AMC 10 Step by Step
Log inSign up
Printable

2011 AMC 10B Problem 1

Problem onlyBack to the problem
2011 AMC 10B Problem 1AMC 10 Step by Step
Name:                             Date: September 13, 2026
Fractions & Decimals · difficulty 1 of 5 · about 1 min · position P1-10

What is 2+4+61+3+5−1+3+52+4+6\dfrac{2+4+6}{1+3+5} - \dfrac{1+3+5}{2+4+6}

  1. A)

    −1-1

  2. B)

    536\frac{5}{36}

  3. C)

    712\frac{7}{12}

  4. D)

    14760\frac{147}{60}

  5. E)

    433\frac{43}{3}

Answer

C

Key insight

The two sums are 12 and 9, so the expression is 4/3 - 3/4, a single subtraction over the common denominator 12.

Solution

Add the numerators and denominators first: 2+4+6=122+4+6 = 12 and 1+3+5=91+3+5 = 9. The expression becomes
129−912=43−34.\frac{12}{9} - \frac{9}{12} = \frac{4}{3} - \frac{3}{4}.
With common denominator 1212 this is
1612−912=712.\frac{16}{12} - \frac{9}{12} = \frac{7}{12}.

The answer is (C) 712\boxed{\textbf{(C)}\ \frac{7}{12}}.

Why this works

A fraction of the form ab−ba\frac{a}{b} - \frac{b}{a} always equals a2−b2ab\frac{a^2 - b^2}{ab}; here that is 144−81108=63108=712\frac{144 - 81}{108} = \frac{63}{108} = \frac{7}{12}. Reduce each fraction before combining so the numbers stay small.

The trap

Subtracting numerators and denominators separately, or reversing the order of subtraction and getting -7/12.

Common mistakes

  • Subtracting numerators and denominators separately, or reversing the order of subtraction and getting −7/12-7/12.
  • Forgetting to reduce 129\frac{12}{9} to 43\frac{4}{3} and then making an arithmetic slip with larger numbers.

Techniques

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