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2009 AMC 10B Problem 2

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2009 AMC 10B Problem 2AMC 10 Step by Step

Name: Date: September 13, 2026

Fractions & Decimals · difficulty 1 of 5 · about 1 min · position P1-10

Which of the following is equal to $\dfrac{\frac{1}{3}-\frac{1}{4}}{\frac{1}{2}-\frac{1}{3}}$ ?

A)
$\frac 14$
B)
$\frac 13$
C)
$\frac 12$
D)
$\frac 23$
E)
$\frac 34$

Answer

Key insight

Both differences of unit fractions collapse to single unit fractions, 1/12 over 1/6, whose quotient is 1/2.

Solution

Simplify numerator and denominator separately:
$\frac13 - \frac14 = \frac{4 - 3}{12} = \frac{1}{12}, \qquad \frac12 - \frac13 = \frac{3 - 2}{6} = \frac16.$
Dividing, $\dfrac{1/12}{1/6} = \dfrac{1}{12} \cdot 6 = \dfrac{1}{2}$ .

The answer is $\boxed{\textbf{(C)}\ \frac12}$ .

Why this works

The difference of consecutive unit fractions is $\frac{1}{n} - \frac{1}{n+1} = \frac{1}{n(n+1)}$ , so each piece is one unit fraction. Dividing by a fraction means multiplying by its reciprocal; simplify top and bottom before dividing to keep the numbers tiny.

Alternative approach

Multiply numerator and denominator by $12$ : the fraction becomes $\dfrac{4 - 3}{6 - 4} = \dfrac{1}{2}$ , with no fraction arithmetic at all.

The trap

Inverting the wrong fraction when dividing, which gives 2 instead of 1/2, or subtracting numerators and denominators separately.

Common mistakes

Inverting the wrong fraction when dividing, which gives 2 instead of 1/2, or subtracting numerators and denominators separately.
Computing $\frac13 - \frac14$ as $\frac{1}{7}$ or $\frac12 - \frac13$ as $\frac15$ by subtracting denominators.

Techniques

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