Problem of the Day

Solution

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On a question like this it is often helpful to focus on the unknown first rather than the given information.  In this question the unknown is “What percent of $a$ is $e$?” Think about how you would represent that symbolically.  If you have trouble with this, think about what you would do if $a$ and $e$ were replaced by real numbers.  For example, if the question were “What percent of $8$ is $2$?”, you would divide $2$ by $8$ and then multiply this by $100$ to convert it to a percentage: $\displaystyle \frac{2}{8} \cdot 100=25 %$.  In this case you would divide $e$ by $a$ and multiply by 100.  In other words, we are looking for the value of $\displaystyle \frac{e}{a} \cdot 100$. Our goal is to find the value of this target expression.

Now, let’s translate the given information into mathematical equations.  The first statement, $a$ percent of $b$ is $c$ translates to $\displaystyle \frac{a}{100}b=c$.  The second statement, $c$ percent of $d$ is $e$ translates to  $\displaystyle \frac{c}{100}d=e$.

Looking back at our target expression, note that we are essentially trying to find the value of $\displaystyle frac{e}{a}$. If we can find the values of $e$ and $a$, we can plug them into this fraction, simplify it, and then multiply by 100 to get our answer.  The second equation is already solved for $e$.  We can solve the first equation for $a$ as follows:

$\displaystyle \frac{a}{100}b=c$

$ab=100c$

$a=\displaystyle \frac{100c}{b}$

Now that we have $\displaystyle e=\frac{cd}{100}$ and $\displaystyle a=\frac{100c}{b}$, we can plug these values in to $\displaystyle \frac{e}{a}$:

$\displaystyle \frac{e}{a}=\frac{\frac{cd}{100}}{{\frac{100c}{b}}}=\frac{cd}{100} \cdot \frac{b}{100c}=\frac{bd}{100\cdot 100}$.

Multiply this value of $\displaystyle \frac{e}{a}$ by $100$ to convert it to a percentage. This gives you the answer, $\displaystyle \frac{db}{100}$.