AMC 10 · 2009 · #3
Grade 6 arithmeticWhich of the following is equal to 1+1+1+111?
Pick an answer.
AMC 10 2009 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: Evaluate the nested (continued) fraction $1 + \frac{1}{1 + \frac{1}{1 + 1}}$ and match its value to one of the five answer choices.
Givens: The expression $1 + \frac{1}{1 + \frac{1}{1 + 1}}$; Three layers of "$1 +$" stacked, with a $1 + 1$ at the very bottom; Answer choices: (A) $\frac{5}{4}$, (B) $\frac{3}{2}$, (C) $\frac{5}{3}$, (D) $2$, (E) $3$
Unknowns: The single numerical value of the whole expression
Understand
Restated: Evaluate the nested (continued) fraction $1 + \frac{1}{1 + \frac{1}{1 + 1}}$ and match its value to one of the five answer choices.
Givens: The expression $1 + \frac{1}{1 + \frac{1}{1 + 1}}$; Three layers of "$1 +$" stacked, with a $1 + 1$ at the very bottom; Answer choices: (A) $\frac{5}{4}$, (B) $\frac{3}{2}$, (C) $\frac{5}{3}$, (D) $2$, (E) $3$
Plan
Primary tool: #7 Identify Subproblems
Secondary: #11 Work Backwards, #3 Eliminate Possibilities
A stacked fraction like this cannot be attacked from the top, because the top $\frac{1}{\dots}$ needs its denominator finished first. Tool #7 (Identify Subproblems) says: break the tower into layers and solve the smallest, innermost one first. Here the innermost subproblem is just $1 + 1$. Once that is a number, the next fraction becomes computable, and so on outward. That inside-out order is also Tool #11 (Work Backwards) — the deepest layer is really the starting point. Tool #3 (Eliminate Possibilities) gives a quick sanity net: the outer "$1 +$ (something between $0$ and $1$)" forces the answer to sit strictly between $1$ and $2$.
Execute — Answer: C
5.OA.A.1 Step 1 Peel to the innermost layer
- A fraction bar works like hidden parentheses, so the very bottom must be evaluated first.
- The deepest denominator is $1 + 1$, which is simply $2$.
💡 Anything sitting under a fraction bar is really inside parentheses, so you finish it before dividing.
5.NF.A.1 Step 2 Climb up one floor
- Now handle the next layer, $1 + \frac{1}{2}$.
- Write $1$ as $\frac{2}{2}$ so the denominators match, then add.
💡 To add a whole number to a fraction, rewrite the whole number with the same denominator so the pieces are the same size.
6.NS.A.1 Step 3 Flip the middle fraction
- The expression is now $1 + \frac{1}{\,3/2\,}$.
- Dividing $1$ by $\frac{3}{2}$ means multiplying by its reciprocal, so $\frac{1}{3/2} = \frac{2}{3}$.
💡 Dividing by a fraction is the same as multiplying by its flip, so $1$ over $\frac{3}{2}$ just turns into $\frac{2}{3}$.
5.NF.A.1 Step 4 Add the outermost 1
- Finish the top layer, $1 + \frac{2}{3}$.
- Rewrite $1$ as $\frac{3}{3}$ and add to reach the final value, which matches choice (C).
💡 One last same-denominator add finishes the climb: $\frac{3}{3} + \frac{2}{3} = \frac{5}{3}$.
5.OA.A.1 A fraction bar works like hidden parentheses, so the very bottom must be evaluat 5.NF.A.1 Now handle the next layer, $1 + \frac{1}{2}$. Write $1$ as $\frac{2}{2}$ so the 6.NS.A.1 The expression is now $1 + \frac{1}{\,3/2\,}$. Dividing $1$ by $\frac{3}{2}$ mea 5.NF.A.1 Finish the top layer, $1 + \frac{2}{3}$. Rewrite $1$ as $\frac{3}{3}$ and add to Review
Reasonableness: The outer part is $1 + (\text{something between } 0 \text{ and } 1)$, since the inner fraction $\frac{2}{3}$ is a proper fraction. So the answer must lie strictly between $1$ and $2$ — this kills (D) $2$ and (E) $3$ immediately. Among the survivors $\frac{5}{4}=1.25$, $\frac{3}{2}=1.5$, $\frac{5}{3}\approx1.67$, only $\frac{5}{3}$ equals $1 + \frac{2}{3}$, confirming $\textbf{(C)}$.
Alternative: Convert everything to decimals from the inside out: $1+1=2$, then $1/2 = 0.5$, then $1+0.5 = 1.5$, then $1/1.5 = 0.6\overline{6}$, then $1 + 0.6\overline{6} = 1.6\overline{6} = \frac{5}{3}$. Same answer, but the repeating decimal is less exact than keeping fractions, so the fraction route is cleaner.
CCSS standards used (min grade 6)
5.OA.A.1Use parentheses, brackets, or braces in numerical expressions and evaluate (Reading each fraction bar as hidden parentheses and evaluating the innermost denominator $1 + 1$ before anything above it.)5.NF.A.1Add and subtract fractions with unlike denominators (Computing $1 + \frac{1}{2} = \frac{3}{2}$ and the final $1 + \frac{2}{3} = \frac{5}{3}$ by rewriting the whole number with a common denominator.)6.NS.A.1Interpret and compute quotients of fractions and solve word problems (Dividing $1$ by $\frac{3}{2}$ by multiplying by the reciprocal to get $\frac{2}{3}$.)
⭐ A stacked fraction is a tower — start at the bottom, finish each denominator before you divide, and climb out one floor at a time to reach $\frac{5}{3}$.
⭐ A stacked fraction is a tower — start at the bottom, finish each denominator before you divide, and climb out one floor at a time to reach $\frac{5}{3}$.
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