AMC 10 · 2010 · #8

Grade 6 arithmetic

Archetype Small-Domain Constrained Search └ Sub-archetype Answer-Choice Plug-In

ratelinear-equations-one-varcasework convert-to-algebra ↑ Prerequisites: linear-equations-one-var

📏 Short solution 💡 2 insights

Problem

Tony works 2 hours each day, and for every full year of his age he is paid $0.50 per hour. Over a six-month period he worked 50 days and earned $630 in all. Find how old Tony was at the end of those six months.

Pick an answer.

(A)

$\ 9$

(B)

$\ 11$

(C)

$\ 12$

(D)

$\ 13$

(E)

$\ 14$

AMC 10 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Introduce a Variable

First untangle the pay rule by tracking units (Tool #8, Analyze the Units): 2 hours times $0.50 per hour per year of age gives$ 1 per year of age each day, so his daily pay in dollars is simply his age in years. Then test the two nearby fixed ages (Tool #3, Eliminate Possibilities): no single unchanging age can produce $630, which forces a birthday partway through. Finally name the number of older-age days with a letter (Tool #4, Introduce a Variable) and solve a one-step equation; a positive whole-number answer confirms the birthday, so the ending age is the older one.

1STEP 1

Turn the pay rule into daily pay

Chase the units: 2 hours × $0.50 per year of age is $1 per year of age each day, so daily pay in dollars equals his age.

2 × 0.50 × a = a dollars per day

2STEP 2

Test a fixed age

A fixed age would give 50 × 12 = 600 or 50 × 13 = 650, but 630 lands strictly between — so his age rose partway through.

50 × 12 = 600, 50 × 13 = 650, 600 ＜ 630 ＜ 650

3STEP 3

Count the days and read off the age

Let d be the days at the older age: 12(50 - d) + 13d = 630 becomes 600 + d = 630, so d = 30 and 20 days sit at 12 — he ends at 13.

12(50-d) + 13d = 630 → 600 + d = 630 → d = 30

Answer

Put the split back in: 20 days at $12 is $240, and 30 days at $13 is $390, and 240 + 390 = 630, matching the total exactly. The day counts 20 and 30 are both positive and add to 50, so the story holds together. The two impossible fixed-age totals, $600 and $650, bracket $630, confirming the age had to cross from 12 to 13. So the ending age is 13, choice (D).

💡Key takeaway

His pay each day is just his age in dollars, so a total stuck between 50×12 and 50×13 means he turned 13 partway through — and that is his age at the end.

Turn the pay rule into daily pay
Test a fixed age
Count the days and read off the age

Turn the pay rule into daily pay

Test a fixed age

Count the days and read off the age

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