## Task 1 - 220731

The consumer cooperative refunds 1.6% of the amount for which consumer vouchers were redeemed each year. For four families $A, B, C$, and $D$, it is known from one year:

$A$ had redeemed for twice the amount as $B$ or, which is the same, for three times the amount as $C$ and for four times the amount as $D$; the four families $A, B, C, D$ together received $336 \mathrm{DM}$ in refunds.

For each of the four families $A, B, C, D$, the following should be determined from this information:

a) For what amount did this family redeem consumer vouchers in this year?

b) What amount did this family therefore receive in refunds?
a) Let the amount for which families $A, B, C$ and $D$ had settled consumer marks be $a, b, c$ and $d$ respectively. Then it holds that

$$
b=\frac{1}{2} a ; \quad c=\frac{1}{3} a ; \quad d=\frac{1}{4} a
$$

The four families thus had a total amount of $a+\frac{1}{2} a+\frac{1}{3} a+\frac{1}{4} a$. Since 1.6% of this is 336 M, it follows that

$$
a+\frac{1}{2} a+\frac{1}{3} a+\frac{1}{4} a=\frac{336 \cdot 100}{1.6} M \quad \Rightarrow \quad a=10080 M
$$

and thus

$$
b=\frac{1}{2} \cdot 10080 M=5040 M \quad ; \quad c=\frac{1}{3} \cdot 10080 M=3360 M \quad ; \quad d=\frac{1}{4} \cdot 10080 M=2520 M
$$

b) Family A received $\frac{10080 \cdot 1.6}{100} M=161.28 M$, Family B received $\frac{5040 \cdot 1.6}{100} M=80.64 M$, Family C received $\frac{3360 \cdot 1.6}{100} M=53.76 M$, Family D received $\frac{2520 \cdot 1.6}{100} M=40.32 M$.