AMC 10 · 2014 · #1
Grade 8 arithmeticWhat is 10⋅(21+51+101)−1?
Pick an answer.
AMC 10 2014 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: We must evaluate $10\cdot\left(\tfrac{1}{2}+\tfrac{1}{5}+\tfrac{1}{10}\right)^{-1}$: add the three fractions inside the parentheses, apply the exponent $-1$, then multiply by $10$.
Givens: The expression $10\cdot\left(\tfrac{1}{2}+\tfrac{1}{5}+\tfrac{1}{10}\right)^{-1}$; Three unit-style fractions to add: $\tfrac{1}{2}$, $\tfrac{1}{5}$, $\tfrac{1}{10}$; An exponent of $-1$ applied to the parenthesized sum; Five answer choices: (A) $3$, (B) $8$, (C) $\frac{25}{2}$, (D) $\frac{170}{3}$, (E) $170$
Unknowns: The single numerical value of the whole expression
Understand
Restated: We must evaluate $10\cdot\left(\tfrac{1}{2}+\tfrac{1}{5}+\tfrac{1}{10}\right)^{-1}$: add the three fractions inside the parentheses, apply the exponent $-1$, then multiply by $10$.
Givens: The expression $10\cdot\left(\tfrac{1}{2}+\tfrac{1}{5}+\tfrac{1}{10}\right)^{-1}$; Three unit-style fractions to add: $\tfrac{1}{2}$, $\tfrac{1}{5}$, $\tfrac{1}{10}$; An exponent of $-1$ applied to the parenthesized sum; Five answer choices: (A) $3$, (B) $8$, (C) $\frac{25}{2}$, (D) $\frac{170}{3}$, (E) $170$
Plan
Primary tool: #7 Identify Subproblems
Secondary: #16 Change Focus / Count the Complement, #3 Eliminate Possibilities
The expression stacks three separate operations, so the clean move is to split it into subproblems (Tool #7): first add the fractions, then handle the $-1$ exponent, then multiply by $10$. The $-1$ is the one trap here — Tool #16 (Change Focus) reframes "raise to the power $-1$" as the simpler idea "flip the fraction over." Because it is multiple-choice, Tool #3 lets us confirm the final value lands exactly on a listed choice.
Execute — Answer: C
5.NF.A.1 Step 1 Add the three fractions
- Start with the innermost subproblem: the sum inside the parentheses.
- The denominators $2$, $5$, and $10$ all divide into $10$, so rewrite each fraction with denominator $10$: $\tfrac{1}{2}=\tfrac{5}{10}$, $\tfrac{1}{5}=\tfrac{2}{10}$, and $\tfrac{1}{10}$ stays.
- Adding the numerators gives $5+2+1=8$, so the sum is $\tfrac{8}{10}$, which simplifies to $\tfrac{4}{5}$.
💡 A common denominator turns three different-sized pieces into one kind of piece you can just count up.
8.EE.A.1 Step 2 Flip for the -1 exponent
- Now handle the exponent.
- Raising any nonzero number to the power $-1$ gives its reciprocal — you turn the fraction upside down.
- So $\left(\tfrac{4}{5}\right)^{-1}=\tfrac{5}{4}$.
- This is the step where the scary-looking $-1$ becomes an easy flip.
💡 A power of $-1$ just means "turn it over": numerator and denominator swap places.
4.NF.B.4 Step 3 Multiply by 10
- Finish with the last subproblem: multiply the $10$ out front by the reciprocal $\tfrac{5}{4}$.
- Multiplying a whole number by a fraction multiplies the numerator: $10\cdot\tfrac{5}{4}=\tfrac{50}{4}=\tfrac{25}{2}$.
- That value matches choice (C).
💡 Ten copies of $\tfrac{5}{4}$ is $\tfrac{50}{4}$, and halving top and bottom by $2$ gives $\tfrac{25}{2}$.
5.NF.A.1 Start with the innermost subproblem: the sum inside the parentheses. The denomin 8.EE.A.1 Now handle the exponent. Raising any nonzero number to the power $-1$ gives its 4.NF.B.4 Finish with the last subproblem: multiply the $10$ out front by the reciprocal $ Review
Reasonableness: Check with decimals: $\tfrac{1}{2}+\tfrac{1}{5}+\tfrac{1}{10}=0.5+0.2+0.1=0.8$, its reciprocal is $\tfrac{1}{0.8}=1.25$, and $10\times1.25=12.5=\tfrac{25}{2}$. This agrees with choice (C). The trap answers come from mishandling the $-1$: forgetting to flip gives $10\cdot\tfrac45=8$ (choice B), and treating $-1$ as multiplying instead of reciprocal-ing leads nowhere near a clean value — so (C) is the sound result.
Alternative: Instead of adding fractions, convert everything to decimals from the start (Tool #15, Organize Information in More Ways): $0.5+0.2+0.1=0.8$, then $10\div 0.8=12.5=\tfrac{25}{2}$. Same answer, reached by reading the $-1$ power directly as "divide by the sum."
CCSS standards used (min grade 8)
5.NF.A.1Add and subtract fractions with unlike denominators (Adding $\tfrac{1}{2}+\tfrac{1}{5}+\tfrac{1}{10}$ by rewriting each over the common denominator $10$.)8.EE.A.1Know and apply the properties of integer exponents (Interpreting the exponent $-1$ as the reciprocal, so $\left(\tfrac{4}{5}\right)^{-1}=\tfrac{5}{4}$.)4.NF.B.4Apply and extend understanding of multiplication to multiply a fraction by a whole number (Multiplying $10\cdot\tfrac{5}{4}=\tfrac{50}{4}=\tfrac{25}{2}$ to finish.)
⭐ A power of $-1$ just means "flip the fraction," and the rest is add-then-multiply.
⭐ A power of $-1$ just means "flip the fraction," and the rest is add-then-multiply.
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