Competition · AMC preparation · step 4 of 4
AMC 8 · 2007 · #5
Grade 6 arithmeticPick an answer.
AMC 8 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Tool #7 (Identify Subproblems) breaks the question into three smaller pieces: (a) total birthday gift money, (b) the gap between gifts and the bike price, and (c) how many 16 chunks fit into that gap. Each piece is a single Grade 4 step. Tool #3 (Write an Equation) shows up at the end as the unit-rate division: weeks = gap ÷16. We deliberately avoid full Tool #13 (Algebra) — no need to set up 500 = 100 + 16w when the same answer falls out of arithmetic.
Add the three gifts
Add the three birthday gifts to get his starting $100 before any paper-route work.
Combining three gift amounts into one running total is a Grade 4 addition step.
4.NBT.B.4Identify SubproblemsSubtract from the bike price
Subtract the $100 from the $500 goal; $400 is what the paper route must still cover.
Splitting the $500 goal into "already have" and "still need" is the standard multi-step word-problem move.
4.OA.A.3Identify SubproblemsDivide by the weekly pay
Divide the $400 gap by the $16 weekly rate — that lands on 25 weeks.
Dollars per week is a unit rate. Dividing total dollars needed by dollars-per-week gives the count of weeks — exactly the Grade 6 unit-rate idea.
The number of weeks of paper-route work equals the $400 the gifts left unpaid divided by the $16 Chandler earns each week.
▸ Why?
After a whole number of weeks, the paper route has added $16 that same number of times, so the earnings are $16 multiplied by the number of weeks.
▸ Why?
The bike's $500 splits with no overlap into the gift money and the paper-route earnings, so the earnings alone must make up the $400 the gifts did not cover.
▸ Why?
Since the weekly $16 times the number of weeks must equal $400, and dividing undoes multiplying, the number of weeks is $400 divided by $16.
When you are saving for a big goal, find what you already have, subtract to see the gap, then divide by what you earn each week. The answer is just one division.
- Add the three gifts
- Subtract from the bike price
- Divide by the weekly pay
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