1985 AJHSME Problems/Problem 10

Problem

The fraction halfway between $\frac{1}{5}$ and $\frac{1}{3}$ (on the number line) is

$[asy] unitsize(12); draw((-1,0)--(20,0),EndArrow); draw((0,-.75)--(0,.75)); draw((10,-.75)--(10,.75)); draw((17,-.75)--(17,.75)); label("$0$",(0,-.5),S); label("$\frac{1}{5}$",(10,-.5),S); label("$\frac{1}{3}$",(17,-.5),S); [/asy]$

$\text{(A)}\ \frac{1}{4} \qquad \text{(B)}\ \frac{2}{15} \qquad \text{(C)}\ \frac{4}{15} \qquad \text{(D)}\ \frac{53}{200} \qquad \text{(E)}\ \frac{8}{15}$

Solution

The fraction halfway between $\frac{1}{5}$ and $\frac{1}{3}$ is simply their average, which is $\begin{align*} \frac{\frac{1}{5}+\frac{1}{3}}{2} &= \frac{\frac{3}{15}+\frac{5}{15}}{2} \\ &= \frac{\frac{8}{15}}{2} \\ &= \frac{4}{15} \\ \end{align*}$

$\boxed{\text{C}}$

For any two fractions that have the property $\frac{a}{b}$ and $\frac{a}{c}$ where $a$ is equal for both fractions;

Set aside the a. Take the average of b and c. For this example, the average of 3 and 5 is 4. Then multiply those numbers. 3*5=15. The number that is exactly in between $\frac{a}{b}$ and $\frac{a}{c}$ will be $\frac{a*((b+c)/2)}{b*c}$ .

In this example, it would be $\frac{1((3+5)/2)}{3*5}$ , which simplifies to $\frac{4}{15}$ .

Therefore, the answer is $\boxed{\text{C}}$

Edit by mathmagical~

Video Solution

https://youtu.be/BcWDI1TvK-Y

~savannahsolver

1985 AJHSME (Problems • Answer Key • Resources)
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1985 AJHSME Problems/Problem 10

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