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Problem 857

AIME late

Algebra Difficulty 5.6 Find the answer

6. Mirna, Ana, Nensi, and Karla bought a gift for their friend Dora, which costs $150 \mathrm{kn}$ . When asked by their mother how much each of them had contributed, they replied:

Mirna: “Nensi, Karla, and I together gave 120 kn.”

Nensi: “I know that Ana, Karla, and I together gave 102 kn.”

Karla: “I remember giving 10 kn more than Nensi.”

Mother was not confused, and neither should you. How much did each girl contribute to the gift?

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

First method:

Let's denote the amounts of money given by Nensi, Ana, Karla, and Mirna as N, A, K, and M, respectively.

Now we can write the conditions from the problem as:

$\begin{aligned} & \mathrm{M}+\mathrm{A}+\mathrm{N}+\mathrm{K}=150 \\ & M+N+K=120 \\ & N+A+K=102 \end{aligned}$

From the first two equations, we conclude that Ana gave $150-120=30$ kuna.

Now from the second or third equation, we conclude that Nensi and Karla together gave 72 kuna $(120-48=72$ or $102-30=72)$ , i.e., $\mathrm{N}+\mathrm{K}=72$ .

In addition, we know that Karla gave 10 kuna more than Nensi, i.e.,

$\mathrm{K}=\mathrm{N}+10$

This means that twice the amount given by Nensi is $72-10=62$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.