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2020 AMC 10A Problem 1

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2020 AMC 10A Problem 1AMC 10 Step by Step
Name:                             Date: September 13, 2026
Linear Equations & Word Problems · difficulty 1 of 5 · about 1 min · position P1-10

What value of xx satisfies x−34=512−13?x- \frac{3}{4} = \frac{5}{12} - \frac{1}{3}?

  1. A)

    −23{-}\frac{2}{3}

  2. B)

    736\frac{7}{36}

  3. C)

    712\frac{7}{12}

  4. D)

    23\frac{2}{3}

  5. E)

    56\frac{5}{6}

Answer

E

Key insight

Simplify the right side to 1/12 first, then add 3/4 = 9/12 to isolate x.

Solution

Work out the right-hand side with denominator 1212:
512−13=512−412=112.\frac{5}{12} - \frac{1}{3} = \frac{5}{12} - \frac{4}{12} = \frac{1}{12}.
So x−34=112x - \frac{3}{4} = \frac{1}{12}. Add 34=912\frac{3}{4} = \frac{9}{12} to both sides:
x=112+912=1012=56.x = \frac{1}{12} + \frac{9}{12} = \frac{10}{12} = \frac{5}{6}.

The answer is (E) 56\boxed{\textbf{(E)}\ \frac{5}{6}}.

Why this works

A one-step linear equation only needs the constant moved to the other side with the opposite sign. Choosing 1212 as the common denominator from the start keeps every fraction in the same units, so the arithmetic is a couple of small integer additions.

The trap

Subtracting 3/4 instead of adding it when moving it across the equals sign, which gives a negative answer like (A).

Common mistakes

  • Subtracting 3/4 instead of adding it when moving it across the equals sign, which gives a negative answer like (A).
  • Combining 512−13\frac{5}{12} - \frac{1}{3} as 49\frac{4}{9} by subtracting numerators and denominators separately.

Techniques

Set up the equation/formula and compute; no special trick needed