AMC 10 · 2003 · #2
Easy mode Grade 5Every day for two weeks, Al takes one green pill and one pink pill. A green pill costs \textdollar1 more than a pink pill. Together, all of his pills cost \textdollar546 for the two weeks. How much does one green pill cost?
Pick an answer.
AMC 10 2003 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: 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.
Givens: The pills are taken every day for two weeks; Each day Al takes exactly one green pill and one pink pill; A green pill costs $\$1$ more than a pink pill; The total cost for the two weeks is $\$546$; Answer choices: (A) $\$7$, (B) $\$14$, (C) $\$19$, (D) $\$20$, (E) $\$39$
Unknowns: The cost of one green pill in dollars
Understand
Restated: 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.
Givens: The pills are taken every day for two weeks; Each day Al takes exactly one green pill and one pink pill; A green pill costs $\$1$ more than a pink pill; The total cost for the two weeks is $\$546$; Answer choices: (A) $\$7$, (B) $\$14$, (C) $\$19$, (D) $\$20$, (E) $\$39$
Plan
Primary tool: #7 Identify Subproblems
Secondary: #8 Analyze the Units, #3 Eliminate Possibilities
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.
Execute — Answer: D
4.MD.A.2 Step 1 Count the days and the pills
- Two weeks is $2\times7=14$ days.
- Each day Al takes one green pill and one pink pill, so the two-week supply is $14$ green pills and $14$ pink pills.
- Group them by day: the $\$546$ pays for $14$ identical daily pairs, each pair being one green pill plus one pink pill.
💡 Bundling the pills into one-day pairs turns a two-week bill into $14$ copies of the same small purchase.
5.NBT.B.6 Step 2 Find one day's cost
The $14$ pairs together cost $\$546$, and every pair costs the same, so one pair costs $546\div14$. Dividing gives $546\div14=39$. So one green pill and one pink pill together cost $\$39$.
💡 Splitting one fixed total evenly among equal groups is just division: total dollars over number of days.
4.OA.A.3 Step 3 Split the pair using the $\$1$ gap
- The two pills in a pair add to $\$39$, and the green one costs $\$1$ more than the pink one.
- Take that extra $\$1$ off the top: $39-1=38$ is what the pair would cost if both pills were at the cheaper pink price — that is, two pink pills. So one pink pill is $38\div2=19$ dollars, and the green pill is $\$1$ more: $19+1=20$ dollars.
💡 Remove the difference first so both parts are equal, halve the rest, then add the difference back to the bigger one.
4.OA.A.3 Step 4 Check and pick the answer
- Confirm the prices fit: green $\$20$ plus pink $\$19$ is $\$39$ per day, and $39\times14=546$ dollars for two weeks — exactly the given total. The green pill costs $\$20$, which is choice (D).
- Watch the trap: $\$39$ (choice (E)) is the cost of the whole daily pair, not the green pill, and $\$19$ (choice (C)) is the pink pill.
💡 A quick multiply-back confirms the split, and naming what each wrong choice really measures rules them out.
4.MD.A.2 Two weeks is $2\times7=14$ days. Each day Al takes one green pill and one pink p 5.NBT.B.6 The $14$ pairs together cost $\$546$, and every pair costs the same, so one pair 4.OA.A.3 The two pills in a pair add to $\$39$, and the green one costs $\$1$ more than t 4.OA.A.3 Confirm the prices fit: green $\$20$ plus pink $\$19$ is $\$39$ per day, and $39 Review
Reasonableness: 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.
Alternative: Use algebra: let the pink pill cost $p$ dollars, so the green pill costs $p+1$. One day's pair is $p+(p+1)=2p+1$, and $14$ days cost $14(2p+1)=546$. Dividing gives $2p+1=39$, so $2p=38$ and $p=19$; then the green pill is $p+1=20$ dollars, choice (D).
CCSS standards used (min grade 5)
4.MD.A.2Solve word problems involving distances, time, liquid volumes, and money (Turning "two weeks, one green and one pink pill a day" into $14$ equal daily pairs that share the $\$546$ total.)5.NBT.B.6Find whole-number quotients with up to four-digit dividends and two-digit divisors (Dividing $546\div14=39$ to get the cost of one day's green-plus-pink pair.)4.OA.A.3Solve multistep word problems using the four operations with whole numbers (Splitting the $\$39$ pair with the $\$1$ difference ($\tfrac{39-1}{2}=19$, then $+1=20$) and checking $39\times14=546$.)
⭐ 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.
⭐ 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.
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