AMC 10 · 2014 · #4

Grade 7 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation

rateratio-proportion dimensional-analysisconvert-to-algebraeasier-related-problem ↑ Prerequisites: ratio-proportion

📏 Short solution 💡 1 insight

Problem

A known output comes from a known count over a known time at a steady rate. Write the output for new numbers.

Pick an answer.

(A)

$\frac{bde}{ac}$

(B)

$\frac{ac}{bde}$

(C)

$\frac{abde}{c}$

(D)

$\frac{bcde}{a}$

(E)

$\frac{abc}{de}$

How to solve

Strategy Analyze the Units

The phrase "at this rate" is the signal for Tool #8: track the units. Milk is measured in gallons, but the thing that stays constant is gallons per cow per day. If we boil the given information down to that one unit rate, the rest is just multiplying it back up by the new number of cows and the new number of days. Tool #4 helps keep the five letters straight — treat each as a placeholder and build the expression piece by piece. Tool #9 is the safety net: if the letters feel slippery, replace them with easy numbers, solve the concrete version, and confirm the formula reproduces that number.

1STEP 1

Find milk per cow per day

Dividing twice gives the unit rate.

rate = (b gallons)/(a cows × c days) = b/ac gallons per cow per day

2STEP 2

Scale up to the new herd and time

Scaling by the new numbers is multiplication.

b/ac × d × e = bde/ac gallons

3STEP 3

Check the units and match a choice

The units check out, choice (A).

gal/(cow·day)·cow·day=gal → bde/ac → (A)

Answer

bde/ac

Sanity-check the shape of the answer: more cows (d) or more days (e) should give more milk, and both appear in the numerator, so the amount grows when they grow — good. The starting cows a and days c appear in the denominator, which is right because if the original a cows had needed more days c to make the same b gallons, each cow would be slower, so the output should shrink. Plugging in a=2,b=3,c=4,d=5,e=6: the rate is 3/8 gallon per cow-day, and 5 cows over 6 days give 3/8×5×6=90/8=11.25 gallons; the formula bde/ac=90/8 agrees. The answer is (A).

💡Key takeaway

For any "at this rate" problem, shrink everything down to one unit (one cow, one day), find that single rate, then multiply it back up by the new amounts.

Find milk per cow per day
Scale up to the new herd and time
Check the units and match a choice

Find milk per cow per day

Scale up to the new herd and time

Check the units and match a choice

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