**Problem of the Week**

Problem C and Solution

Sibling Rivalry

**Problem**

Akira and Hideo are twins with different jobs. Akira earns five-eighths of what Hideo earns, but Akira’s expenses are half of Hideo’s. Akira ends up saving 40% of his income. What percentage of his income does Hideo save?

**Solution**

__Solution 1__: Using only one variable

Let \(h\) represent Hideo’s income. Then Akira’s income is \(\frac{5}{8}h\).

Since Akira saves 40% of his income, his expenses are \(100\%-40\%=60\%\) of his income. Therefore, Akira’s expenses are \(60\%\times \frac{5}{8}h=\frac{60}{100}\times \frac{5}{8}h=\frac{3}{8}h\).

Akira’s expenses are one-half of Hideo’s expenses so Hideo’s expenses are twice Akira’s expenses. Therefore, Hideo’s expenses are \(2\times \frac{3}{8}h=\frac{3}{4}h=0.75h=75\% \mbox{ of } h\). Since Hideo’s expenses are 75% of his income, he saves \(100\%-75\%=25\%\) of his income.

Therefore, Hideo saves 25% of his income.

__Solution 2__: Using two variables

Let \(x\) represent Hideo’s income and \(y\) represent Hideo’s expenses.

Then Akira’s income is \(\frac{5}{8}x\) and his expenses are \(\frac{1}{2}y\).

Since Akira saves 40% of his income, his expenses are 60% of his income.

\[\begin{aligned}
\frac{1}{2}y&=0.60\times \frac{5}{8}x\\
\frac{1}{2}y&=\frac{6}{10}\times \frac{5}{8}x\\
\frac{1}{2}y&=\frac{3}{8}x\\
y&=\frac{3}{4}x\\[-4mm]\end{aligned}\] Hideo saves whatever is left of his income after expenses. Therefore Hideo saves \[x-y=x-\frac{3}{4}x=\frac{1}{4}x=0.25x=25\% \mbox{ of } x.\] Therefore, Hideo saves 25% of his income.

__Solution 3__: Using two variables a bit differently

Let \(8x\) represent Hideo’s income and \(2y\) represent Hideo’s expenses.

Then Akira’s income is \(\frac{5}{8}(8x)=5x\) and his expenses are \(\frac{1}{2}(2y)=y\).

Since Akira saves 40% of his income, his expenses are 60% of his income.

\[\begin{aligned}
y&=0.60\times 5x\\
y&=\frac{6}{10}\times 5x\\
y&=3x\\[-4mm]\end{aligned}\] Hideo earns \(8x\) and his expenses are \(2y\) so his savings are \(8x-2y\). We want the ratio of his savings to his income, \(\dfrac{8x-2y}{8x}=\dfrac{8x-2(3x)}{8x}=\dfrac{2x}{8x}=\dfrac{1}{4}\) or 25% .

Therefore, Hideo saves 25% of his income.