AMC 8 · 2006 · #3

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

ratefraction-arithmeticfraction-decimal-conversion identify-subproblems ↑ Prerequisites: fraction-arithmeticmulti-digit-arithmetic

📏 Short solution 💡 2 insights

Problem

Elisa used to swim 10 laps in 25 minutes. Now she swims 12 laps in 24 minutes. How many minutes per lap has she shaved off?

Pick an answer.

(A)

$\frac{1}{2}$

(B)

$\frac{3}{4}$

(C)

(D)

(E)

AMC 8 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Analyze the Units

The question asks for an improvement in "minutes per lap," but the data is given as totals (laps and minutes mixed together). Tool #8 (Analyze the Units) tells us to convert each pace to the requested unit first — minutes ÷ laps gives minutes per lap. Tool #7 (Identify Subproblems) splits the work into three small steps: find the old per-lap time, find the new per-lap time, and subtract. Each subproblem is a single division or subtraction.

1STEP 1

Old pace: divide the old minutes by the old laps to get 2.5 min per lap.

\frac{25 min}{10 laps}

= 2.5 min/lap

2STEP 2

New pace: the same division gives 2 min per lap.

\frac{24 min}{12 laps}

= 2 min/lap

3STEP 3

Both are in minutes per lap, so subtract old minus new: she shaved off $\frac{1}{2}$ min per lap.

2.5 - 2 = 0.5 =

\frac{1}{2}

min/lap → (A)

Answer

\frac{1}{2}

Sanity check: Elisa went from 10 laps to 12 laps in roughly the same time window, so her per-lap time should drop only a little — half a minute fits that picture. The other choices are too big: an improvement of 1, 2, or 3 minutes per lap would mean the old time was at least 3 minutes per lap, but 25 ÷ 10 = 2.5 rules those out. Choice (B)

\frac{3}{4}

would require the new time to be 2.5 - 0.75 = 1.75 min/lap, but 24 ÷ 12 = 2 exactly, so (A) is the only consistent answer.

💡Key takeaway

When a problem asks "per lap" or "per minute," convert both rates to that unit first — once the units match, subtraction tells the whole story.

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