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Ratios, Rates & Proportions — Same ratio in two places.

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About this topic

A ratio compares two quantities by division. The notation '2 : 3' (read 'two to three') says 'for every 2 boys there are 3 girls' — it tells you the shape of the comparison, not the actual numbers.

A rate is a ratio with units — miles per hour, dollars per pound, problems per minute.

A proportion is the statement that two ratios are equal: a/b = c/d. Solving a proportion (cross-multiplying) is the single most common move in AMC 8 rates problems.

If you understand ratios, you understand percents (a percent is just a ratio out of 100), unit conversions (1 yard / 3 feet is a ratio), and the entire D = S × T family of problems. This lesson covers nine ideas: parts of a whole, proportions, the D = S × T formula, average speed, unit conversion, relative speed, reading graphs, exponential growth, and work-rate (when rates add).

CHAPTER 1

Ratios — parts of a whole

THEORY

A ratio of 2 : 3 doesn't mean '2 boys and 3 girls'. It means 'for every 2 boys, 3 girls' — the actual numbers can be 4 and 6, or 20 and 30, or 200 and 300. Same ratio.

The cleanest mental model: each side of the ratio is a count of parts.

RATIO PARTS

A ratio a : b means there are a + b total parts.

Total quantity ÷ total parts = size of one part.

Then each side's actual count = (its parts) × (size of one part).

Walkthrough. 30 students are split 2 : 3 boys to girls. Find boys and girls.

  • Total parts = 2 + 3 = 5.
  • Size of one part = 30 ÷ 5 = 6 students.
  • Boys = 2 × 6 = 12. Girls = 3 × 6 = 18.
  • Check: 12 + 18 = 30 ✓.

For three-part ratios, same idea. A ratio of 1 : 2 : 3 has 1+2+3 = 6 total parts. If the total quantity is 18 cookies, then 1 part = 3 cookies, so the three people get 3, 6, and 9 cookies.

THE TRICK

For 'find the difference' questions, you can short-cut: the difference between two sides of a ratio is just (difference in part counts) × (size of one part).

For 2 : 3 with each part = 6: difference is (3 − 2) × 6 = 6.

WORKED EXAMPLE
PROBLEM · 1985 #16

The ratio of boys to girls in Mr. Brown's math class is 2 : 3. If there are 30 students in the class, how many more girls than boys are in the class?

A) 10 B) 5 C) 3 D) 6 E) 2

Ratio 2 : 3, total 30. Total parts = 5, so each part = 6.

Boys = 2 × 6 = 12. Girls = 3 × 6 = 18. Difference = 6.

Even faster: difference in parts (3 − 2 = 1) times the size of a part (6) = 6.

Don't compute both sides if the question only asks for the difference. The 'difference in parts' is itself a small number — multiply it by the part-size and stop.

Answer: D — 6.
RULE OF THUMB

For any ratio, add all parts to get total parts. Total ÷ total parts = size of one part. Then multiply each part count.

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1989 · #9 There are 2 boys for every 3 girls in Ms. Johnson's math class. If there are 30 students in her class, what percent of them are boys?

There are 2 boys for every 3 girls in Ms. Johnson's math class. If there are 30 students in her class, what percent of them are boys?

Show answer
Answer: C — 40%.
Show hints
Hint 1 of 2
The ratio 2 : 3 makes 5 equal parts in all.
Still stuck? Show hint 2 →
Hint 2 of 2
Find how many students are in each part, then count the boys.
Show solution
Approach: split into ratio parts
  1. 5 parts make 30 students, so each part is 6 and the boys (2 parts) number 12.
  2. 12 of 30 is 40%.
2003 · #9 Art, Roger, Paul, and Trisha bake cookies that are all the same thickness, in the shapes shown below (dimensions in inches). Each friend...

Art, Roger, Paul, and Trisha bake cookies that are all the same thickness, in the shapes shown below (dimensions in inches). Each friend uses the same amount of dough, and Art's batch makes exactly 12 cookies.

Art's cookies sell for 60 cents each. To bring in the same total from one batch, how much should one of Roger's cookies cost, in cents?

ArtRogerPaulTrisha533423243
Show answer
Answer: C — 40 cents.
Show hints
Hint 1 of 2
Same total money from one batch means each cookie's price scales with its size.
Still stuck? Show hint 2 →
Hint 2 of 2
Compare Roger's cookie area to Art's, then scale 60 cents by that ratio.
Show solution
Approach: price per cookie scales with cookie area
  1. For a fixed batch and a fixed total price, each cookie's price is proportional to its area.
  2. Roger's cookie is 8 in² and Art's is 12 in², a ratio of 8/12 = 2/3.
  3. So Roger should charge 60 × 2/3 = 40 cents.
Another way — count the cookies:
  1. Art: 12 cookies at 60¢ = 720¢ per batch, using 12 × 12 = 144 in² of dough.
  2. Roger's 8 in² cookies: 144 ÷ 8 = 18 per batch.
  3. 720 ÷ 18 = 40 cents each.
2011 · #14 There are 270 students at Colfax Middle School, where the ratio of boys to girls is 5 : 4. There are 180 students at Winthrop Middle...

There are 270 students at Colfax Middle School, where the ratio of boys to girls is 5 : 4. There are 180 students at Winthrop Middle School, where the ratio of boys to girls is 4 : 5. The two schools hold a dance and all students from both schools attend. What fraction of the students at the dance are girls?

Show answer
Answer: C — 22/45.
Show hint
Hint 1
Each ratio uses 9 parts. Compute girls at each school, then total girls / total students.
Show solution
Approach: ratio-of-9 parts per school
  1. Colfax: girls = (4/9)(270) = 120.
  2. Winthrop: girls = (5/9)(180) = 100.
  3. Total girls: 220 of 450 students ⇒ 22/45.
2018 · #1 An amusement park has a collection of scale models, with a ratio of 1 : 20, of buildings and other sights from around the country. The...

An amusement park has a collection of scale models, with a ratio of 1 : 20, of buildings and other sights from around the country. The height of the United States Capitol is 289 feet. What is the height in feet of its replica at this park, rounded to the nearest whole number?

Show answer
Answer: A — 14 feet.
Show hint
Hint 1
Scale 1:20 means the replica is 1/20 of the original. Divide.
Show solution
Approach: divide by the scale factor
  1. Replica height = 289 / 20 = 14.45.
  2. Rounded: 14 feet.
2020 · #1 Luka is making lemonade to sell at a school fundraiser. His recipe requires 4 times as much water as sugar and twice as much sugar as...

Luka is making lemonade to sell at a school fundraiser. His recipe requires 4 times as much water as sugar and twice as much sugar as lemon juice. He uses 3 cups of lemon juice. How many cups of water does he need?

Show answer
Answer: E — 24 cups.
Show hints
Hint 1 of 2
Don't solve for sugar first — chain the two scalings into one trip from lemon juice to water.
Still stuck? Show hint 2 →
Hint 2 of 2
Water is 4 × sugar, sugar is 2 × lemon. So water is 4 × 2 = 8 times the lemon juice.
Show solution
Approach: chain the ratios
  1. Water is 4 × sugar, and sugar is 2 × lemon juice, so water is 4 × 2 = 8 times the lemon juice.
  2. With 3 cups of lemon juice, water = 8 × 3 = 24 cups.
Another way — step by step: lemon → sugar → water (MAA):
  1. Sugar is twice the lemon juice: 2 × 3 = 6 cups.
  2. Water is four times the sugar: 4 × 6 = 24 cups.
2023 · #5 A lake contains 250 trout, along with a variety of other fish. When a marine biologist catches and releases a sample of 180 fish from...

A lake contains 250 trout, along with a variety of other fish. When a marine biologist catches and releases a sample of 180 fish from the lake, 30 are identified as trout. Assume the ratio of trout to the total number of fish is the same in both the sample and the lake. How many fish are there in the lake?

Show answer
Answer: B — 1500 fish.
Show hints
Hint 1 of 2
The fraction of trout should be the same in the sample as in the whole lake.
Still stuck? Show hint 2 →
Hint 2 of 2
The fraction of the sample that is trout should equal the fraction of the whole lake that is trout. Set the two fractions equal.
Show solution
  1. In the sample, 30 of 180 are trout: 30 ÷ 180 = 16.
  2. So trout make up 16 of the whole lake too.
  3. If 250 trout are 16 of the fish, the total is 250 × 6 = 1500.
CHAPTER 2

Proportions — same ratio over here as over there

THEORY

A proportion is just two equal fractions. The whole game: find the missing piece.

The move is called cross-multiplication. Picture an X drawn across the fractions. The two diagonals each give you a product, and those products must be equal.

4 cups of flour make 6 cookies. How many cookies from 6 cups?46=6?cupscookiescupscookies

Multiply along the diagonals:

4 × ? = 6 × 6 ⇒ ? = 36 ÷ 4 = 9 cookies

WHEN TO USE A PROPORTION

Any time the problem says “at this rate” or “in the same ratio”:

  • Recipes — double the flour, double the cookies.
  • Map scales — 1 inch on map = 50 real miles, so 4.2 inches = 210 miles.
  • Similar triangles — corresponding sides are in the same ratio.
  • Per-unit prices — 3 candies for $1, 9 candies for $3.

The trap: when MORE means LESS (inverse proportion)

Not every “scaling” problem is a proportion. If 6 painters take 4 hours to paint a house, will 12 painters take 8 hours? No — more painters means FASTER, not slower.

DIRECT (ratios stay equal): more cups → more cookies4 cups → 6 cookiesRATIO 4:68 cups → 12 cookiesRATIO 8:12 = 4:6 ✓ sameINVERSE (product stays equal): more painters → less time6 painters, 4 hoursPRODUCT 6×4 = 2412 painters, 2 hoursPRODUCT 12×2 = 24 ✓ same
Quick check: “If I DOUBLE one, does the other DOUBLE (direct) or HALVE (inverse)?”
TypeWhat stays constantSet up
DirectRATIO (a/b)a/b = c/d → cross-multiply
InversePRODUCT (a·b)a · b = c · d
THE TRICK

For 'inverse proportion' problems ('more workers, less time'), set up a · b = c · d rather than a / b = c / d. The direction of the relationship is the whole game.

WORKED EXAMPLE
PROBLEM · 1985 #6

A stack of paper containing 500 sheets is 5 cm thick. Approximately how many sheets of this type of paper would there be in a stack 7.5 cm high?

A) 250 B) 550 C) 667 D) 750 E) 1250

500 sheets of paper are 5 cm thick. How many sheets in a 7.5 cm stack?

Per-unit approach. 500 sheets ÷ 5 cm = 100 sheets per cm. For 7.5 cm: 100 × 7.5 = 750 sheets.

Or equivalent proportion: 500/5 = x/7.5 → cross-multiply → 5x = 500 × 7.5 = 3750 → x = 750.

Same answer either way; the per-unit approach is usually faster because you compute the rate once and then it's just one multiplication.

Answer: D — 750.
RULE OF THUMB

Direct proportion: a/b = c/d, cross-multiply. Inverse proportion: a·b = c·d. Ask 'double one, does the other double or halve?'

MORE LIKE THIS
1993 · #6 A can of soup can feed 3 adults or 5 children. If there are 5 cans of soup and 15 children are fed, then how many adults would the...

A can of soup can feed 3 adults or 5 children. If there are 5 cans of soup and 15 children are fed, then how many adults would the remaining soup feed?

Show answer
Answer: B — 6 adults.
Show hints
Hint 1 of 2
First find how many cans the 15 children use.
Still stuck? Show hint 2 →
Hint 2 of 2
The leftover cans each feed 3 adults.
Show solution
Approach: use cans for children, then convert the rest
  1. 15 children need 15 ÷ 5 = 3 cans, leaving 5 − 3 = 2 cans.
  2. Those 2 cans feed 2 × 3 = 6 adults.
1995 · #8 An American traveling in Italy wishes to exchange American dollars for Italian lire. If 3000 lire = $1.60, how much lire will the...

An American traveling in Italy wishes to exchange American dollars for Italian lire. If 3000 lire = $1.60, how much lire will the traveler receive for $1.00?

Show answer
Answer: D — 1875 lire.
Show hints
Hint 1 of 2
Find how many lire equal one dollar.
Still stuck? Show hint 2 →
Hint 2 of 2
Divide 3000 by 1.60.
Show solution
Approach: lire per dollar
  1. $1.60 buys 3000 lire, so $1.00 buys 3000 ÷ 1.60.
  2. That is 1875 lire.
1997 · #4 Julie is preparing a speech. It must last between one-half hour and three-quarters of an hour, and her ideal rate is 150 words per...

Julie is preparing a speech. It must last between one-half hour and three-quarters of an hour, and her ideal rate is 150 words per minute. If she speaks at that rate, which of the following word counts is an appropriate length?

Show answer
Answer: E — 5650 words.
Show hints
Hint 1 of 2
Turn each time limit into a word count at 150 words per minute.
Still stuck? Show hint 2 →
Hint 2 of 2
The answer must fall between those two counts.
Show solution
Approach: convert the time bounds to word counts
  1. At 150 words per minute, 30 minutes is 4500 words and 45 minutes is 6750 words.
  2. The only choice between 4500 and 6750 is 5650.
2018 · #12 The clock in Sri's car, which is not accurate, gains time at a constant rate. One day as he begins shopping he notes that his car clock...

The clock in Sri's car, which is not accurate, gains time at a constant rate. One day as he begins shopping he notes that his car clock and his watch (which is accurate) both say 12:00 noon. When he is done shopping, his watch says 12:30 and his car clock says 12:35. Later that day, Sri loses his watch. He looks at his car clock and it says 7:00. What is the actual time?

Show answer
Answer: B — 6:00.
Show hints
Hint 1 of 2
30 real minutes correspond to 35 car-clock minutes. Ratio: actual / car = 30/35 = 6/7.
Still stuck? Show hint 2 →
Hint 2 of 2
Car clock shows 7 hours = 420 car-minutes past noon. Actual = 420 × 6/7.
Show solution
Approach: convert car-time to actual via the rate
  1. 30 real minutes = 35 car-clock minutes ⇒ actual = (6/7) × car-clock time.
  2. Car shows 7:00, which is 420 car-minutes past noon. Actual = 420 × 6/7 = 360 minutes = 6 hours.
  3. Actual time: 6:00.
2022 · #7 When the World Wide Web first became popular in the 1990s, download speeds reached a maximum of about 56 kilobits per second....

When the World Wide Web first became popular in the 1990s, download speeds reached a maximum of about 56 kilobits per second. Approximately how many minutes would the download of a 4.2-megabyte song have taken at that speed? (Note that there are 8000 kilobits in a megabyte.)

Show answer
Answer: B — 10 minutes.
Show hints
Hint 1 of 2
Convert the song to kilobits first (units of the speed), then time = size ÷ speed.
Still stuck? Show hint 2 →
Hint 2 of 2
4.2 MB × 8000 kbits/MB = 33,600 kbits. Divide by 56 kbits/sec.
Show solution
Approach: convert to matching units, then divide
  1. Song size: 4.2 × 8000 = 33,600 kilobits.
  2. Time = 33,600 ÷ 56 = 600 seconds = 10 minutes.
1995 · #16 Students from three middle schools worked on a summer project. Seven students from Allen school worked for 3 days, four students from...

Students from three middle schools worked on a summer project. Seven students from Allen school worked for 3 days, four students from Balboa school worked for 5 days, and five students from Carver school worked for 9 days. The total amount paid for the students' work was $774. Assuming each student received the same amount for a day's work, how much did the students from Balboa school earn altogether?

Show answer
Answer: C — 180.00 dollars.
Show hints
Hint 1 of 2
Count total student-days across all schools.
Still stuck? Show hint 2 →
Hint 2 of 2
Find the pay per student-day, then multiply by Balboa's student-days.
Show solution
Approach: pay per student-day
  1. Student-days: Allen 7·3 = 21, Balboa 4·5 = 20, Carver 5·9 = 45, totaling 86.
  2. Each student-day pays $774 ÷ 86 = $9, so Balboa earned 20 × $9 = $180.
CHAPTER 3

D = S × T — the master rate formula

THEORY

The master rate formula:

DISTANCE-SPEED-TIME

Distance = Speed × Time

Rearranged:

Speed = Distance ÷ Time

Time = Distance ÷ Speed

The D / S × T triangle. A handy memory aid: write D on top, S and T on the bottom of a triangle. Cover the letter you want — what's left is the formula.

DST

Cover D → see S × T below it. Cover S → see D / T. Cover T → see D / S.

This single formula powers an entire category of AMC problems. Always identify which two of (D, S, T) you're given, solve for the third.

Units matter. If speed is in mph but time is in minutes, convert one of them. 30 minutes at 60 mph = (½ hour) × 60 mph = 30 miles, not 1800 miles.

For two-leg trips (walk then run, drive then bike, etc.), compute distance per leg separately and add. Do not average the speeds — see chapter 4 for why that fails.

THE TRICK

When a problem gives time in minutes and speed in mph, convert time to hours by dividing by 60. Then D = S × T works directly. Match the time unit to the speed.

WORKED EXAMPLE
PROBLEM · 1985 #13

If you walk for 45 minutes at a rate of 4 mph and then run for 30 minutes at a rate of 10 mph, how many miles will you have gone at the end of one hour and 15 minutes?

A) 3.5 miles B) 8 miles C) 9 miles D) 25 1⁄3 miles E) 480 miles

Walk for 45 minutes at 4 mph, then run for 30 minutes at 10 mph. Total miles?

  • Walk leg: 4 mph × (45/60) hr = 3 miles.
  • Run leg: 10 mph × (30/60) hr = 5 miles.
  • Total: 3 + 5 = 8 miles.

The two legs are independent applications of D = S × T. The 'one hour and 15 minutes' total is a sanity check — 45 + 30 = 75 minutes ✓.

Answer: B — 8 miles.
RULE OF THUMB

Always know which two of (D, S, T) are given. Convert all times to a consistent unit (usually hours). For two-leg trips, compute each leg separately.

MORE LIKE THIS
2016 · #4 When Cheenu was a boy he could run 15 miles in 3 hours and 30 minutes. As an old man he can now walk 10 miles in 4 hours. How many...

When Cheenu was a boy he could run 15 miles in 3 hours and 30 minutes. As an old man he can now walk 10 miles in 4 hours. How many minutes longer does it take for him to travel a mile now compared to when he was a boy?

Show answer
Answer: B — 10 minutes longer.
Show hints
Hint 1 of 2
Convert each to minutes-per-mile, then subtract.
Still stuck? Show hint 2 →
Hint 2 of 2
Boy: 210 min / 15 mi = 14 min/mi. Old: 240 min / 10 mi = 24 min/mi.
Show solution
Approach: minutes per mile, then subtract
  1. Boy: 3 h 30 min = 210 minutes for 15 miles ⇒ 14 min/mile.
  2. Old: 4 h = 240 minutes for 10 miles ⇒ 24 min/mile.
  3. Difference: 24 − 14 = 10 minutes.
2016 · #14 Karl's car uses a gallon of gas every 35 miles, and his gas tank holds 14 gallons when it is full. One day, Karl started with a full...

Karl's car uses a gallon of gas every 35 miles, and his gas tank holds 14 gallons when it is full. One day, Karl started with a full tank of gas, drove 350 miles, bought 8 gallons of gas, and continued driving to his destination. When he arrived, his gas tank was half full. How many miles did Karl drive that day?

Show answer
Answer: A — 525 miles.
Show hint
Hint 1
Track gallons. Start 14, drive 350 = use 10, left 4. Buy 8 → 12. End half full = 7. So extra used = 5 gallons = 175 miles.
Show solution
Approach: track gallons through the trip
  1. Start: 14 gal. After 350 mi: used 350/35 = 10 gal, leaving 4.
  2. Bought 8 gal → 12 gal in tank. Arrived half full (7 gal), so used 12 − 7 = 5 more gallons = 5 × 35 = 175 miles.
  3. Total: 350 + 175 = 525 miles.
2014 · #17 George walks 1 mile to school. He leaves home at the same time each day, walks at a steady speed of 3 miles per hour, and arrives just...

George walks 1 mile to school. He leaves home at the same time each day, walks at a steady speed of 3 miles per hour, and arrives just as school begins. Today he was distracted by the pleasant weather and walked the first 12 mile at a speed of only 2 miles per hour. At how many miles per hour must George run the last 12 mile in order to arrive just as school begins today?

Show answer
Answer: B — 6 mph.
Show hint
Hint 1
Total time on a normal day = 1/3 hour. How much of that did the slow first half use? Whatever's left must cover the second half-mile.
Show solution
Approach: subtract used time from total time
  1. Normal total time = 1 / 3 hr.
  2. First half-mile at 2 mph took (1/2) / 2 = 1/4 hr.
  3. Time remaining for the second half = 1/3 − 1/4 = 1/12 hr.
  4. Required speed = (1/2) / (1/12) = 6 mph.
2018 · #6 On a trip to the beach, Anh traveled 50 miles on the highway and 10 miles on a coastal access road. He drove three times as fast on the...

On a trip to the beach, Anh traveled 50 miles on the highway and 10 miles on a coastal access road. He drove three times as fast on the highway as on the coastal road. If Anh spent 30 minutes driving on the coastal road, how many minutes did his entire trip take?

Show answer
Answer: C — 80 minutes.
Show hints
Hint 1 of 2
Coastal speed = 10 mi / 30 min = 1/3 mile/minute. Highway is 3× faster = 1 mile/minute.
Still stuck? Show hint 2 →
Hint 2 of 2
Highway time = 50 miles ÷ 1 mile/min = 50 min. Add the 30 min coastal.
Show solution
Approach: find each leg's time separately
  1. Coastal: 10 miles in 30 min ⇒ 1/3 mile per minute. Highway is 3× that: 1 mile/min.
  2. Highway time: 50 / 1 = 50 min.
  3. Total: 30 + 50 = 80 minutes.
2022 · #10 (figure problem)
amc8-2022-10
Show answer
Answer: E — Graph (E).
Show hints
Hint 1 of 2
Compute the peak distance (when she reaches the trail) and the timing of the three phases.
Still stuck? Show hint 2 →
Hint 2 of 2
Drive 2 hr at 45 mph = 90 mi. Hike 3 hr at the trail (distance flat). Drive back at 60 mph → 90/60 = 1.5 hr.
Show solution
Approach: compute peak distance and return time
  1. Outbound: 2 hr × 45 mph = 90 miles, reaching the trail at 10am.
  2. Hike 3 hr (distance stays at 90 miles), so she leaves the trail at 1pm.
  3. Return: 90 miles ÷ 60 mph = 1.5 hr, so she's home at 2:30pm.
  4. The graph that peaks at 90 miles between 10am and 1pm and comes back to 0 at 2:30pm is choice E.
2001 · #1 Casey's shop class is making a golf trophy. He has to paint 300 dimples on a golf ball. If it takes him 2 seconds to paint one dimple,...

Casey's shop class is making a golf trophy. He has to paint 300 dimples on a golf ball. If it takes him 2 seconds to paint one dimple, how many minutes will he need to do his job?

Show answer
Answer: D — 10 minutes.
Show hint
Hint 1
Find the total seconds first, then convert to minutes.
Show solution
Approach: total time, then change units
  1. Painting takes 2 × 300 = 600 seconds.
  2. 600 ÷ 60 = 10 minutes.
★ MINI-QUIZ

Ratios, proportions, D = ST

Three problems on parts-of-a-whole and the master rate formula.

2020 · #1 Luka is making lemonade to sell at a school fundraiser. His recipe requires 4 times as much water as sugar and twice as much sugar as...

Luka is making lemonade to sell at a school fundraiser. His recipe requires 4 times as much water as sugar and twice as much sugar as lemon juice. He uses 3 cups of lemon juice. How many cups of water does he need?

Show answer
Answer: E — 24 cups.
Show hints
Hint 1 of 2
Don't solve for sugar first — chain the two scalings into one trip from lemon juice to water.
Still stuck? Show hint 2 →
Hint 2 of 2
Water is 4 × sugar, sugar is 2 × lemon. So water is 4 × 2 = 8 times the lemon juice.
Show solution
Approach: chain the ratios
  1. Water is 4 × sugar, and sugar is 2 × lemon juice, so water is 4 × 2 = 8 times the lemon juice.
  2. With 3 cups of lemon juice, water = 8 × 3 = 24 cups.
Another way — step by step: lemon → sugar → water (MAA):
  1. Sugar is twice the lemon juice: 2 × 3 = 6 cups.
  2. Water is four times the sugar: 4 × 6 = 24 cups.
2023 · #5 A lake contains 250 trout, along with a variety of other fish. When a marine biologist catches and releases a sample of 180 fish from...

A lake contains 250 trout, along with a variety of other fish. When a marine biologist catches and releases a sample of 180 fish from the lake, 30 are identified as trout. Assume the ratio of trout to the total number of fish is the same in both the sample and the lake. How many fish are there in the lake?

Show answer
Answer: B — 1500 fish.
Show hints
Hint 1 of 2
The fraction of trout should be the same in the sample as in the whole lake.
Still stuck? Show hint 2 →
Hint 2 of 2
The fraction of the sample that is trout should equal the fraction of the whole lake that is trout. Set the two fractions equal.
Show solution
  1. In the sample, 30 of 180 are trout: 30 ÷ 180 = 16.
  2. So trout make up 16 of the whole lake too.
  3. If 250 trout are 16 of the fish, the total is 250 × 6 = 1500.
2002 · #24 Miki has a dozen oranges of the same size and a dozen pears of the same size. Miki uses her juicer to extract 8 ounces of pear juice...

Miki has a dozen oranges of the same size and a dozen pears of the same size. Miki uses her juicer to extract 8 ounces of pear juice from 3 pears and 8 ounces of orange juice from 2 oranges. She makes a pear-orange juice blend from an equal number of pears and oranges. What percent of the blend is pear juice?

Show answer
Answer: B — 40%.
Show hints
Hint 1 of 2
Work out how much juice one pear gives and how much one orange gives.
Still stuck? Show hint 2 →
Hint 2 of 2
With equal numbers of each fruit, just compare those per-fruit yields.
Show solution
Approach: compare juice per fruit
  1. One pear gives 8/3 oz of juice; one orange gives 8/2 = 4 oz.
  2. Equal numbers of each fruit make the pear-to-orange juice ratio 8/3 : 4 = 2 : 3.
  3. Pear's share of the blend is 2 ÷ (2 + 3) = 2/5 = 40%.
CHAPTER 4

Average speed — never average the speeds

THEORY

If you drive 60 mph for an HOUR, then 30 mph for ANOTHER HOUR, your average is 45 mph. Equal times — simple average works.

But what if the question gives equal DISTANCES, not equal times? Say, 60 miles at 60 mph, then 60 miles back at 30 mph. The average is NOT 45 — it’s 40 mph. Watch why:

Round trip: 60 mi out at 60 mph, 60 mi back at 30 mphOutbound (60 mph, 60 mi)1 hour← fast leg, shorter timeReturn (30 mph, 60 mi)2 hours← slow leg, DOUBLE the time0 hr1 hr3 hr (total)Plug into the formula:avg = total dist ÷ total time = 120 mi ÷ 3 hr = 40 mph(NOT 45 = (60+30)/2)Why? You spend MORE TIME at the slower speed.2 of the 3 total hours are at 30 mph — so the average gets pulled toward 30, not stuck in the middle.Equal distances ⇒ the slower leg always dominates.

AVERAGE SPEED — ONE FORMULA, ALWAYS

average speed = TOTAL distance ÷ TOTAL time

Never average two speeds directly unless you spent equal TIME at each.

Round-trip shortcut

For a round trip where each leg has the same distance but different speeds a and b:

average = 2ab / (a + b)

(This is the harmonic mean.) For 60 and 30: 2·60·30 / (60+30) = 3600/90 = 40. ✓

Two speedsSimple avg (TRAP)Harmonic mean (CORRECT)
60 & 30 mph4540
40 & 60 mph5048
3 & 6 mph4.54
10 & 90 mph5018 (slow really dominates!)
The harmonic mean is ALWAYS lower than the simple average. The bigger the speed gap, the more dramatic the drop.
THE TRICK

When you see 'equal distance each leg', the average is always less than the simple average. When you see 'equal time each leg', the simple average works.

WORKED EXAMPLE
PROBLEM · 1992 #18

On a trip, a car traveled 80 miles in an hour and a half, then was stopped in traffic for 30 minutes, then traveled 100 miles during the next 2 hours. What was the car's average speed in miles per hour for the 4-hour trip?

A) 45 B) 50 C) 60 D) 75 E) 90

The car drives 80 miles in 1.5 hours, sits in traffic for 0.5 hours, and drives 100 more miles in 2 hours. What's the average speed for the whole 4-hour trip?

Don't average the speeds — the legs have different times and there's a stop. Go straight to the definition:

  • Total distance = 80 + 100 = 180 miles.
  • Total time = 1.5 + 0.5 + 2 = 4 hours (the 30-minute stop counts!).
  • Average speed = 180 / 4 = 45 mph (choice A).

Don't fall for the 60 mph trap. If you forget the stop, you'd compute 180 / 3.5 ≈ 51.4 mph. If you average the two driving speeds (80/1.5 ≈ 53 and 100/2 = 50), you'd land somewhere near 51. The stop is the whole point — it's part of the trip's elapsed time.

The setters put the stop in the middle on purpose. They want you to either (a) forget it and use only driving time, or (b) try to average the speeds. Total distance over total elapsed time avoids both traps. The 'time stopped' counts as time at speed 0 — it pulls the average down.

Answer: A — 45.
RULE OF THUMB

Average speed = total distance ÷ total time. Never average two speeds directly unless times are equal. For equal-distance round trips: 2ab/(a+b).

MORE LIKE THIS
2008 · #5 Barney Schwinn notices that the odometer on his bicycle reads 1441, a palindrome, because it reads the same forward and backward. After...

Barney Schwinn notices that the odometer on his bicycle reads 1441, a palindrome, because it reads the same forward and backward. After riding 4 more hours that day and 6 the next, he notices that the odometer shows another palindrome, 1661. What was his average speed in miles per hour?

Show answer
Answer: E — 22 mph.
Show hint
Hint 1
Distance = 1661 − 1441 = 220 over 10 hours.
Show solution
Approach: distance / time
  1. Distance: 220 miles. Time: 4 + 6 = 10 hours.
  2. Average: 220 / 10 = 22 mph.
2011 · #9 Carmen takes a long bike ride on a hilly highway. The graph indicates the miles traveled during the time of her ride. What is Carmen's...
amc8-2011-09
Show answer
Answer: E — 5 mph.
Show hint
Hint 1
Average speed = total distance / total time. The graph's endpoints give you both.
Show solution
Approach: total miles / total hours
  1. Total: 35 miles in 7 hours.
  2. Average speed: 35 / 7 = 5 mph.
2014 · #17 George walks 1 mile to school. He leaves home at the same time each day, walks at a steady speed of 3 miles per hour, and arrives just...

George walks 1 mile to school. He leaves home at the same time each day, walks at a steady speed of 3 miles per hour, and arrives just as school begins. Today he was distracted by the pleasant weather and walked the first 12 mile at a speed of only 2 miles per hour. At how many miles per hour must George run the last 12 mile in order to arrive just as school begins today?

Show answer
Answer: B — 6 mph.
Show hint
Hint 1
Total time on a normal day = 1/3 hour. How much of that did the slow first half use? Whatever's left must cover the second half-mile.
Show solution
Approach: subtract used time from total time
  1. Normal total time = 1 / 3 hr.
  2. First half-mile at 2 mph took (1/2) / 2 = 1/4 hr.
  3. Time remaining for the second half = 1/3 − 1/4 = 1/12 hr.
  4. Required speed = (1/2) / (1/12) = 6 mph.
2019 · #16 Qiang drives 15 miles at an average speed of 30 miles per hour. How many additional miles will he have to drive at 55 miles per hour to...

Qiang drives 15 miles at an average speed of 30 miles per hour. How many additional miles will he have to drive at 55 miles per hour to average 50 miles per hour for the entire trip?

Show answer
Answer: D — 110 miles.
Show hints
Hint 1 of 2
Average speed = total distance / total time. Write that equation with x = additional miles.
Still stuck? Show hint 2 →
Hint 2 of 2
Time so far: 15/30 = 1/2 hr. Set (15 + x)/(1/2 + x/55) = 50.
Show solution
Approach: set total-distance / total-time = 50
  1. Time so far: 15/30 = 1/2 hour. After x additional miles at 55 mph, the new total is (15 + x) miles and (1/2 + x/55) hours.
  2. (15 + x) / (1/2 + x/55) = 50 ⇒ 15 + x = 25 + 10x/11.
  3. Multiply by 11: 165 + 11x = 275 + 10x ⇒ x = 110.
2001 · #5 On a dark and stormy night Snoopy suddenly saw a flash of lightning. Ten seconds later he heard the sound of thunder. The speed of sound...

On a dark and stormy night Snoopy suddenly saw a flash of lightning. Ten seconds later he heard the sound of thunder. The speed of sound is 1088 feet per second and one mile is 5280 feet. Estimate, to the nearest half-mile, how far Snoopy was from the flash of lightning.

Show answer
Answer: C — 2 miles.
Show hints
Hint 1 of 2
Distance = speed × time gives the feet; then compare with a mile.
Still stuck? Show hint 2 →
Hint 2 of 2
Two miles is about 10,560 feet.
Show solution
Approach: distance in feet, then convert to miles
  1. In 10 seconds the sound travels 10 × 1088 = 10,880 feet.
  2. Since 2 miles is 2 × 5280 = 10,560 feet, the distance is closest to 2 miles.
2001 · #19 (figure problem)
amc8-2001-19
Show answer
Answer: D — Graph D.
Show hints
Hint 1 of 2
Twice the speed over the same distance means half the time.
Still stuck? Show hint 2 →
Hint 2 of 2
So Car N's line should sit twice as high and run half as long as Car M's.
Show solution
Approach: twice the speed over the same distance halves the time
  1. Car N is twice as fast, so its (horizontal) speed line sits at twice the height of Car M's.
  2. The same distance at double speed takes half the time, so N's line is half as long — only graph D shows both.
CHAPTER 5

Unit conversion — multiply by 1

THEORY

To convert units, multiply by a fraction equal to 1 — one where the numerator and denominator are the SAME quantity written in different units (like 5280 ft / 1 mi).

This is the factor-label method. The trick: write the units and let them cancel like algebra. The numbers fall out at the end.

Convert 60 mph → ft/s using the factor-label methodEach fraction equals 1 (the numerator equals the denominator in different units).60mihr×5280 ft1 mi(= 1)×1 hr3600 s(= 1)=88ft / smi cancelshr cancelsonly the numbers left: 60 × 5280 × 1 ÷ 1 ÷ 3600 = 316,800 ÷ 3600 = 88only the units left: ft / s (everything else cancelled). So 60 mph = 88 ft/s.

FACTOR-LABEL RECIPE

  1. Write the starting value with its units (e.g., 60 mi/hr).
  2. Multiply by conversion fractions, each equal to 1 (e.g., 5280 ft / 1 mi).
  3. Arrange so the unit you DON’T want is in the OPPOSITE part of the fraction (it will cancel).
  4. The unit you DO want should survive.
  5. Multiply all numerators, divide by all denominators — the answer’s units are what didn’t cancel.

Conversion facts to know cold

FromToMultiply by
1 milefeet5280
1 mileyards1760
1 yardfeet3
1 footinches12
1 hourminutes60
1 minuteseconds60
1 hourseconds3600
1 mphft/s≈ 1.467 (so 60 mph = 88 ft/s)
1 kmmeters1000
1 lbounces16
If your final answer has the WRONG units, you flipped a conversion fraction. Re-do that fraction upside down.
THE TRICK

Don't memorize 'when to multiply by 60 vs divide.' Write the conversion factors with units and let the cancellation tell you. The units are guardrails.

WORKED EXAMPLE
PROBLEM · 1993 #8

To control her blood pressure, Jill's grandmother takes one half of a pill every other day. If one supply of medicine contains 60 pills, then the supply would last approximately

A) 1 month B) 4 months C) 6 months D) 8 months E) 1 year

Jill's grandmother takes ½ pill every other day. The bottle has 60 pills. About how many months will it last?

Stack up the conversion factors with their units, and let cancellation do the work:

60 pills × (1 dose / ½ pill) × (2 days / 1 dose) × (1 month / 30 days)

Cancel 'pills', then 'doses', then 'days':

  • 60 × (1/0.5) × 2 = 60 × 2 × 2 = 240 days.
  • 240 days × (1 month / 30 days) ≈ 8 months (choice D).

The units are the proof: we started with 'pills' and ended with 'months', and every other unit cancelled along the way.

The 'every other day' phrase hides a conversion (1 dose ↔ 2 days). 'Half a pill per dose' hides another (1 dose ↔ ½ pill). Writing each as a fraction with units forces those hidden conversions into the open, so cancellation can finish the job.

Answer: D — 8 months.
RULE OF THUMB

Multiply by 1 = (target unit) / (source unit). Units cancel like algebra. The number falls out.

MORE LIKE THIS
1987 · #14 A computer can do 10,000 additions per second. How many additions can it do in one hour?

A computer can do 10,000 additions per second. How many additions can it do in one hour?

Show answer
Answer: B — 36 million.
Show hint
Hint 1
1 hour = 3600 seconds.
Show solution
Approach: multiply per-second by seconds-per-hour
  1. 10,000 × 3600 = 36,000,000.
  2. = 36 million.
1994 · #5 Given that 1 mile = 8 furlongs and 1 furlong = 40 rods, the number of rods in one mile is

Given that 1 mile = 8 furlongs and 1 furlong = 40 rods, the number of rods in one mile is

Show answer
Answer: B — 320.
Show hints
Hint 1 of 2
Convert miles to furlongs, then furlongs to rods.
Still stuck? Show hint 2 →
Hint 2 of 2
Just multiply the two conversion factors.
Show solution
Approach: chain the conversions
  1. 1 mile = 8 furlongs, and each furlong = 40 rods.
  2. So 1 mile = 8 × 40 = 320 rods.
2022 · #7 When the World Wide Web first became popular in the 1990s, download speeds reached a maximum of about 56 kilobits per second....

When the World Wide Web first became popular in the 1990s, download speeds reached a maximum of about 56 kilobits per second. Approximately how many minutes would the download of a 4.2-megabyte song have taken at that speed? (Note that there are 8000 kilobits in a megabyte.)

Show answer
Answer: B — 10 minutes.
Show hints
Hint 1 of 2
Convert the song to kilobits first (units of the speed), then time = size ÷ speed.
Still stuck? Show hint 2 →
Hint 2 of 2
4.2 MB × 8000 kbits/MB = 33,600 kbits. Divide by 56 kbits/sec.
Show solution
Approach: convert to matching units, then divide
  1. Song size: 4.2 × 8000 = 33,600 kilobits.
  2. Time = 33,600 ÷ 56 = 600 seconds = 10 minutes.
CHAPTER 6

Relative speed — meet, chase, lap

THEORY

When two things move on the same line, the question “when do they meet?” or “when does one catch the other?” reduces to a tiny formula. The trick is figuring out the closing speed — how fast the gap between them shrinks.

Case 1: heading TOWARD each other (speeds ADD)

Two trains, 100 miles apart, headed at each other (40 mph and 60 mph)A40 mphB60 mphgap = 100 miles
Closing speed = 40 + 60 = 100 mph. Time to meet = 100 mi ÷ 100 mph = 1 hour.

Why add? Each mile the gap shrinks gets credited to BOTH trains — A walks toward B by 40, B walks toward A by 60, so the gap loses 100 mi every hour.

Case 2: same direction, faster behind (speeds SUBTRACT)

Bob (8 mph) chasing Alice (4 mph). Alice is 1 mile ahead.B8 mphA4 mphgap = 1 mile
Closing speed = 8 − 4 = 4 mph. Time = 1 mi ÷ 4 mph = 1/4 hr = 15 minutes.

Why subtract? Bob runs forward at 8 mph but Alice is ALSO moving forward at 4 mph, taking 4 miles of progress with her each hour. Bob only nets 4 mph of catching-up.

RELATIVE-SPEED SUMMARY

SituationClosing speedTime
Toward each otherv₁ + v₂gap ÷ (v₁ + v₂)
Same direction (faster behind)vfast − vslowgap ÷ (vfast − vslow)
Round track, same directionvfast − vslowtrack length ÷ that (next lap)
Round track, opposite directionv₁ + v₂track length ÷ that (next meet)
THE TRICK

Subtract speeds for same-direction; add speeds for opposite. The trap is forgetting that 'catching up' is a relative-speed problem, not a single-speed problem.

WORKED EXAMPLE
PROBLEM · 2010 #8

As Emily is riding her bicycle on a long straight road, she spots Emerson skating in the same direction 1/2 mile in front of her. After she passes him, she can see him in her rear mirror until he is 1/2 mile behind her. Emily rides at a constant rate of 12 miles per hour, and Emerson skates at a constant rate of 8 miles per hour. For how many minutes can Emily see Emerson?

A) 6 B) 8 C) 12 D) 15 E) 16

Emily (12 mph) and Emerson (8 mph) are heading the same direction. The instant Emily spots Emerson, he is ½ mile ahead. She loses sight of him when he is ½ mile behind in her mirror.

The relative shift between them — from ½ mile ahead to ½ mile behind — is 1 mile, not ½. (Picture Emily standing still and Emerson 'rolling backward' past her: he has to travel a full mile relative to her.)

  • Closing speed = Emily − Emerson = 12 − 8 = 4 mph.
  • Time = (1 mile) / (4 mph) = 1/4 hour = 15 minutes (choice D).

The trap is reading 'gap = ½ mile' twice and treating it as a single ½-mile chase. It's really two halves stitched together: ½ mile to catch up, then ½ mile to pull away. Both happen at the same 4-mph relative speed, so the time is 1 mile / 4 mph = 15 minutes, not 7.5.

Answer: D — 15 minutes.
RULE OF THUMB

Toward each other: speeds add. Same direction: speeds subtract. Time to meet/catch = gap / relative speed.

MORE LIKE THIS
1995 · #11 (figure problem)
ajhsme-1995-11
Show answer
Answer: D — Point D.
Show hints
Hint 1 of 2
Jane is twice as fast, so she covers twice the distance Hector does.
Still stuck? Show hint 2 →
Hint 2 of 2
Together they cover the whole 18-block perimeter, so Hector covers one-third of it.
Show solution
Approach: split the perimeter by their speed ratio
  1. Going opposite ways, together they cover the full 18-block loop. Jane (twice as fast) covers 12 blocks and Hector covers 6.
  2. Hector's 6 blocks from the start (right, then up the side) land him exactly at corner D.
2019 · #16 Qiang drives 15 miles at an average speed of 30 miles per hour. How many additional miles will he have to drive at 55 miles per hour to...

Qiang drives 15 miles at an average speed of 30 miles per hour. How many additional miles will he have to drive at 55 miles per hour to average 50 miles per hour for the entire trip?

Show answer
Answer: D — 110 miles.
Show hints
Hint 1 of 2
Average speed = total distance / total time. Write that equation with x = additional miles.
Still stuck? Show hint 2 →
Hint 2 of 2
Time so far: 15/30 = 1/2 hr. Set (15 + x)/(1/2 + x/55) = 50.
Show solution
Approach: set total-distance / total-time = 50
  1. Time so far: 15/30 = 1/2 hour. After x additional miles at 55 mph, the new total is (15 + x) miles and (1/2 + x/55) hours.
  2. (15 + x) / (1/2 + x/55) = 50 ⇒ 15 + x = 25 + 10x/11.
  3. Multiply by 11: 165 + 11x = 275 + 10x ⇒ x = 110.
2020 · #11 (figure problem)
amc8-2020-11
Show answer
Answer: E — 24 mph.
Show hints
Hint 1 of 2
Read each girl's total distance (6 miles) and the time she took. Convert each to mph and subtract.
Still stuck? Show hint 2 →
Hint 2 of 2
Naomi: 6 miles in ~10 min → 36 mph. Maya: 6 miles in 30 min → 12 mph.
Show solution
Approach: average speed = total distance / total time
  1. Naomi covers 6 miles in 10 minutes = 1/6 hour, so average speed = 6 ÷ (1/6) = 36 mph.
  2. Maya covers 6 miles in 30 minutes = 1/2 hour, so average speed = 6 ÷ (1/2) = 12 mph.
  3. Difference: 36 − 12 = 24 mph.
2023 · #11 NASA's Perseverance Rover was launched on July 30, 2020. After traveling 292,526,838 miles, it landed on Mars in Jezero Crater about 6.5...

NASA's Perseverance Rover was launched on July 30, 2020. After traveling 292,526,838 miles, it landed on Mars in Jezero Crater about 6.5 months later. Which of the following is closest to the Rover's average interplanetary speed in miles per hour?

Show answer
Answer: C — About 60,000 mph.
Show hints
Hint 1 of 2
The choices span orders of magnitude, so round generously. Distance ≈ 3 × 108 miles.
Still stuck? Show hint 2 →
Hint 2 of 2
6.5 months ≈ 200 days ≈ 5000 hours. Then speed ≈ distance ÷ time.
Show solution
Approach: round to easy numbers, divide
  1. Distance ≈ 3 × 108 miles.
  2. 6.5 months × 30 days/month ≈ 195 days ≈ 200 days ≈ 200 × 24 = 4800 hours ≈ 5000 hours.
  3. Speed ≈ (3 × 108) ÷ 5000 = 60,000 mph.
2023 · #15 (figure problem)
amc8-2023-15
Show answer
Answer: B — 4.2 mph.
Show hints
Hint 1 of 2
Find Viswam's usual speed in mph, then figure out how many blocks vs minutes he has left after the detour starts.
Still stuck? Show hint 2 →
Hint 2 of 2
After 5 blocks he has 5 min left. Detour makes the remaining distance 5 + (3 − 1) = 7 blocks instead of 5. Distance per block: 0.05 mile.
Show solution
Approach: convert blocks to miles, time to hours
  1. Usual: 10 blocks = 0.5 mile in 10 min = 1/6 hr → speed = 3 mph. Each block is 0.05 mile.
  2. After 5 blocks, 5 minutes are left. Detour replaces 1 block with 3, so remaining distance becomes 5 + 2 = 7 blocks = 0.35 mile.
  3. 5 minutes = 1/12 hr. Speed = 0.35 ÷ (1/12) = 0.35 × 12 = 4.2 mph.
2025 · #19 (figure problem)
amc8-2025-19
Show answer
Answer: D — 8.5 miles from A.
Show hints
Hint 1 of 2
Once both cars are in the middle section, they're going the same speed — the asymmetry is in how they got there.
Still stuck? Show hint 2 →
Hint 2 of 2
Car A reaches the middle in 5/25 = 1/5 hr; car B in 5/20 = 1/4 hr. A has a 1/20-hr head start in the middle section — that's 2 miles.
Show solution
Approach: let the cars meet in the middle 40-mph section
  1. Car A's left section is 5 miles at 25 mph → reaches the middle in 5/25 = 1/5 hr. Car B's right section is 5 miles at 20 mph → reaches the middle in 5/20 = 1/4 hr. A enters the middle 1/20 hr before B.
  2. In that 1/20 hr, A travels 40 × 1/20 = 2 miles, so when B enters the middle, A is at mile 7, B at mile 10 — a 3-mile gap.
  3. Both now go 40 mph, closing at 80 mph. They split the 3-mile gap equally: each covers 1.5 miles. A is at 7 + 1.5 = 8.5 miles from A.
★ MINI-QUIZ

Speeds and conversions

Three problems on average speed, unit conversion, and relative speed.

2026 · #5 Casey went on a road trip that covered 100 miles, stopping only for a lunch break along the way. The trip took 3 hours in total and her...

Casey went on a road trip that covered 100 miles, stopping only for a lunch break along the way. The trip took 3 hours in total and her average speed while driving was 40 miles per hour. In minutes, how long was the lunch break?

Show answer
Answer: B — 30 minutes.
Show hints
Hint 1 of 2
The 3 hours includes the break. Find the driving time first.
Still stuck? Show hint 2 →
Hint 2 of 2
Time = distance ÷ speed gives only the driving time. Whatever's left of the 3 hours is the break.
Show solution
  1. Driving time = 100 ÷ 40 = 2.5 hours.
  2. Break = total − driving = 3 − 2.5 = 0.5 hour.
  3. 0.5 hour = 30 minutes.
2022 · #22 A bus takes 2 minutes to drive from one stop to the next, and waits 1 minute at each stop to let passengers board. Zia takes 5 minutes...

A bus takes 2 minutes to drive from one stop to the next, and waits 1 minute at each stop to let passengers board. Zia takes 5 minutes to walk from one bus stop to the next. As Zia reaches a bus stop, if the bus is at the previous stop or has already left the previous stop, then she will wait for the bus. Otherwise she will start walking toward the next stop. Suppose the bus and Zia start at the same time toward the library, with the bus 3 stops behind. After how many minutes will Zia board the bus?

Show answer
Answer: A — 17 minutes.
Show hints
Hint 1 of 2
Zia only checks at multiples of 5 minutes (when she reaches a stop). At each check, see where the bus is and whether to wait.
Still stuck? Show hint 2 →
Hint 2 of 2
Bus cycle: 2 min driving + 1 min stopped = 3 min per stop. Track stop indexes at t = 0, 5, 10, 15.
Show solution
Approach: simulate at Zia's 5-minute checkpoints
  1. Index stops by the bus's start (stop 0). At t = 0, Zia is at stop 3, bus at stop 0.
  2. t = 5: Zia at stop 4. Bus took 5 min → finished stop 1 (arrived at 2 min, left at 3 min, arrived at stop 2 at 5 min). Bus is at stop 2 — not yet at the previous stop (3), so Zia walks on.
  3. t = 10: Zia at stop 5. Bus: from t = 5 (at stop 2) waits 1 min (leaves at 6), drives 2 min to stop 3 (arrives at 8), waits till 9, drives to stop 4 (arrives at 11). So at t = 10, bus is mid-drive between stops 3 and 4 — not at the previous stop (4), so Zia walks on.
  4. t = 15: Zia at stop 6. Bus: arrives at stop 4 at 11, waits till 12, drives to stop 5 (arrives 14, waits till 15). At t = 15, bus is at stop 5 — the previous stop. Zia waits.
  5. Bus leaves stop 5 at t = 15 and drives 2 min to stop 6: arrives at t = 17.
2014 · #17 George walks 1 mile to school. He leaves home at the same time each day, walks at a steady speed of 3 miles per hour, and arrives just...

George walks 1 mile to school. He leaves home at the same time each day, walks at a steady speed of 3 miles per hour, and arrives just as school begins. Today he was distracted by the pleasant weather and walked the first 12 mile at a speed of only 2 miles per hour. At how many miles per hour must George run the last 12 mile in order to arrive just as school begins today?

Show answer
Answer: B — 6 mph.
Show hint
Hint 1
Total time on a normal day = 1/3 hour. How much of that did the slow first half use? Whatever's left must cover the second half-mile.
Show solution
Approach: subtract used time from total time
  1. Normal total time = 1 / 3 hr.
  2. First half-mile at 2 mph took (1/2) / 2 = 1/4 hr.
  3. Time remaining for the second half = 1/3 − 1/4 = 1/12 hr.
  4. Required speed = (1/2) / (1/12) = 6 mph.
CHAPTER 7

Reading rates from graphs

THEORY

The AMC loves cumulative-total graphs, speed-vs-time graphs, distance-vs-time graphs, and pie charts. Each has a different way to read 'rate':

  • Cumulative-total (like $ spent vs month): the slope at any point is the rate. Differences in heights between two times = the amount over that period.
  • Distance-vs-time: slope is speed.
  • Speed-vs-time: slope is acceleration; area under is distance traveled.
  • Pie chart: each slice's angle / 360° = its fraction of the total.

The single most common AMC graph question: 'how much was spent during June, July, August?' You read the y-value at end-of-August and subtract the y-value at end-of-May. Don't read each month individually.

THE TRICK

For cumulative graphs, your answer is always a difference of two readings. For per-period bar charts, sum or read directly. For pie charts, multiply the slice fraction by total.

WORKED EXAMPLE
PROBLEM · 1989 #19

Cumulative dollars spent. Read total at end-of-August (≈ 4.7 million). Read total at end-of-May (≈ 2.2 million). Summer total = 4.7 − 2.2 = 2.5 million.

The graph is doing the addition for you — each month's height already includes everything before it. Just subtract two heights for a window of months.

Answer: B — 2.5.
RULE OF THUMB

Identify cumulative vs per-period before reading. Cumulative answers are differences; per-period answers are direct reads or sums.

MORE LIKE THIS
1999 · #4 (figure problem)
amc8-1999-04
Show answer
Answer: A — About 15 miles.
Show hints
Hint 1 of 2
Read each rider's distance straight off the graph at the 4-hour mark.
Still stuck? Show hint 2 →
Hint 2 of 2
Then just subtract.
Show solution
Approach: read the two values at 4 hours and subtract
  1. At 4 hours the graph shows Alberto at about 60 miles and Bjorn at about 45 miles.
  2. Alberto is ahead by 60 − 45 = 15 miles.
CHAPTER 8

Exponential growth — when a quantity multiplies each step

THEORY

There are two ways a quantity can grow:

  • Linear growth: the SAME amount is ADDED each step. (You save $5 every week.)
  • Exponential growth: the value is MULTIPLIED by the same factor each step. (Bacteria double every hour. Your savings earn 10% interest each year.)

They start out close, but exponential rockets up. Watch what happens with $1 over 10 steps:

$0$250$500$750$1024012345678910step numberlinear: ends at $11(+$1 each step)exponential: ends at $1024(×2 each step)The exponential curve barely moves for 6 steps… then it rockets past linear and never looks back.

Both start at $1. By step 10, linear is at $11 (still nearly on the x-axis); exponential has hit $1024 (off the top). The gap is 93×.

EXPONENTIAL FORMULA

Starting value V₀, multiplier r per step, after n steps:

V(n) = V₀ × rn

r > 1 means growing. 0 < r < 1 means shrinking (like 0.9 = lose 10% each step).

Common growth factors to recognize

PhraseMultiplier per step5 steps gives …
doubles each year× 2× 32
triples each year× 3× 243
grows 50% each year× 1.5× 7.59
grows 10% each year× 1.1× 1.61 (not 1.5!)
shrinks 10% each year× 0.9× 0.59
half-life: halves each step× 0.5× 0.03 (1/32)
Compound interest: $100 at 5% for 20 years = 100 × (1.05)20 ≈ $265. Linear estimate gives $200 — off by $65 of “interest on interest.”
THE TRICK

For 'doubles every N hours' problems: starting with 1 at hour 0, after kN hours there are 2^k. Watch the off-by-one — at hour 0 you have 1, at hour N you have 2, at hour 2N you have 4.

WORKED EXAMPLE
PROBLEM · 1998 #17

Nisos Isles. In 1998 the islands have 200 people, and the population triples every 25 years. The total area is 24,900 square miles, and the Queen requires at least 1.5 square miles per person. In about how many years from 1998 will the population reach the maximum the islands can support?

A) 50 years B) 75 years C) 100 years D) 125 years E) 150 years

In 1998 the Nisos Isles have 200 people and the population triples every 25 years. The total area is 24,900 sq mi, and the Queen requires at least 1.5 sq mi per person. In about how many years will the islands hit their cap?

First, find the cap: 24,900 ÷ 1.5 = 16,600 people. That's about 16,600 / 200 = 83 times today's count.

Now count triplings. Powers of 3: 3, 9, 27, 81. Four triplings give 3⁴ = 81 — just shy of 83. So it takes about four triplings.

Four triplings × 25 years = 100 years.

Don't solve 200·3n ≥ 16,600 with logarithms — just memorize 3¹, 3², 3³, 3⁴ = 3, 9, 27, 81 and find the smallest n that crosses 83.

Answer: C — About 100 years.
RULE OF THUMB

Exponential growth = ×r per step. After n steps: V₀ × r^n. Don't mistake it for linear (additive) growth.

MORE LIKE THIS
1998 · #15 Nisos Isles. In 1998 the islands have 200 people, and the population triples every 25 years. Estimate the population in the year 2050.

Nisos Isles. In 1998 the islands have 200 people, and the population triples every 25 years. Estimate the population in the year 2050.

Show answer
Answer: D — About 2000.
Show hints
Hint 1 of 2
From 1998 to about 2050 spans two 25-year tripling periods.
Still stuck? Show hint 2 →
Hint 2 of 2
Triple once, then triple again.
Show solution
Approach: triple twice over two periods
  1. By 2048 (two 25-year periods) the population triples twice: 200 × 3 × 3 = 1800.
  2. That is closest to 2000.
1998 · #16 Nisos Isles. In 1998 the islands have 200 people, and the population triples every 25 years. Estimate the year in which the population...

Nisos Isles. In 1998 the islands have 200 people, and the population triples every 25 years. Estimate the year in which the population will be about 6000.

Show answer
Answer: B — About 2075.
Show hints
Hint 1 of 2
6000 is about 30 times the starting 200.
Still stuck? Show hint 2 →
Hint 2 of 2
Each tripling multiplies by 3 — how many triples reach about 30×?
Show solution
Approach: count triplings to reach ~30×
  1. 6000 ÷ 200 = 30, and three triplings give ×27 ≈ 30.
  2. Three 25-year periods is 75 years after 1998, about 2075.
CHAPTER 9

Work-rate problems — when rates add

THEORY

Two pipes fill a tank. Two painters paint a fence. Two helpers fold the laundry. The question is always: working together, how long does it take?

Students often try to average the times — that's the trap. Times don't add (and they don't average). Rates add.

RATES ADD; TIMES DON'T

If A finishes the job alone in t_a units of time, A's rate is 1/t_a jobs per unit. Same for B at 1/t_b. Together their rates add:

combined rate = 1/t_a + 1/t_b

Combined time is the reciprocal:

t_combined = 1 / (1/t_a + 1/t_b) = (t_a · t_b) / (t_a + t_b)

Why rates add (but times don't). Think of 'job per hour' as a speed. If two cars are driving the same direction, their speeds don't average — they keep their individual speeds. Rates work the same way. Each worker is doing their own fraction of the job per hour; the fractions add.

tank (1 full job)Hose Arate 1/4 /hrHose Brate 1/6 /hr1/4 + 1/6= 3/12 + 2/12= 5/12 /hrtime = 12/5 hr

Together the hoses fill 5/12 of the tank every hour, so the full job takes 12/5 = 2.4 hours.

Worked walkthrough. Hose A fills a tank in 4 hours, Hose B fills the same tank in 6 hours. Both run at once. How long?

  • A's rate: 1/4 tank per hour.
  • B's rate: 1/6 tank per hour.
  • Combined rate: 1/4 + 1/6. Common denominator 12: 3/12 + 2/12 = 5/12 tank per hour.
  • Combined time: 1 ÷ (5/12) = 12/5 = 2.4 hours.

The wrong answer (averaging 4 and 6 to get 5) is plausible but always too big. The combined time is always less than the faster worker alone — that's a quick sanity check.

Special case: multiple identical workers. If k identical workers can do the job, and one alone would take t hours, then together they take t/k hours. (Their rates simply multiply by k.)

Work = Rate × Time. Same formula as D = S × T — just a different story. “Job” replaces “distance,” “jobs per hour” replaces “mph.” The triangle still works: cover the letter you want.
THE TRICK

For 'A alone takes a hours, B alone takes b hours, how long together?': memorize ab/(a+b). It's the same harmonic-mean formula from average-speed chapter 4.

For 3 or more workers, just add all the rates: 1/a + 1/b + 1/c + …, then flip.

The trap reframe. If a problem says 'A and B together take 3 hours, A alone takes 5 hours, how long does B alone take?' — set up B's rate as the unknown: 1/5 + 1/x = 1/3, solve for x.

WORKED EXAMPLE
PROBLEM · 2009 #6

Steve's empty swimming pool will hold 24,000 gallons of water when full. It will be filled by 4 hoses, each of which supplies 2.5 gallons of water per minute. How many hours will it take to fill Steve's pool?

A) 40 B) 42 C) 44 D) 46 E) 48

Steve's pool holds 24,000 gallons. Four hoses, each at 2.5 gallons/min. How long to fill?

Combined rate: 4 × 2.5 = 10 gallons per minute. Or in hours: 10 × 60 = 600 gallons per hour.

Time = 24,000 ÷ 600 = 40 hours.

This is the easiest flavor — all four hoses have the same rate, so we just multiply (rather than adding distinct rates). The deeper formula 1/(1/t_a + 1/t_b + …) reduces to t/k when all k rates are equal.

Identify the unit. Hoses are given in gallons/minute; the question asks for hours. Convert at the start (× 60) or at the end (÷ 60). Doing it once at the right place avoids unit confusion.

Answer: A — 40 hours.
RULE OF THUMB

Rates add; times don't. For two workers with times a and b: combined time = ab/(a+b). For k identical workers each taking time t: combined = t/k. Sanity check: combined time is always less than the fastest worker alone.

MORE LIKE THIS
2001 · #15 Homer began peeling a pile of 44 potatoes at the rate of 3 potatoes per minute. Four minutes later Christen joined him and peeled at the...

Homer began peeling a pile of 44 potatoes at the rate of 3 potatoes per minute. Four minutes later Christen joined him and peeled at the rate of 5 potatoes per minute. When they finished, how many potatoes had Christen peeled?

Show answer
Answer: A — 20.
Show hints
Hint 1 of 2
First find how many potatoes are left when Christen starts.
Still stuck? Show hint 2 →
Hint 2 of 2
Then they peel together at a combined rate — find how long that takes.
Show solution
Approach: head start, then combined rate
  1. In the first 4 minutes Homer peels 3 × 4 = 12, leaving 44 − 12 = 32 potatoes.
  2. Together they peel 3 + 5 = 8 per minute, so the rest takes 32 ÷ 8 = 4 minutes.
  3. In those 4 minutes Christen peels 5 × 4 = 20 potatoes.
2009 · #6 Steve's empty swimming pool will hold 24,000 gallons of water when full. It will be filled by 4 hoses, each of which supplies 2.5...

Steve's empty swimming pool will hold 24,000 gallons of water when full. It will be filled by 4 hoses, each of which supplies 2.5 gallons of water per minute. How many hours will it take to fill Steve's pool?

Show answer
Answer: A — 40 hours.
Show hint
Hint 1
Combined rate = 4 · 2.5 = 10 gal/min. Convert minutes to hours.
Show solution
Approach: combined rate, total volume, convert
  1. Rate: 10 gal/min ⇒ 600 gal/hour.
  2. Time: 24,000 / 600 = 40 hours.
⬢ FINAL TEST

Stretch test

Five harder rate problems combining D=ST with average-speed and exponential growth.

2001 · #15 Homer began peeling a pile of 44 potatoes at the rate of 3 potatoes per minute. Four minutes later Christen joined him and peeled at the...

Homer began peeling a pile of 44 potatoes at the rate of 3 potatoes per minute. Four minutes later Christen joined him and peeled at the rate of 5 potatoes per minute. When they finished, how many potatoes had Christen peeled?

Show answer
Answer: A — 20.
Show hints
Hint 1 of 2
First find how many potatoes are left when Christen starts.
Still stuck? Show hint 2 →
Hint 2 of 2
Then they peel together at a combined rate — find how long that takes.
Show solution
Approach: head start, then combined rate
  1. In the first 4 minutes Homer peels 3 × 4 = 12, leaving 44 − 12 = 32 potatoes.
  2. Together they peel 3 + 5 = 8 per minute, so the rest takes 32 ÷ 8 = 4 minutes.
  3. In those 4 minutes Christen peels 5 × 4 = 20 potatoes.
1999 · #22 In a far-off land three fish can be traded for two loaves of bread, and a loaf of bread can be traded for four bags of rice. How many...

In a far-off land three fish can be traded for two loaves of bread, and a loaf of bread can be traded for four bags of rice. How many bags of rice is one fish worth?

Show answer
Answer: D — 2⅔ bags.
Show hints
Hint 1 of 2
Convert bread into rice first, so everything is measured in rice.
Still stuck? Show hint 2 →
Hint 2 of 2
Then 3 fish equals that many bags — divide by 3.
Show solution
Approach: convert to a common good (rice), then divide
  1. Since 1 loaf = 4 bags of rice, 2 loaves = 8 bags, and 3 fish trade for those 2 loaves.
  2. So 3 fish = 8 bags, making one fish 8 ÷ 3 = 2⅔ bags of rice.
2019 · #16 Qiang drives 15 miles at an average speed of 30 miles per hour. How many additional miles will he have to drive at 55 miles per hour to...

Qiang drives 15 miles at an average speed of 30 miles per hour. How many additional miles will he have to drive at 55 miles per hour to average 50 miles per hour for the entire trip?

Show answer
Answer: D — 110 miles.
Show hints
Hint 1 of 2
Average speed = total distance / total time. Write that equation with x = additional miles.
Still stuck? Show hint 2 →
Hint 2 of 2
Time so far: 15/30 = 1/2 hr. Set (15 + x)/(1/2 + x/55) = 50.
Show solution
Approach: set total-distance / total-time = 50
  1. Time so far: 15/30 = 1/2 hour. After x additional miles at 55 mph, the new total is (15 + x) miles and (1/2 + x/55) hours.
  2. (15 + x) / (1/2 + x/55) = 50 ⇒ 15 + x = 25 + 10x/11.
  3. Multiply by 11: 165 + 11x = 275 + 10x ⇒ x = 110.
2023 · #15 (figure problem)
amc8-2023-15
Show answer
Answer: B — 4.2 mph.
Show hints
Hint 1 of 2
Find Viswam's usual speed in mph, then figure out how many blocks vs minutes he has left after the detour starts.
Still stuck? Show hint 2 →
Hint 2 of 2
After 5 blocks he has 5 min left. Detour makes the remaining distance 5 + (3 − 1) = 7 blocks instead of 5. Distance per block: 0.05 mile.
Show solution
Approach: convert blocks to miles, time to hours
  1. Usual: 10 blocks = 0.5 mile in 10 min = 1/6 hr → speed = 3 mph. Each block is 0.05 mile.
  2. After 5 blocks, 5 minutes are left. Detour replaces 1 block with 3, so remaining distance becomes 5 + 2 = 7 blocks = 0.35 mile.
  3. 5 minutes = 1/12 hr. Speed = 0.35 ÷ (1/12) = 0.35 × 12 = 4.2 mph.
2025 · #19 (figure problem)
amc8-2025-19
Show answer
Answer: D — 8.5 miles from A.
Show hints
Hint 1 of 2
Once both cars are in the middle section, they're going the same speed — the asymmetry is in how they got there.
Still stuck? Show hint 2 →
Hint 2 of 2
Car A reaches the middle in 5/25 = 1/5 hr; car B in 5/20 = 1/4 hr. A has a 1/20-hr head start in the middle section — that's 2 miles.
Show solution
Approach: let the cars meet in the middle 40-mph section
  1. Car A's left section is 5 miles at 25 mph → reaches the middle in 5/25 = 1/5 hr. Car B's right section is 5 miles at 20 mph → reaches the middle in 5/20 = 1/4 hr. A enters the middle 1/20 hr before B.
  2. In that 1/20 hr, A travels 40 × 1/20 = 2 miles, so when B enters the middle, A is at mile 7, B at mile 10 — a 3-mile gap.
  3. Both now go 40 mph, closing at 80 mph. They split the 3-mile gap equally: each covers 1.5 miles. A is at 7 + 1.5 = 8.5 miles from A.
APPENDIX

Rates quick-reference

Memorize these

FORMULAS / FACTS TO KNOW COLD

  • D = S × T (and the two cousins). Same as Work = Rate × Time.
  • Average speed = total distance ÷ total time. NEVER the average of the speeds (unless times are equal).
  • 1 hour = 3600 seconds = 60 minutes.
  • 1 mile = 5280 feet. 1 km = 1000 m. 1 yard = 3 feet.
  • 1 mph ≈ 1.467 ft/s (so 60 mph = 88 ft/s).
  • Equal-distance round trip avg speed: 2ab / (a+b) (harmonic mean).
  • Equal-time legs avg speed: (a+b)/2 (simple average).
  • Together-time (A alone takes a, B alone takes b): T = ab / (a+b).
  • Same-direction closing speed: faster − slower.
  • Opposite-direction closing speed: sum of speeds.
  • Exponential growth: V(n) = V₀ · rn. Compound interest: V₀ · (1 + p/100)n.
Common traps
  • Averaging speeds for equal-distance trips. The slow leg dominates the time, so the average pulls toward the slow speed.
  • Forgetting to convert minutes to hours before applying D = S × T with mph.
  • Inverse vs direct proportion confusion. 'More workers, less time' is inverse; 'more time, more distance' is direct.
  • Cumulative vs per-period graphs. Cumulative answers are differences; per-period answers are direct reads.
  • Linear vs exponential growth. A 10% annual raise four times compounds to 46.4%, not 40%.
Warm-ups

Drill these:

  • 60 mph for 90 minutes = how many miles? (90)
  • Faucet A fills tub in 6 min, faucet B in 12 min; both: rates add → 1/6 + 1/12 = 1/4 → 4 min.
  • Mix 30% of 10 L with 70% of 20 L. Combined %? ((0.3·10 + 0.7·20) / 30 ≈ 56.7%)
  • 4 painters take 9 hours, how long for 6? (Inverse: 4·9 = 36 painter-hours; ÷6 = 6 hours.)
  • Round-trip 60 mph and 30 mph (equal distance) average? (40 mph via 2·60·30/(60+30).)
Want to climb higher? — advanced rate ideas (#22–#25 territory)
  • Three-worker together-time (A, B, C alone take a, b, c hours): T = 1 / (1/a + 1/b + 1/c). Same pattern, three terms.
  • Filling AND draining at once. If pipe A fills at rate 1/a and drain D empties at rate 1/d, the net rate is 1/a − 1/d. The drain SUBTRACTS from the filling.
  • Mixture problems. Mixing x liters of A% solution with y liters of B% solution gives a (x+y)-liter mix with concentration (xA + yB) / (x + y). Same as a weighted average.
  • “Train passes a person” vs “train passes a station.” To pass a person (a point): train travels its own length. To pass a station (also has length): train travels train length + station length. Time = (relevant length) / train speed.
  • Continuous compound interest (rare on AMC 8): ert grows faster than discrete (1+r)t for the same r, t.
→ Practice Ratios, Rates & Proportions problems