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# Choir Boys

##### Age 14 to 16 ShortChallenge Level

Answer: $99$

Two unknowns

School A School B Total
Last Year $a$ $b$ $a+b$
This Year $+20\%$ $+5\%$ $+10\%$

Using fractions:
School A School B Total
Last Year $a$ $b$ $a+b$
This Year $\frac{12}{10}a$ $\frac{21}{20}b$ $\frac{11}{10}(a+b)$
\begin{align} \therefore \frac{24}{20}a+\frac{21}{20}b&=\frac{22}{20}(a+b)\\ \Rightarrow 24a+21b=&22(a+b)\\ \Rightarrow 24a+21b=&22a+22b\\ \Rightarrow 2a=&b\end{align}
And $b$ is $30$ more than $a$, so $a+a=a+30$, so $a=30$ and $b=60$.

So last year there were $90$ students in the choir, and this year there are $10\%$ more, so there are $99$.

One unknown

School A School B Total
Last Year $a$ $a+30$ $2a+30$
This Year $+20\%$ $+5\%$ $+10\%$

Using decimals:
School A School B Total
Last Year $a$ $a+30$ $2a+30$
This Year $1.2a$ $1.05(a+30)$
$=1.05a+31.5$
$1.1(2a+30)$
$=2.2a+33$

\begin{align}1.2a+1.05a+31.5&=2.2a+33\\ \Rightarrow 2.25a+31.5&=2.2a+33\\ \Rightarrow 0.05a&=1.5\\ \Rightarrow a&=1.5\div 0.05\\ \Rightarrow a&=30\end{align}
So last year, there were $2\times30+30=90$ students in the choir.

This year there are $10\%$ more, so there are $99$ choir members.

You can find more short problems, arranged by curriculum topic, in our short problems collection.