100 → 75 → 93.75. CC stays 100.
  2. So 93.75 < 96 < 100, i.e. B < A < C.
1985 · #21 Mr. Green receives a 10% raise every year. His salary after four such raises has gone up by what percent?

Mr. Green receives a 10% raise every year. His salary after four such raises has gone up by what percent?

Show answer
Answer: E — more than 45%.
Show hints
Hint 1 of 2
Each year multiplies by 1.10 — apply that four times.
Still stuck? Show hint 2 →
Hint 2 of 2
1.10⁴ is more than just 1 + 4(0.10) = 1.40 because compounding earns interest on interest.
Show solution
Approach: compute (1.10)⁴
  1. (1.10)⁴ = 1.4641 — that's about 46.4% above the original.
  2. Greater than 45%, so the answer is more than 45%.
1989 · #21 Jack had a bag of 128 apples. He sold 25% of them to Jill. Next he sold 25% of those remaining to June. Of those apples still in his...

Jack had a bag of 128 apples. He sold 25% of them to Jill. Next he sold 25% of those remaining to June. Of those apples still in his bag, he gave the shiniest one to his teacher. How many apples did Jack have then?

Show answer
Answer: D — 71.
Show hints
Hint 1 of 2
Selling 25% means keeping 75% = 3⁄4.
Still stuck? Show hint 2 →
Hint 2 of 2
Multiply by 3⁄4 twice, then subtract 1 for the apple given to the teacher.
Show solution
Approach: keep ¾ twice, then subtract 1
  1. After Jill: 128 × 3⁄4 = 96. After June: 96 × 3⁄4 = 72.
  2. Give 1 to the teacher: 72 − 1 = 71.
★ MINI-QUIZ

Percent basics — multipliers and compounds

Three problems on +p% = ×(1+p/100) and stacking percent changes.

2020 · #5 Three fourths of a pitcher is filled with pineapple juice. The pitcher is emptied by pouring an equal amount of juice into each of 5...

Three fourths of a pitcher is filled with pineapple juice. The pitcher is emptied by pouring an equal amount of juice into each of 5 cups. What percent of the total capacity of the pitcher did each cup receive?

Show answer
Answer: C — 15%.
Show hints
Hint 1 of 2
Split 34 into 5 equal shares — what fraction of the whole pitcher is each share?
Still stuck? Show hint 2 →
Hint 2 of 2
Each cup gets 34 ÷ 5 = 320 of the pitcher. Now turn that into a percent.
Show solution
Approach: fraction split equally into percent
  1. Each cup gets 34 ÷ 5 = 320 of the pitcher.
  2. 320 = 15100 = 15%.
2025 · #17 (figure problem)
amc8-2025-17
Show answer
Answer: D — 115 people.
Show hints
Hint 1 of 2
Workers in A come from all three cities. Tally each city's contribution.
Still stuck? Show hint 2 →
Hint 2 of 2
From A, the workers who stay are everyone not leaving for B or C. From B and C, just multiply by the labeled fraction.
Show solution
Approach: sum the contribution from each home city
  1. From A → A: those who don't leave. 100 − 100×14 − 100×15 = 100 − 25 − 20 = 55.
  2. From B → A: 120 × 13 = 40.
  3. From C → A: 160 × 18 = 20.
  4. Total working in A: 55 + 40 + 20 = 115.
2019 · #8 Gilda has a bag of marbles. She gives 20% of them to her friend Pedro. Then Gilda gives 10% of what is left to another friend, Ebony....

Gilda has a bag of marbles. She gives 20% of them to her friend Pedro. Then Gilda gives 10% of what is left to another friend, Ebony. Finally, Gilda gives 25% of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?

Show answer
Answer: E — 54%.
Show hints
Hint 1 of 2
Each transfer leaves a fraction behind. Multiply the "keep" fractions together.
Still stuck? Show hint 2 →
Hint 2 of 2
Keeps: 0.8 × 0.9 × 0.75.
Show solution
Approach: chain the leftover fractions
  1. After Pedro: 80% remains. After Ebony: 90% of that. After Jimmy: 75% of what's left.
  2. 0.8 × 0.9 × 0.75 = 0.54 = 54%.
CHAPTER 4

Comparing fractions without computing them

THEORY

If a problem asks 'which fraction is largest?' and you reach for a calculator, you're working too hard. Three faster habits:

FRACTION-COMPARISON TRICKS

  • Compare to ½. A fraction a/b is bigger than ½ exactly when 2a > b. Test mentally.
  • Compare two fractions by cross-multiplying. a/b > c/d exactly when ad > bc (assuming b, d positive). No common denominator needed.
  • Compare to 1. Top bigger than bottom → fraction > 1; top smaller → < 1.

The number-line picture. Imagine a number line from 0 to 1, with ½ marked in the middle. Every fraction lives somewhere on this line. Comparing to ½ tells you which half it's in.

0½13/74/917/35100/201151/301

Four of these five fractions sit just below ½. Only 151/301 crosses over.

How to compare to ½ in your head. Double the top; if it's bigger than the bottom, the fraction beats ½.

  • Is 151/301 bigger than ½? Double 151 = 302. Compare to 301: 302 > 301, so YES, 151/301 > ½.
  • Is 100/201 bigger than ½? Double 100 = 200. Compare to 201: 200 < 201, so 100/201 < ½.

For five-way comparisons, use the landmark idea: compare each candidate to ½ first. Any below ½ is eliminated immediately if a candidate above ½ exists.

THE TRICK

For a fraction a/b close to ½, look at 2a − b: positive means > ½, negative means < ½. Quick mental check.

WORKED EXAMPLE
PROBLEM · 1987 #13

Which of the following fractions has the largest value?

A) 3⁄7 B) 4⁄9 C) 17⁄35 D) 100⁄201 E) 151⁄301

Check each fraction against ½ (i.e., is 2a > b?):

  • 3/7: 2·3 = 6 < 7 → less than ½.
  • 4/9: 2·4 = 8 < 9 → less than ½.
  • 17/35: 2·17 = 34 < 35 → less than ½.
  • 100/201: 2·100 = 200 < 201 → less than ½.
  • 151/301: 2·151 = 302 > 301 → bigger than ½ ✓

Only one fraction exceeds ½ — that's the biggest. Answer: 151/301. Took 30 seconds, no division.

Each comparison to ½ is one mental multiplication and one comparison. Five quick mental checks beat any actual division.

Answer: E — 151⁄301.
RULE OF THUMB

To compare fractions, compare to ½ first as a sieve. Then cross-multiply pairs if needed. Decimals as a last resort.

MORE LIKE THIS
1992 · #2 Which of the following is not equal to 54?

Which of the following is not equal to 54?

Show answer
Answer: D — 1 1/5.
Show hints
Hint 1 of 2
5/4 = 1.25 — convert each choice and compare.
Still stuck? Show hint 2 →
Hint 2 of 2
A mixed number like 1 1/5 equals 1.2.
Show solution
Approach: convert everything to a decimal
  1. 5/4 = 1.25. The choices 10/8, 1¼, 1 3/12, and 1 10/40 all equal 1.25.
  2. But 1 1/5 = 1.2, so 1 1/5 is the one that's not equal.
CHAPTER 5

Complex fractions — top first, bottom next, divide last

THEORY

A complex fraction has fractions stacked inside fractions:

(1 − 1/3) ÷ (1 − 1/2)

The temptation is to bash everything with one giant common denominator across all four pieces. Don’t. The clean habit is THREE small steps:

Simplify: (1 − 1/3) ÷ (1 − 1/2)Step 1: simplify the TOP1 − 1/3= 3/3 − 1/3= 2/3Step 2: simplify the BOTTOM1 − 1/2= 2/2 − 1/2= 1/2Now stack:2/31/2Step 3: divide (Keep, Change, Flip)2/3 ÷ 1/2start→2/3 × 2/1flip the second fraction=4/3multiply across=1⅓Three small operations beat one big one. NEVER try to find a common denominator across all four pieces at once.

THE RECIPE

  1. Simplify the TOP to one fraction.
  2. Simplify the BOTTOM to one fraction.
  3. Divide: Keep the first, Change ÷ to ×, Flip the second. (“KCF.”)

Why “flip the second” works

Dividing by 1/2 is the SAME as multiplying by 2. Because asking “how many halves fit?” doubles whatever you started with.

Division= Multiplication by flipped secondAnswer
5 ÷ 1/35 × 315
6 ÷ 2/36 × 3/29
(3/4) ÷ (2/5)(3/4) × (5/2)15/8
10 ÷ 1/410 × 440
The TRAP: dividing by a fraction makes the number BIGGER (when the divisor is < 1). Kids expect division to make things smaller.
THE TRICK

Dividing by a fraction is multiplying by its flip (reciprocal). ÷ (1/n) = × n. So 5 ÷ (1/3) = 15, not 5/3.

Memorize the rule: Keep, Change, Flip (KCF). Keep the first fraction, change ÷ to ×, flip the second fraction. (3/4) ÷ (2/5) = (3/4) × (5/2) = 15/8.

WORKED EXAMPLE
PROBLEM · 1989 #12
1 − 131 − 12=
A) 13 B) 23 C) 34 D) 32 E) 43

The expression is (1 − 1/3) / (1 − 1/2).

Top: 1 − 1/3 = 2/3. Bottom: 1 − 1/2 = 1/2.

Divide using Keep-Change-Flip: (2/3) ÷ (1/2) = (2/3) × (2/1) = 4/3.

Three small operations beat one big one. Never try to combine the top and bottom into a single ratio in your head — you'll lose a sign or factor.

Answer: E — 4⁄3.
RULE OF THUMB

Simplify top and bottom separately. Then KCF (Keep-Change-Flip) the bottom.

MORE LIKE THIS
1986 · #6 2 ⁄ (1 − 2⁄3) =

2 ⁄ (1 − 2⁄3) =

Show answer
Answer: E — 6.
Show hints
Hint 1 of 2
Simplify the denominator first.
Still stuck? Show hint 2 →
Hint 2 of 2
1 − 2⁄3 = 1⁄3.
Show solution
Approach: simplify denominator, then divide
  1. Denominator: 1 − 2⁄3 = 1⁄3.
  2. 2 ÷ 1⁄3 = 2 × 3 = 6.
CHAPTER 6

Multiplying fractions — telescoping products

THEORY

When you multiply many fractions in a row, you can cancel across: any factor on the top of one fraction cancels with the same factor on the bottom of another, even fractions away.

Example: (2/3) × (3/4) × (4/5). The 3s cancel; the 4s cancel. You're left with 2/5. Two mental cancellations beat three full multiplications.

TELESCOPING

When a chain of fractions has each numerator matching the previous denominator, all the middle terms cancel and you're left with just first-top divided by last-bottom.

Classic case: the product (1 − 1/n) from n = 2 to N.

(1 − 1/2)(1 − 1/3)(1 − 1/4) ⋯ (1 − 1/N) = (1/2)(2/3)(3/4) ⋯ ((N−1)/N) = 1/N

Why? Look at consecutive factors: the 2 on top of 2/3 cancels the 2 on the bottom of 1/2. The 3 on top of 3/4 cancels the 3 on the bottom of 2/3. The 4 on top of 4/5 cancels the 4 on the bottom of 3/4. Each numerator (except the first) kills the denominator on its left, so everything in the middle cancels — leaving the very first numerator (1) over the very last denominator (N).

So a product of 1000 fractions of this kind collapses to a single fraction in one move.

Sums can telescope too

The same cancellation idea works when you’re adding fractions, not just multiplying them — you just need to rewrite each fraction as a difference of two pieces first. Here’s the most useful identity:

SPLIT IDENTITY (the partial-fraction crack)

1 / (n · (n+1)) = 1/n − 1/(n+1)

Each fraction of the form “one over consecutive product” splits into a difference of two unit fractions.

Don’t take it on faith — just compute a few:

  • 1/(1·2) = 1/2 — and 1/1 − 1/2 = 1/2 ✓
  • 1/(2·3) = 1/6 — and 1/2 − 1/3 = 1/6 ✓
  • 1/(3·4) = 1/12 — and 1/3 − 1/4 = 1/12 ✓

So a sum like 1/(1·2) + 1/(2·3) + 1/(3·4) + … + 1/(99·100) rewrites as:

(1/1 − 1/2) + (1/2 − 1/3) + (1/3 − 1/4) + … + (1/99 − 1/100).

Now look closely: every term in the middle cancels. The +1/2 cancels the −1/2, the +1/3 cancels the −1/3, and so on, all the way down. The only survivors are the very first piece (1/1 = 1) and the very last piece (−1/100).

Sum = 1 − 1/100 = 99/100.

Adding 99 fractions becomes a single subtraction. Whenever you see a sum that looks like “one over (consecutive product),” split each piece and let the cancellation do the work.

THE TRICK

Before multiplying any chain of fractions, write each as something/something and look for matches across. If the chain has structure like (k)/(k+1), it telescopes — only the very first top and very last bottom survive.

WORKED EXAMPLE
PROBLEM · 1985 #9

The product of the 9 factors (1 − 1⁄2)(1 − 1⁄3)(1 − 1⁄4) ⋯ (1 − 1⁄10) =

A) 1⁄10 B) 1⁄9 C) 1⁄2 D) 10⁄11 E) 11⁄2

Rewrite each factor:

(1 − 1/2)(1 − 1/3)(1 − 1/4) ⋯ (1 − 1/10) = (1/2)(2/3)(3/4)(4/5)(5/6)(6/7)(7/8)(8/9)(9/10).

Each numerator matches the previous denominator. Cancel down the line. The only survivors are the 1 in the very first numerator and the 10 in the very last denominator:

= 1/10.

Once you see this pattern, problems of this form become 5-second answers regardless of how many factors there are. A product (1−1/2)…(1−1/100) is just 1/100.

Answer: A — 1⁄10.
RULE OF THUMB

In a product of fractions, cancel across before you multiply. If the chain has the (k)/(k+1) shape, only first-top and last-bottom remain.

MORE LIKE THIS
1992 · #25 One half of the water is poured out of a full container. Then one third of the remainder is poured out. Continue the process: one fourth...

One half of the water is poured out of a full container. Then one third of the remainder is poured out. Continue the process: one fourth of the remainder for the third pouring, one fifth of the remainder for the fourth pouring, and so on. After how many pourings does exactly one tenth of the original water remain?

Show answer
Answer: D — 9.
Show hints
Hint 1 of 2
After each pouring, multiply by what's left: 1/2, then 2/3, then 3/4, …
Still stuck? Show hint 2 →
Hint 2 of 2
These products telescope to a simple fraction.
Show solution
Approach: multiply the surviving fractions (they telescope)
  1. After k pourings the remaining fraction is ½ · ⅔ · ¾ · … · k/(k+1), which telescopes to 1/(k+1).
  2. Setting 1/(k+1) = 1/10 gives k = 9.
★ MINI-QUIZ

Fraction arithmetic

Three fraction problems: a comparison, a complex fraction, and a telescoping product. Resist long-multiplying.

1992 · #2 Which of the following is not equal to 54?

Which of the following is not equal to 54?

Show answer
Answer: D — 1 1/5.
Show hints
Hint 1 of 2
5/4 = 1.25 — convert each choice and compare.
Still stuck? Show hint 2 →
Hint 2 of 2
A mixed number like 1 1/5 equals 1.2.
Show solution
Approach: convert everything to a decimal
  1. 5/4 = 1.25. The choices 10/8, 1¼, 1 3/12, and 1 10/40 all equal 1.25.
  2. But 1 1/5 = 1.2, so 1 1/5 is the one that's not equal.
1986 · #6 2 ⁄ (1 − 2⁄3) =

2 ⁄ (1 − 2⁄3) =

Show answer
Answer: E — 6.
Show hints
Hint 1 of 2
Simplify the denominator first.
Still stuck? Show hint 2 →
Hint 2 of 2
1 − 2⁄3 = 1⁄3.
Show solution
Approach: simplify denominator, then divide
  1. Denominator: 1 − 2⁄3 = 1⁄3.
  2. 2 ÷ 1⁄3 = 2 × 3 = 6.
1992 · #25 One half of the water is poured out of a full container. Then one third of the remainder is poured out. Continue the process: one fourth...

One half of the water is poured out of a full container. Then one third of the remainder is poured out. Continue the process: one fourth of the remainder for the third pouring, one fifth of the remainder for the fourth pouring, and so on. After how many pourings does exactly one tenth of the original water remain?

Show answer
Answer: D — 9.
Show hints
Hint 1 of 2
After each pouring, multiply by what's left: 1/2, then 2/3, then 3/4, …
Still stuck? Show hint 2 →
Hint 2 of 2
These products telescope to a simple fraction.
Show solution
Approach: multiply the surviving fractions (they telescope)
  1. After k pourings the remaining fraction is ½ · ⅔ · ¾ · … · k/(k+1), which telescopes to 1/(k+1).
  2. Setting 1/(k+1) = 1/10 gives k = 9.
CHAPTER 7

Fraction of an unknown — keep-fractions

THEORY

Half the people in a room left. One third of those remaining started to dance. Twelve people weren't dancing. How many people started?

This kind of story chains fractions of fractions of an unknown. There are two clean ways through.

Forward (keep-fractions): at each step, track the fraction that remains (or whatever subgroup you're after). Multiply those fractions to get the fraction of the original that's left at the end. Then divide the final count by that.

Backward: start from the end and undo each operation. If 'one third started dancing' means 'two thirds didn't', then 12 non-dancers ÷ (2/3) = 18 of the remaining people. Then undoing 'half left' (so half remained) gives 18 × 2 = 36 original people.

Both work. Pick whichever has fewer arithmetic steps.

THE TRICK

When tracking a subgroup through a story, multiply the fractions that preserve it. Not-dancers = (½ remained) × (⅔ didn't dance) = ⅓ of original.

WORKED EXAMPLE
PROBLEM · 1987 #18

Half the people in a room left. One third of those remaining started to dance. There were then 12 people who were not dancing. The original number of people in the room was what?

A) 24 B) 30 C) 36 D) 42 E) 72

After half leaves: half remain. Of those, ⅔ are not dancing. So non-dancers = (½)(⅔) = ⅓ of the original.

If non-dancers = 12 people, then (1/3) × N = 12, so N = 36.

Notice you never had to compute the intermediate count (how many remained after the first half left). The chain of fractions handles it. This generalizes: any 'fraction of fraction of fraction' story collapses to one multiplication.

Answer: C — 36.
RULE OF THUMB

Multiply keep-fractions to track a subgroup through several stages. Then use the final count to solve for the unknown start.

MORE LIKE THIS
1997 · #18 Last week small boxes of facial tissue were priced at 4 boxes for $5. This week they are on sale at 5 boxes for $4. The percent decrease...

Last week small boxes of facial tissue were priced at 4 boxes for $5. This week they are on sale at 5 boxes for $4. The percent decrease in the price per box during the sale was closest to

Show answer
Answer: B — About 35%.
Show hints
Hint 1 of 2
Find the price of one box before and during the sale.
Still stuck? Show hint 2 →
Hint 2 of 2
Percent decrease compares the drop to the original price.
Show solution
Approach: per-box price before vs. during the sale
  1. A box cost $5 ÷ 4 = $1.25 before and $4 ÷ 5 = $0.80 on sale.
  2. The decrease is $0.45 ÷ $1.25 = 36%, closest to 35%.
1992 · #14 When four gallons are added to a tank that is one-third full, the tank is then one-half full. The capacity of the tank in gallons is

When four gallons are added to a tank that is one-third full, the tank is then one-half full. The capacity of the tank in gallons is

Show answer
Answer: D — 24.
Show hints
Hint 1 of 2
The 4 gallons raised the level from 1/3 to 1/2 — what fraction is that?
Still stuck? Show hint 2 →
Hint 2 of 2
Then scale up to the full tank.
Show solution
Approach: the added fraction gives the whole
  1. 1/2 − 1/3 = 1/6 of the tank equals 4 gallons.
  2. So the full tank is 6 × 4 = 24 gallons.
1996 · #10 When Walter drove up to the gasoline pump, his tank was 1/8 full. He bought 7.5 gallons, after which the tank was 5/8 full. How many...

When Walter drove up to the gasoline pump, his tank was 1/8 full. He bought 7.5 gallons, after which the tank was 5/8 full. How many gallons does the tank hold when it is full?

Show answer
Answer: D — 15 gallons.
Show hints
Hint 1 of 2
The 7.5 gallons raised the level from 1/8 to 5/8 — what fraction of the tank is that?
Still stuck? Show hint 2 →
Hint 2 of 2
Then scale up to a full tank.
Show solution
Approach: fraction added gives the whole
  1. Going from 1/8 to 5/8 is 4/8 = half the tank, filled by 7.5 gallons.
  2. So a full tank holds 2 × 7.5 = 15 gallons.
1999 · #7 The third exit on a highway is located at milepost 40 and the tenth exit is at milepost 160. There is a service center on the highway...

The third exit on a highway is located at milepost 40 and the tenth exit is at milepost 160. There is a service center on the highway located three-fourths of the way from the third exit to the tenth exit. At what milepost would you expect to find this service center?

Show answer
Answer: E — Milepost 130.
Show hints
Hint 1 of 2
The two exits are 160 − 40 = 120 mileposts apart.
Still stuck? Show hint 2 →
Hint 2 of 2
Go three-fourths of that distance past milepost 40.
Show solution
Approach: fraction of the gap, added to the start
  1. From milepost 40 to 160 is 120 miles, and three-fourths of 120 is 90.
  2. So the center sits at 40 + 90 = 130.
1999 · #18 Cookies for a Crowd. The recipe makes a pan of 15 cookies, and only full recipes are made. Normally 108 students each eat 2 cookies, but...

Cookies for a Crowd. The recipe makes a pan of 15 cookies, and only full recipes are made. Normally 108 students each eat 2 cookies, but a concert cuts attendance by 25%. How many recipes should Walter and Gretel make for the smaller party?

Show answer
Answer: E — 11 recipes.
Show hints
Hint 1 of 2
A 25% drop leaves three-fourths of the 108 students.
Still stuck? Show hint 2 →
Hint 2 of 2
Find their cookies, then round up to whole pans of 15.
Show solution
Approach: fewer guests → cookies → round up pans
  1. Three-fourths of 108 is 81 students, eating 81 × 2 = 162 cookies.
  2. That needs 162 ÷ 15 = 10.8 → 11 full recipes.
2026 · #7 Mika wants to estimate how far a new electric bike goes on a full charge. She made two trips totaling 40 miles: the first used 12 of the...

Mika wants to estimate how far a new electric bike goes on a full charge. She made two trips totaling 40 miles: the first used 12 of the battery and the second used 310 of the battery. How many miles can the bike go on a fully charged battery?

Show answer
Answer: C — 50 miles.
Show hints
Hint 1 of 2
Add the two battery fractions to see what share of a full charge covered the 40 miles.
Still stuck? Show hint 2 →
Hint 2 of 2
Then scale up from that share to a whole battery.
Show solution
Approach: scale up from the fraction used
  1. The two trips used ½ + 3/10 = 4/5 of the battery for 40 miles.
  2. A full battery covers 40 ÷ (4/5) = 50 miles.
CHAPTER 8

Weighted averages — when group sizes differ

THEORY

Two classes take the same test. Class A (20 kids) averages 80. Class B (30 kids) averages 70. What's the combined average?

The trap: 75 (the simple average of 80 and 70). Wrong, because the two classes aren't the same size.

WEIGHTED AVERAGE

Combined average = (total of all values) / (total count).

For two groups: (n_A × avg_A + n_B × avg_B) / (n_A + n_B).

For our example: (20·80 + 30·70) / (20+30) = (1600 + 2100)/50 = 3700/50 = 74. Closer to 70 (because Class B is bigger and pulls the average toward 70).

The combined average always lies between the two group averages, pulled toward the larger group. Two checks: the answer can't be less than 70 or more than 80, and it should be closer to whichever group is larger.

THE TRICK

For 'X% are boys, average height A; Y% are girls, average height B' problems, weighted average works in percent: combined avg = X% · A + Y% · B (since X% + Y% = 100%, no extra divide).

WORKED EXAMPLE
PROBLEM · 1995 #17

Annville Junior High has 100 students; 11% are in grade 6. Cleona Junior High has 200 students; 17% are in grade 6. What percent of the combined 300 students are in grade 6?

Convert each percent to a head count:

  • Annville: 11% × 100 = 11 grade-6 students.
  • Cleona: 17% × 200 = 34 grade-6 students.

Combined: 11 + 34 = 45 grade-6 students out of 300 total = 45/300 = 15%.

The simple average of 11% and 17% is 14% — close, but wrong. Cleona is the bigger school, so it pulls the combined percent toward its 17%. The right answer (15%) sits closer to 17% than to 11%, which is your sanity check.

Answer: D — 15%.
RULE OF THUMB

Combined average = combined total ÷ combined count. The answer is always between the two group averages, weighted toward the bigger group.

MORE LIKE THIS
1996 · #19 (figure problem)
ajhsme-1996-19
Show answer
Answer: C — 32%.
Show hints
Hint 1 of 2
Find the actual number of tennis fans at each school, not the percents.
Still stuck? Show hint 2 →
Hint 2 of 2
Combine and divide by the total of all students.
Show solution
Approach: count tennis fans, then divide by everyone
  1. Tennis fans: East 22% of 2000 = 440, West 40% of 2500 = 1000, totaling 1440.
  2. Out of 2000 + 2500 = 4500 students, that's 1440/4500 = 32%.
1997 · #13 Three bags of jelly beans contain 26, 28, and 30 beans. The fractions of yellow beans in the bags are 50%, 25%, and 20%, respectively....

Three bags of jelly beans contain 26, 28, and 30 beans. The fractions of yellow beans in the bags are 50%, 25%, and 20%, respectively. All three bags are poured into one bowl. Which of the following is closest to the percent of yellow beans in the bowl?

Show answer
Answer: A — About 31%.
Show hints
Hint 1 of 2
Count the yellow beans in each bag, then total them.
Still stuck? Show hint 2 →
Hint 2 of 2
Divide the yellow total by the grand total of beans.
Show solution
Approach: yellow total over grand total
  1. Yellow beans: 13 + 7 + 6 = 26, out of 26 + 28 + 30 = 84 beans.
  2. 26/84 ≈ 30.9%, closest to 31%.
2010 · #9 Ryan got 80% of the problems correct on a 25-problem test, 90% on a 40-problem test, and 70% on a 10-problem test. What percent of all...

Ryan got 80% of the problems correct on a 25-problem test, 90% on a 40-problem test, and 70% on a 10-problem test. What percent of all the problems did Ryan answer correctly?

Show answer
Answer: D — 84%.
Show hint
Hint 1
Count correct on each test, then divide by the total number of problems.
Show solution
Approach: total correct / total problems
  1. Correct: 0.8 · 25 + 0.9 · 40 + 0.7 · 10 = 20 + 36 + 7 = 63.
  2. Total: 25 + 40 + 10 = 75.
  3. 63 / 75 = 84%.
CHAPTER 9

Percent OF vs percent INCREASE

THEORY

One of the most consistent AMC #6–#15 traps is mixing up “A is what percent OF B” with “A is what percent MORE THAN B.” Two different sentences, two different formulas, two different answers.

Pictures kill the confusion. Take A = 50 and B = 40 (so A is bigger):

A = 50,   B = 40Q1: “A is what PERCENT OF B?”Compare A directly to B as a fraction of B.B = 40A = 50⇒ A ÷ B = 50/40= 125%Q2: “A is what percent MORE THAN B?”Compare only the EXTRA (A − B) to B.B = 40 (baseline)+10only this counts⇒ 10 ÷ 40= 25%Both correct — they answer different questions. The trap is picking the wrong one.
Same A and B. 125% of and 25% more than are both correct — they just answer different questions. The trap is picking the wrong one.

Phrase ↔ formula cheat-sheet

PhraseFormulaWith A=50, B=40
A is x% OF BA = (x/100) · B125% of 40 = 50 ✓
A is x% MORE than BA = (1 + x/100) · B25% more than 40 = 50 ✓
A is x% LESS than BA = (1 − x/100) · B(if A<B; e.g. 20% less than 50 = 40)
What percent IS A of B?= A/B × 10050/40 = 125%
What percent GREATER is A than B?= (A−B)/B × 10010/40 = 25%
The key word: “OF” means full ratio. “MORE/LESS” means the EXTRA over baseline.
THE TRICK

'A is what percent of B?' is just A/B × 100. 'A is what percent more than B?' is (A − B)/B × 100. Always identify which phrasing before computing.

WORKED EXAMPLE
PROBLEM · 2020 #15

Suppose 15% of x equals 20% of y. What percentage of x is y?

A) 5 B) 35 C) 75 D) 133⅓ E) 300

'Suppose 15% of x equals 20% of y. What percent of x is y?'

Translate the sentence into a multiplier equation: 0.15·x = 0.20·y. We want y/x.

Divide: y/x = 0.15 / 0.20 = 0.75 = 75%.

The trap is reading 'what percent of x is y' as 'how much MORE/LESS than x'. It's neither — it's the ratio y/x directly. 75% means y equals 0.75·x. (If the question had asked 'percent less than x', the answer would be 25%. Same numbers, different formula.)

Answer: C — 75%.
RULE OF THUMB

'of' = direct ratio. 'more than' / 'less than' = difference over original. Two phrasings, two formulas.

MORE LIKE THIS
2002 · #14 A merchant offers a large group of items at 30% off. Later, the merchant takes 20% off these sale prices and claims that the final price...

A merchant offers a large group of items at 30% off. Later, the merchant takes 20% off these sale prices and claims that the final price of these items is 50% off the original price. The total discount is

Show answer
Answer: B — 44%.
Show hints
Hint 1 of 2
Discounts don't simply add — track the fraction of the price you still pay.
Still stuck? Show hint 2 →
Hint 2 of 2
After 30% off you pay 0.7 of the price; the next 20% off pays 0.8 of that.
Show solution
Approach: multiply the fractions of price still paid
  1. After 30% off you pay 0.70 of the original; taking another 20% off pays 0.80 of that.
  2. So you pay 0.70 × 0.80 = 0.56 of the original — a 44% total discount, not 50%.
2017 · #14 Chloe and Zoe are both students in Ms. Demeanor's math class. Last night they each solved half of the problems in their homework...

Chloe and Zoe are both students in Ms. Demeanor's math class. Last night they each solved half of the problems in their homework assignment alone and then solved the other half together. Chloe had correct answers to only 80% of the problems she solved alone, but overall 88% of her answers were correct. Zoe had correct answers to 90% of the problems she solved alone. What was Zoe's overall percentage of correct answers?

Show answer
Answer: C — 93%.
Show hints
Hint 1 of 2
Pretend there are 100 problems — 50 alone, 50 together. The "together" half is the same for both girls. Find that.
Still stuck? Show hint 2 →
Hint 2 of 2
Chloe: 80% of 50 = 40 alone; 88% total = 88 correct; so together = 48 of 50.
Show solution
Approach: use 100 problems, isolate the joint half
  1. Set 100 problems: 50 alone + 50 together. Chloe alone: 0.8 × 50 = 40 correct. Chloe total: 88. So together: 88 − 40 = 48 correct.
  2. Zoe alone: 0.9 × 50 = 45 correct. Zoe together: same 48 (joint work).
  3. Zoe total: 45 + 48 = 93 out of 100 = 93%.
2026 · #4 Brynn's savings decreased by 20% in July, then increased by 50% of the new amount in August. Brynn's savings are now what percent of the...

Brynn's savings decreased by 20% in July, then increased by 50% of the new amount in August. Brynn's savings are now what percent of the original amount?

Show answer
Answer: E — 120%.
Show hints
Hint 1 of 2
Don't pick a starting amount — that's extra work. Each percent change is something you can just multiply by.
Still stuck? Show hint 2 →
Hint 2 of 2
Each percent change is just a multiplier — you don't even need a starting amount. Multiply the two.
Show solution
  1. Down 20% means × 0.8; up 50% means × 1.5.
  2. Multiply the changes: 0.8 × 1.5 = 1.2.
  3. 1.2 = 120% of the original.
2020 · #13 Jamal has a drawer containing 6 green socks, 18 purple socks, and 12 orange socks. After adding more purple socks, Jamal noticed that...

Jamal has a drawer containing 6 green socks, 18 purple socks, and 12 orange socks. After adding more purple socks, Jamal noticed that there is now a 60% chance that a sock randomly selected from the drawer is purple. How many purple socks did Jamal add?

Show answer
Answer: B — 9 purple socks added.
Show hints
Hint 1 of 2
If s purple socks are added, what fraction of the new drawer is purple? Set that equal to 0.6.
Still stuck? Show hint 2 →
Hint 2 of 2
(18 + s) / (36 + s) = 0.6.
Show solution
Approach: non-purple count is fixed at 18 — and that's 40% of the new total
  1. The green + orange socks (6 + 12 = 18) don't change. After adding purple, 60% of the drawer is purple, so the non-purple 18 socks make up the remaining 40%.
  2. New total = 18 / 0.4 = 45 socks. Started with 36, so Jamal added 45 − 36 = 9 purple socks.
Another way — set up the new probability and solve:
  1. If s purple are added, (18 + s) / (36 + s) = 0.6.
  2. Cross-multiply: 18 + s = 21.6 + 0.6s, so 0.4s = 3.6 and s = 9.
2009 · #8 The length of a rectangle is increased by 10% and the width is decreased by 10%. What percent of the old area is the new area?

The length of a rectangle is increased by 10% and the width is decreased by 10%. What percent of the old area is the new area?

Show answer
Answer: B — 99%.
Show hint
Hint 1
Multiply the two factors: 1.1 · 0.9 = ?
Show solution
Approach: multiply multipliers
  1. New area / old area = 1.1 · 0.9 = 0.99 = 99%.
2024 · #19 Jordan owns 15 pairs of sneakers. Three fifths of the pairs are red and the rest are white. Two thirds of the pairs are high-top and the...

Jordan owns 15 pairs of sneakers. Three fifths of the pairs are red and the rest are white. Two thirds of the pairs are high-top and the rest are low-top. The red high-top sneakers make up a fraction of the collection. What is the least possible value of this fraction?

Show answer
Answer: C — 4/15.
Show hints
Hint 1 of 2
To minimize the overlap of "red AND high-top", let the OTHER kind (white) soak up as many high-tops as it can.
Still stuck? Show hint 2 →
Hint 2 of 2
9 red, 6 white. 10 high-top, 5 low-top. Make all 6 whites high-top; only 4 high-top spots remain — those must be red.
Show solution
Approach: push white pairs into high-top to crowd out red
  1. Counts: 35 × 15 = 9 red, 6 white. 23 × 15 = 10 high-top, 5 low-top.
  2. Minimize red high-tops by letting all 6 white pairs be high-top. That accounts for 6 of the 10 high-top spots.
  3. The remaining 10 − 6 = 4 high-top pairs must be red. Fraction = 415.
⬢ FINAL TEST

Stretch test

Five harder FDP problems combining percent reasoning and fraction manipulation.

2016 · #12 Jefferson Middle School has the same number of boys and girls. 34 of the girls and 23 of the boys went on a field trip. What fraction of...

Jefferson Middle School has the same number of boys and girls. 34 of the girls and 23 of the boys went on a field trip. What fraction of the students on the field trip were girls?

Show answer
Answer: B — 9/17.
Show hint
Hint 1
Common denominator: 3/4 = 9/12 and 2/3 = 8/12. So per 12 girls there are 9 on the trip; per 12 boys, 8.
Show solution
Approach: common denominator gives the ratio
  1. Same group sizes ⇒ girls : boys = 1 : 1. Their trip fractions: 3/4 vs 2/3 → 9/12 vs 8/12.
  2. Ratio of girls : boys on the trip = 9 : 8.
  3. Fraction girls = 9/17.
2019 · #17 What is the value of the product(1·32·2)(2·43·3)(3·54·4) … (97·9998·98)(98·10099·99) ?

What is the value of the product

(1·32·2)(2·43·3)(3·54·4) … (97·9998·98)(98·10099·99) ?
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Answer: B — 50/99.
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Hint 1 of 2
Regroup. Each fraction is k(k+2)(k+1)2. Pull out a 1/2 from the first term and look for telescoping.
Still stuck? Show hint 2 →
Hint 2 of 2
Rewrite as (1/2)(3/2 · 2/3)(4/3 · 3/4) … (100/99) — most factors cancel.
Show solution
Approach: telescoping after pairing
  1. Each factor is k(k+2)(k+1)2 = kk+1 · k+2k+1.
  2. Product across k = 1 to 98: (1/2 · 3/2)(2/3 · 4/3)…(98/99 · 100/99). Pair telescopes: the kk+1 chain → 1/99 (left telescoping); the k+2k+1 chain → 100/2 (right telescoping).
  3. Product = (1/99) · (100/2) = 50/99.
2023 · #13 (figure problem)
amc8-2023-13
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Answer: D — 48 miles.
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Hint 1 of 2
Where, as a fraction of the route, are the 1st repair station and the 3rd water station?
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Hint 2 of 2
Repair stations split into thirds: 1st is at 1/3. Water stations split into eighths: 3rd is at 3/8. Difference is 2 miles.
Show solution
Approach: convert station positions to fractions of the route
  1. Let L be the race length. 2 repair stations evenly spaced between start and finish divide the route into thirds → the 1st repair is at L/3.
  2. 7 water stations evenly spaced divide the route into eighths → the 3rd water is at 3L/8.
  3. Their gap: 3L/8 − L/3 = (9L − 8L)/24 = L/24 = 2 miles.
  4. So L = 48.
2019 · #22 A store increased the original price of a shirt by a certain percent and then decreased the new price by the same amount. Given that the...

A store increased the original price of a shirt by a certain percent and then decreased the new price by the same amount. Given that the resulting price was 84% of the original price, by what percent was the price increased and decreased?

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Answer: E — 40%.
Show hints
Hint 1 of 2
Multiplying by (1+p) then (1−p) gives 1 − p2. That equals 0.84.
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Hint 2 of 2
p2 = 0.16.
Show solution
Approach: compose the two changes
  1. (1 + p)(1 − p) = 1 − p2 = 0.84.
  2. p2 = 0.16 ⇒ p = 0.4 = 40%.
2022 · #9 A cup of boiling water (212°F) is placed to cool in a room whose temperature remains constant at 68°F. Suppose the difference between...

A cup of boiling water (212°F) is placed to cool in a room whose temperature remains constant at 68°F. Suppose the difference between the water temperature and the room temperature is halved every 5 minutes. What is the water temperature, in degrees Fahrenheit, after 15 minutes?

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Answer: B — 86°F.
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Hint 1 of 2
Don't track the temperature directly — track the difference from room temp.
Still stuck? Show hint 2 →
Hint 2 of 2
Initial gap: 212 − 68 = 144. Halved three times in 15 minutes: 144 / 23 = 18. Add to room temp.
Show solution
Approach: track the gap, halve it each interval
  1. Initial gap above room: 212 − 68 = 144°F.
  2. 15 minutes = three 5-minute halvings: 144 → 72 → 36 → 18.
  3. Final temperature = 68 + 18 = 86°F.
APPENDIX

FDP quick-reference

Memorize these

CONVERSIONS TO MEMORIZE

  • 1/2 = 0.5 = 50%; 1/3 ≈ 0.333 = 33⅓%; 2/3 ≈ 0.667 = 66⅔%
  • 1/4 = 0.25 = 25%; 3/4 = 0.75 = 75%
  • 1/5 = 0.2 = 20%; 2/5 = 0.4 = 40%; 3/5 = 0.6 = 60%; 4/5 = 0.8 = 80%
  • 1/6 ≈ 0.167; 5/6 ≈ 0.833
  • 1/8 = 0.125 = 12.5%; 3/8 = 0.375; 5/8 = 0.625; 7/8 = 0.875
  • 1/9 ≈ 0.111; 1/11 ≈ 0.0909; 1/12 ≈ 0.0833
  • KCF for division: Keep, Change, Flip — keep the first, change ÷ to ×, flip the second.
  • Telescoping product: (1−1/2)(1−1/3)…(1−1/N) = 1/N.
  • Telescoping sum: 1/(n(n+1)) = 1/n − 1/(n+1).
  • +25% then −20% returns to start (because 1.25 × 0.8 = 1).
  • +1/n undoes with −1/(n+1). +25% (=+1/4) undoes with −20% (=−1/5).
Common traps
  • Adding percents from successive applications. 50% off then 20% off ≠ 70% off (it's 60% off). Multiply the multipliers.
  • +25% then −25% returns to less than the start. +25% then −20% returns exactly (because ×1.25 × ×0.8 = 1).
  • Averaging two averages without weighting. Use total ÷ count, not (a+b)/2 when groups differ.
  • Confusing 'A is x% of B' with 'A is x% more than B'. The first is A = x%·B; the second is A = (1+x%)·B.
  • Forgetting to KCF when dividing fractions. Dividing by a fraction is multiplying by the flipped fraction.
Warm-ups

Drill these:

  • What is 20% of 75? (15)
  • What is the result of $200 raised by 30%? ($260)
  • Price drops from $50 to $40, percent decrease? (20%)
  • Price rises from $40 to $50, percent increase? (25%)
  • Two consecutive 10% raises: net multiplier? (1.21, so 21% raise)
  • (1/2) ÷ (3/4) using KCF: (1/2)(4/3) = 4/6 = 2/3.
Want to climb higher? — telescoping with bigger gaps
  • Two-apart split: 1 / (n · (n+2)) = ½[1/n − 1/(n+2)]. So 1/(1·3) + 1/(3·5) + 1/(5·7) + … telescopes too, just with a 1/2 out front.
  • Product telescoping with (1 + 1/k): (1 + 1/1)(1 + 1/2)(1 + 1/3) … (1 + 1/n) = (2/1)(3/2)(4/3) … ((n+1)/n) = n + 1. Each numerator kills the next denominator. The whole product collapses to just n + 1.
  • The harmonic series. 1 + 1/2 + 1/3 + … does NOT telescope — it grows without bound (slowly). Don’t try.
→ Practice Fractions, Decimals & Percents problems