AMC 10 · 2004 · #3
Grade 6 arithmeticAlicia earns 20 dollars per hour, of which 1.45% is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes?
Pick an answer.
AMC 10 2004 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: Alicia earns $20$ dollars for each hour she works. Of that hourly pay, $1.45\%$ is taken out for local taxes. Find how many cents of each hour's pay go to local taxes.
Givens: Hourly wage is $20$ dollars.; Local taxes take $1.45\%$ of the hourly wage.; The answer must be reported in cents, not dollars.; Answer choices: (A) $0.0029$, (B) $0.029$, (C) $0.29$, (D) $2.9$, (E) $29$
Unknowns: The number of cents per hour that go to local taxes.
Understand
Restated: Alicia earns $20$ dollars for each hour she works. Of that hourly pay, $1.45\%$ is taken out for local taxes. Find how many cents of each hour's pay go to local taxes.
Givens: Hourly wage is $20$ dollars.; Local taxes take $1.45\%$ of the hourly wage.; The answer must be reported in cents, not dollars.; Answer choices: (A) $0.0029$, (B) $0.029$, (C) $0.29$, (D) $2.9$, (E) $29$
Plan
Primary tool: #8 Analyze the Units
Secondary: #3 Eliminate Possibilities
The whole problem turns on units (Tool #8): the wage is given in dollars but the answer is wanted in cents, and the tax rate is a percent. If you convert the wage to cents first and then take the percent, the answer comes out in cents directly. The trap is to compute $1.45\%$ of $\$20$ and stop at $\$0.29$, forgetting to turn dollars into cents — that lands on choice (C). Tool #3 (Eliminate Possibilities) helps because every choice is the same digits $29$ shifted by a power of ten, so only the correct unit handling picks the right one.
Execute — Answer: E
4.MD.A.1 Step 1 Convert the wage to cents
- Because the answer is asked for in cents, change the hourly wage from dollars to cents before doing anything else.
- Since $1$ dollar is $100$ cents, $20$ dollars is $20\times100 = 2000$ cents.
- Working in cents from the start means the taxes will also come out in cents.
💡 Match your units to the question first, so the final number is already in the units you were asked for.
6.RP.A.3 Step 2 Take 1.45% of the wage
- A percent is a rate per hundred, so $1.45\%$ means $\tfrac{1.45}{100}=0.0145$.
- Take that fraction of the $2000$ cents: $0.0145\times2000$.
- Multiplying $2000$ by $0.0145$ is the same as $2000\times\tfrac{1.45}{100}=20\times1.45=29$.
- So $29$ cents of each hour's pay go to local taxes.
💡 Percent means "out of a hundred", so a percent of a quantity is that many hundredths of it.
4.MD.A.1 Step 3 Match to a choice and rule out the trap
- The result is $29$ cents, which is choice (E).
- Every other choice is the digits $29$ moved by a power of ten and comes from a units slip: (C) $0.29$ is $1.45\%$ of $\$20$ left in dollars instead of cents, and (A), (B), (D) each drop or move the decimal further.
- Only converting dollars to cents gives a whole number of cents, so the answer is (E).
💡 When the choices are the same digits at different decimal places, getting the units right is the whole problem.
4.MD.A.1 Because the answer is asked for in cents, change the hourly wage from dollars to 6.RP.A.3 A percent is a rate per hundred, so $1.45\%$ means $\tfrac{1.45}{100}=0.0145$. T 4.MD.A.1 The result is $29$ cents, which is choice (E). Every other choice is the digits Review
Reasonableness: A sanity check on size: $1.45\%$ is close to $1\%$, and $1\%$ of $2000$ cents is $20$ cents, so the tax should be a little over $20$ cents. The value $29$ cents fits that estimate, while $0.29$ or $2.9$ cents would be far too small for a percent of $2000$ cents. A tax that is only a few hundredths of a cent (choices A and B) is not sensible for a rate near $1\%$ of two thousand cents.
Alternative: Work in dollars, then convert at the end: $1.45\%$ of $\$20$ is $0.0145\times20 = \$0.29$. Converting the dollar amount to cents at the last step, $\$0.29 = 0.29\times100 = 29$ cents, again giving (E). This shows the only way to miss is to forget the final dollars-to-cents conversion.
CCSS standards used (min grade 6)
4.MD.A.1Know relative sizes of measurement units and convert larger to smaller units (Converting $20$ dollars into $2000$ cents and recognizing that the final answer must be reported in cents.)6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems (Reading $1.45\%$ as $\tfrac{1.45}{100}$ and finding that percent of the $2000$-cent wage.)
⭐ Convert to the units the question asks for first, then take the percent — here that turns $20 dollars into 2000 cents, and 1.45% of 2000 is 29 cents.
⭐ Convert to the units the question asks for first, then take the percent — here that turns $20 dollars into 2000 cents, and 1.45% of 2000 is 29 cents.
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