6. Four friends, Ana, Dora, Marta, and Tea, buy a birthday gift together for their fifth friend. Ana contributed $40 \%$ of the total cost of the gift, Dora contributed one-third of the amount the other three friends contributed, and Marta contributed 25 \% of the amount the other three friends contributed. Tea contributed 51 euros. What is the price of the gift?
6. Four friends, Ana, Dora, Marta, and Tea, buy a birthday gift for a fifth friend. Ana contributed $40\%$ of the total cost of the gift, Dora contributed one-third of the amount the other three friends contributed, and Marta contributed $25\%$ of the total amount the other three friends contributed. Tea contributed 51 euros. What is the cost of the gift?

## Solution.

Let the cost of the gift be $x$ euros. Let $A, D, M$, and $T$ be the amounts contributed by the four friends, respectively.

Dora contributed one-third of the amount the other three friends contributed, i.e., $D=\frac{1}{3}(A+M+T)$.

The four friends together contributed $A+M+T+\frac{1}{3}(A+M+T)=x$, which implies:

$\frac{4}{3}(A+M+T)=x$

$A+M+T=\frac{3}{4} x$

Thus, Dora contributed $D=\frac{1}{3} \cdot \frac{3}{4} x=\frac{1}{4} x$, which is $25\%$ of the total amount.

1 POINT

Marta contributed $25\%$ or one-fourth of the amount the other three friends contributed, so similarly, we conclude that Marta contributed $M=\frac{1}{4} \cdot \frac{4}{5} x=\frac{1}{5} x$, which is $20\%$ of the total cost of the gift.

4 POINTS

According to the problem, Ana contributed $40\%$ of the total cost of the gift, so Tea contributed $100\%-40\%-25\%-20\%=15\%$ of the total cost of the gift.

1 POINT

Since Tea contributed 51 euros, we have:

$15\% \cdot x=51$

$x=340$

The cost of the gift is 340 euros.

1 POINT

TOTAL 10 POINTS

Note 1: If a student does not use equations but graphically or verbally explains the solution process leading to the correct result, evaluate according to the provided point distribution.

Note 2: The conclusion that Dora contributed $25\%$ of the total amount can also be obtained by writing the ratio $3D = A + M + T$ and adding $D$ to both sides. This results in $4D = A + M + T + D = x$, i.e., $D = \frac{1}{4} x$.