Choir Boys
Can you work out how many members this choir has from these percentages?
Age
14 to 16
Challenge level
Two schools share a choir.
Last year there were 30 more students from school B than students from school A in the choir.
This year the number of choir members has increased by 10%.
The number of members from school A has increased by 20%.
The number of members from school B has increased by 5%.
How many members does the choir have this year?
This problem is taken from the UKMT Mathematical Challenges.
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 $ | $\frac{21}{20}b$ | $\frac{11}{10}(a+b)$ |
\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.