# 1951 AHSME Problems/Problem 10

## Problem

Of the following statements, the one that is incorrect is: $\textbf{(A)}\ \text{Doubling the base of a given rectangle doubles the area.}$ $\textbf{(B)}\ \text{Doubling the altitude of a triangle doubles the area.}$ $\textbf{(C)}\ \text{Doubling the radius of a given circle doubles the area.}$ $\textbf{(D)}\ \text{Doubling the divisor of a fraction and dividing its numerator by 2 changes the quotient.}$ $\textbf{(E)}\ \text{Doubling a given quantity may make it less than it originally was.}$

## Solution

The well-known area formula for a circle is $A = \pi r^2$, so doubling the radius will result it quadrupling the area (since $A' = \pi (2r)^2 = 4 \pi r^2 = 4A$). Statement $\boxed{\textbf{(C)}}$ is therefore incorrect, and is the correct answer choice.

Statements $\textbf{(A)}$ and $\textbf{(B)}$ are evidently correct, since in triangles the area is directly proportional to both the base and the height ( $A = \frac{1}{2} bh$). Statement $\textbf{(D)}$ is also correct: let $q = \frac{a}{b}$. Then $q' = \frac{a \div 2}{2b} = \frac{a}{4b} = \frac{q}{4}$, which is a change from $q$. Finally, statement $\textbf{(E)}$ is true: $2x < x$ if $x < 0$, as doubling a negative number makes it even more negative (and therefore less than it originally was).

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