AMC 10 · 2020 · #1

Grade 5 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

fraction-arithmeticlinear-equations-one-varwork-backwards work-backwards ↑ Prerequisites: fraction-arithmetic

📏 Short solution 💡 1 insight

Problem

Find the number x that makes x -

\frac{3}{4}

\frac{5}{12}

\frac{1}{3}

true.

Pick an answer.

(A)

${-}\frac{2}{3}$

(B)

$\frac{7}{36}$

(C)

$\frac{7}{12}$

(D)

$\frac{2}{3}$

(E)

$\frac{5}{6}$

AMC 10 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Work Backwards

Tool #11 (Work Backwards) fits perfectly: x went through one operation (subtract $\frac{3}{4}$ ) to reach the right-hand value, so we undo it by adding $\frac{3}{4}$ . The arithmetic on the right is a small fraction sum on the common denominator 12. Tool #3 (Eliminate) is a quick sanity check — the right side is clearly small and positive after adding back $\frac{3}{4}$ , so negative answer (A) and tiny $\frac{7}{36}$ (B) can be eliminated by feel before the final compute.

1STEP 1

First compute the right side: rewrite $\frac{1}{3}$ = $\frac{4}{12}$ and subtract to get $\frac{1}{12}$ .

\frac{5}{12}

\frac{1}{3}

\frac{5}{12}

\frac{4}{12}

\frac{1}{12}

2STEP 2

Undo the - $\frac{3}{4}$ by adding $\frac{3}{4}$ to both sides, so x = $\frac{1}{12}$ + $\frac{3}{4}$ .

x -

\frac{3}{4}

\frac{1}{12}

→ x =

\frac{1}{12}

\frac{3}{4}

3STEP 3

Add over denominator 12: $\frac{3}{4}$ = $\frac{9}{12}$ , so x = $\frac{1}{12}$ + $\frac{9}{12}$ = $\frac{10}{12}$ = $\frac{5}{6}$ .

x =

\frac{1}{12}

\frac{9}{12}

\frac{10}{12}

\frac{5}{6}

→ (E)

4STEP 4

Elimination check: the right side sits near 0.83, so only $\frac{5}{6}$ ≈ 0.83 fits; the others all miss.

\frac{1}{12}

\frac{3}{4}

≈ 0.83 → (E)

\frac{5}{6}

Answer

\frac{5}{6}

Plug x =

\frac{5}{6}

back in:

\frac{5}{6}

\frac{3}{4}

\frac{10}{12}

\frac{9}{12}

\frac{1}{12}

, and the right side was

\frac{5}{12}

\frac{4}{12}

\frac{1}{12}

. Both sides match, so (E) is correct.

💡Key takeaway

This AMC 10 problem only needs Grade 5 "add and subtract fractions with unlike denominators" you already know — write everything over 12, undo the - $\frac{3}{4}$ by adding it back, and the answer falls out as $\frac{5}{6}$ .

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