How to find a solution to a compound fraction

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ACT Math › How to find a solution to a compound fraction

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John's shadow is six feet long, and Mary's shadow is five feet long. If John is four feet tall, which of the following is closest to Mary's height in inches?

$33\ inches$

$12\ inches$

$40\ inches$

CORRECT

$48\ inches$

$60\ inches$

Explanation

First we set up a proportion. Mary's shadow (5 feet) to John's shadow (6 feet) is equal to Mary's height (x feet) to John's height (4 feet), i.e. 5 / 6 = x / 4.

Solve for x by first cross-mulitplying: 20 = 6x.

Divide both sides by 6: x = 20 / 6 feet

Multiply by 12 to find this height in inches: 20 * 12 / 6 = 20 * 2 = 40 inches

When two resistors ( $R_{1}$ and $R_{2}$ ) are added together in a parallel circuit to create an equivalent single resistor, the equivalent resistor ( $R_{eq}$ ) has a resistance defined by the following equation:

$\frac{1}{R_{eq}}= \frac{1}{R_{1}}+\frac{1}{R_{2}}$

If $R_{1}=10$ and $R_{2}=20$ , what does $R_{eq}$ equal?

$\frac{20}{3}$

CORRECT

$\frac{3}{20}$

None of the other answers

Explanation

Plugging $R_{1}=10$ and into the equation gives:

$\frac{1}{R_{eq}}=\frac{1}{10}+\frac{1}{20}$

Create a common denominator by multiplying $\frac{1}{10}$ by in the numerator and denominator:

$\frac{1}{R_{eq}}=\frac{2}{20}+\frac{1}{20}$

$\frac{1}{R_{eq}}=\frac{3}{20}$

Finally:

$R_{eq} = \frac{20}{3}$

What does $\small \frac{\frac{2}{3}\times \frac{1}{4}}{\frac{2}{9}+\frac{1}{3}}$ equal?

$\frac{3}{10}$

CORRECT

$\frac{18}{35}$

$\frac{5}{54}$

$\frac{11}{17}$

$\frac{1}{2}$

Explanation

Simplify the numerator first: $\small \frac{2}{3}\times \frac{1}{4}=\frac{2}{12}=\frac{1}{6}$ .

Then, simplify the denominator: $\frac{2}{9}+\frac{1}{3}=\frac{2}{9}+\frac{3}{9}=\frac{5}{9}$ .

Next, you have to do $\frac{1}{6}\div \frac{5}{9}$ , which is the same as $\small \small \frac{1}{6}\times \frac{9}{5}=\frac{9}{30}=\frac{3}{10}$ , when solved for and simplified.

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