1999 CEMC Gauss (Grade 7) Problems/Problem 4

Problem

$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8}$ is equal to

$\text{(A)}\ \frac{15}{8} \qquad \text{(B)}\ 1\frac{3}{14} \qquad \text{(C)}\ \frac{11}{8} \qquad \text{(D)}\ 1\frac{3}{4} \qquad \text{(E)}\ \frac{7}{8}$

Solution

We have the following:

$1 = \frac{8}{8}$

$\frac{1}{2} = \frac{1 \times 4}{2 \times 4} = \frac{4}{8}$

$\frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8}$

This means summing the terms would give:

$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} = \frac{8}{8} + \frac{4}{8} + \frac{2}{8} + \frac{1}{8} = \frac{8 + 4 + 2 + 1}{8} = \boxed{\textbf{(A)} \frac{15}{8}}$

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1999 CEMC Gauss (Grade 7) Problems/Problem 4

Problem

Solution