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2013 AMC 10B Problem 1

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2013 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 $\frac{2+4+6}{1+3+5} - \frac{1+3+5}{2+4+6}$ ?

A)
$-1$
B)
$\frac{5}{36}$
C)
$\frac{7}{12}$
D)
$\frac{49}{20}$
E)
$\frac{43}{3}$

Answer

Key insight

The two sums are just 12 and 9, so the expression is 4/3 minus 3/4, which is 7/12 over the common denominator 12.

Solution

Evaluate the two sums first: $2 + 4 + 6 = 12$ and $1 + 3 + 5 = 9$ . The expression becomes
$\frac{12}{9} - \frac{9}{12} = \frac{4}{3} - \frac{3}{4}.$
Over the common denominator $12$ :
$\frac{16}{12} - \frac{9}{12} = \frac{7}{12}.$

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

Why this works

Simplify the pieces before combining: reducing $\frac{12}{9}$ to $\frac{4}{3}$ keeps the common denominator small. A fraction minus its reciprocal, $\frac{p}{q} - \frac{q}{p} = \frac{p^2 - q^2}{pq}$ , is a pattern worth recognizing; here it gives $\frac{16 - 9}{12}$ immediately.

The trap

Subtracting in the wrong order and getting a negative value, or subtracting numerators and denominators separately.

Common mistakes

Subtracting in the wrong order and getting a negative value, or subtracting numerators and denominators separately.
Forgetting to reduce $\frac{12}{9}$ and then making an arithmetic slip with denominator $108$ .

Techniques

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