Tennis training
After tennis training, Andy, Roger and Maria collect up the balls. Can you work out how many Andy collects?
Age
14 to 16
Challenge level
Problem
After tennis training, Andy collected twice as many balls as Roger, and five more than Maria. If they collected $35$ balls in total, how many did Andy collect?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
Answer: Andy collected $16$ balls.
In terms of $a$
Suppose Andy collected $a$ balls. Then Roger collected $\frac 12 a$ and Maria collected $a-5$.
In total they collected $35$, so: $$\begin{align}&a + \tfrac 12 a + a - 5 = 35\\
\Rightarrow&\tfrac 52 a - 5 = 35\\
\Rightarrow&\tfrac 52 a = 40\\
\Rightarrow&a = 16\end{align}$$
In terms of $R$
Suppose Andy collected $A$ balls, Roger collected $R$ balls and Maria collected $M$ balls. Then $A=2R=M+5$.
Suppose Andy collected $A$ balls, Roger collected $R$ balls and Maria collected $M$ balls. Then $A=2R=M+5$.
In total they collected $35$ balls, so: $$A+R+M=35$$
Putting this in terms of $R$ gives: $$\begin{align}&2R+R+(2R-5)=35\\
\Rightarrow&5R=40\\
\Rightarrow&R=8\end{align}$$
\Rightarrow&5R=40\\
\Rightarrow&R=8\end{align}$$
Since Andy collected twice as many balls as Roger, he collected $2R=2\times8=16$ balls.