AMC 10 · 2003 · #2

Grade 5 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation

multi-digit-arithmeticlinear-equations-one-var identify-subproblems ↑ Prerequisites: multi-digit-arithmetic

📏 Medium solution 💡 2 insights

Problem

For two weeks Al takes one green pill and one pink pill every day. A green pill costs $1 more than a pink pill, and the whole two-week supply costs $546. Find the cost of one green pill.

Pick an answer.

(A)

$\textdollar 7$

(B)

$\textdollar 14$

(C)

$\textdollar 19$

(D)

$\textdollar 20$

(E)

$\textdollar 39$

AMC 10 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The $546 covers 14 days at once, which is too big to reason about directly, so Tool #7 (Identify Subproblems) breaks it into three easy layers: how many days, the cost of one day's pair, then splitting that pair into the two pills. Tool #8 (Analyze the Units) keeps the middle layer honest — the money is dollars-per-day for a green-plus-pink pair, so dividing the total by the number of days gives the daily pair cost. Tool #3 (Eliminate Possibilities) catches the built-in trap that$ 39 (choice (E)) is the cost of the pair, not of the green pill alone.

1STEP 1

Count the days and the pills

Two weeks is 2×7=14 days, so the $546 buys 14 identical daily pairs — one green pill plus one pink pill each.

2 weeks=14 days → 14 (green + pink) pairs

2STEP 2

Find one day's cost

All 14 pairs cost the same, so one pair is 546÷14 = $39.

546÷14=39 → green+pink=$39

3STEP 3

Split the pair using the $1 gap

Take the extra $1 off first: 39-1=38 is two pink pills, so pink is 38÷2=19 and green is 19+1 = $20.

pink=(39-1)/2=19, green=19+1=20

4STEP 4

Check and pick the answer

Check: 20+19=39 a day and 39×14=546. The green pill is $20, choice (D); $39 is the whole pair.

39×14=546 ✓ → green=$20=(D)

Answer

$ 20

The daily pair costs $39, and the two pills are almost equal (they differ by only $1), so each must be very close to half of $39, near $19.50. A green pill of $20 and a pink pill of $19 sit right around that half and differ by exactly $1, so the answer is sensible. The small choices (A) $7 and (B) $14 are far below half the pair and cannot be right, while (E) $39 is the full pair — three times too big for a single pill.

💡Key takeaway

When two things add to a known total and you know their difference, take the difference off first, split the rest in half, then give the extra back to the bigger one.

Count the days and the pills
Find one day's cost
Split the pair using the $1 gap
Check and pick the answer

Count the days and the pills

Find one day's cost

Split the pair using the $1 gap

Check and pick the answer

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