AMC 10 · 2014 · #8
Grade 6 arithmeticA truck travels 6b feet every t seconds. There are 3 feet in a yard. How many yards does the truck travel in 3 minutes?
Pick an answer.
AMC 10 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: A truck moves at a fixed rate of \frac{b}{6} feet for each t seconds. Using that 3 feet make 1 yard, find how many yards it travels in 3 minutes, written as an expression in b and t.
Givens: The truck travels \frac{b}{6} feet every t seconds.; 3 feet = 1 yard.; The time interval of interest is 3 minutes.
Unknowns: The number of yards the truck travels in 3 minutes, in terms of b and t.
Understand
Restated: A truck moves at a fixed rate of \frac{b}{6} feet for each t seconds. Using that 3 feet make 1 yard, find how many yards it travels in 3 minutes, written as an expression in b and t.
Givens: The truck travels \frac{b}{6} feet every t seconds.; 3 feet = 1 yard.; The time interval of interest is 3 minutes.
Plan
Primary tool: #8 Analyze the Units
Secondary: #13 Convert to Algebra, #3 Eliminate Possibilities
The problem mixes feet with yards and seconds with minutes, so tracking units is the backbone: line every quantity up in one unit, then multiply and divide. The letters b and t are carried along as an algebraic expression, and the answer choices let you sanity-check by where b and t land.
Execute — Answer: E
6.RP.A.3 Step 1 Convert minutes to seconds
- The rate is given per t seconds, but the trip length is given in minutes.
- Put both times in the same unit.
- One minute is 60 seconds, so 3 minutes is 3 times 60 = 180 seconds.
💡 You can only combine quantities once they share the same unit.
6.RP.A.2 Step 2 Total the feet traveled
- The truck covers \frac{b}{6} feet in each t-second chunk.
- In 180 seconds there are \frac{180}{t} such chunks.
- Multiply the number of chunks by the feet in one chunk to get the total feet.
💡 Total distance is how many equal chunks you travel times the length of one chunk.
6.EE.A.2 Step 3 Simplify the feet expression
- Reduce the fraction by dividing 180 by 6.
- The letters b and t just stand for numbers, so the usual rules for simplifying fractions apply.
💡 Simplifying an expression with letters works the same as with plain numbers.
6.RP.A.3 Step 4 Convert feet to yards
- There are 3 feet in a yard, so divide the number of feet by 3 to get yards.
- Dividing \frac{30b}{t} by 3 gives \frac{10b}{t}.
- That matches choice (E).
💡 It takes more feet to make each yard, so the yard count is smaller — you divide.
6.RP.A.3 The rate is given per t seconds, but the trip length is given in minutes. Put bo 6.RP.A.2 The truck covers \frac{b}{6} feet in each t-second chunk. In 180 seconds there a 6.EE.A.2 Reduce the fraction by dividing 180 by 6. The letters b and t just stand for num 6.RP.A.3 There are 3 feet in a yard, so divide the number of feet by 3 to get yards. Divi Review
Reasonableness: Check the structure and a test case. A bigger b (faster truck) should mean more distance, and \frac{10b}{t} grows with b — good. A bigger t means each chunk takes longer, so fewer chunks and less distance; \frac{10b}{t} shrinks as t grows — also good. Test b=6, t=1: the rate is 1 ft/s, so 180 s gives 180 ft = 60 yards, and \frac{10 \times 6}{1} = 60. It matches.
Alternative: Set up a proportion: \frac{b}{6} feet per t seconds equals x feet per 180 seconds, so x = \frac{180}{t} \times \frac{b}{6} = \frac{30b}{t} feet, then divide by 3 to get \frac{10b}{t} yards. Or reason by units alone: distance grows with b and shrinks with t, which leaves only (D) and (E); the constant then selects (E).
CCSS standards used (min grade 6)
6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems (Converting minutes to seconds and feet to yards, and reasoning about the rate across the full time interval)6.RP.A.2Understand the concept of a unit rate and use rate language (Treating \frac{b}{6} feet per t seconds as a rate and scaling it up to 180 seconds)6.EE.A.2Write, read, and evaluate expressions in which letters stand for numbers (Building and simplifying the expression \frac{180b}{6t} into \frac{30b}{t} with b and t standing for numbers)
⭐ Get everything into the same units first, and then the multiplying and dividing takes care of itself.
⭐ Get everything into the same units first, and then the multiplying and dividing takes care of itself.
More like this
Same archetype — closest grade level first.