## 9. Balls

In a box, there are only red and blue balls. If five red balls were drawn from the box, one seventh of the remaining balls in the box would be red. If ten blue balls were drawn from the box instead of five red ones, one fifth of the remaining balls in the box would be red. How many balls are there in the box in total?

## Result: $\quad 110$
Solution. Let $c$ be the number of red, and $p$ the number of blue balls in the box.

If five red balls were drawn from the box, one seventh of the remaining balls in the box would be red. This means that the number of blue balls $p$ would be six times the number of red balls $c-5$, i.e.,

$$
p=6 \cdot(c-5) .
$$

If instead of five red balls, ten blue balls were drawn from the box, one fifth of the remaining balls in the box would be red. This means that the number of blue balls $p-10$ would be four times the number of red balls, i.e.,

$$
p-10=4 \cdot c
$$

From $p=6 \cdot(c-5)$ and $p=4 \cdot c+10$ we get $6 \cdot(c-5)=4 \cdot c+10$, so $c=20$, and $p=90$.

There are $20+90=110$ balls in the box.