In our factory, the production index numbers for television sets and washing machines are as follows:

|  | 1962 | 1963 | 1964 |
| :--- | :---: | :---: | :---: |
| TV set | 100 | 119 | 128 |
| washing machine | 100 | 127 | 132 |
| combined | 100 | 122 | $?$ |

a) In 1964, how many times was the combined production of the two items compared to that of 1962?
b) In 1962, how many times more TV sets were produced compared to washing machines?
According to our table, the number of TV sets produced in 1963 is 119% of the number in 1962, or 1.19 times. Let the number of TV sets and washing machines produced in 1962 be $t$ and $m$, respectively, and convert the rows of (1) into quantities of units:

|  | 1962 | 1963 | 1964 |
| :--- | :---: | :--- | :--- |
| TV set | $t$ | $1.19 t$ | $1.28 t$ |
| washing machine | $m$ | $1.27 m$ | $1.32 m$ |
| total | $t+m$ | $1.22(t+m)$ | $?$ |

Thus, the second data in the total row is equal to the sum of the data above it:

$$
1.19 t + 1.27 m = 1.22(t + m), \quad \text{from which} \quad \frac{t}{m} = \frac{5}{3}
$$

This answer is for part b) of the question.

Similarly, for the number $x$ to be written in place of the question mark in (1), we have $0.01 \cdot x(t + m) = 1.28 t + 1.32 m$, from which

$$
x = \frac{1.28 t + 1.32 m}{0.01(t + m)} = \frac{128 \frac{t}{m} + 132}{\frac{t}{m} + 1} = 129.5
$$

With this, we have solved the problem.

Péter Komjáth (8th grade, Budapest, XII., Mártonhegyi út)

Note. We could have chosen the ratio $t / m = x$ as the unknown right away, in which case the rows of the table would be: $x, 1.19 x, 1.28 x$; and $1, 1.27, 1.32$; and $1 + x, 1.19 x + 1.27 = 1.22(1 + x)$, so again only ratios would stand, but every data point would be relative to the same reference, the number of washing machines produced in 1962.