## Task 2 - 080622

During the summer holidays, Monika visited the capital of the USSR. For her mathematics study group, she brought the following problem:

In the "Gorky" ring of the Moscow underground, there are four escalators of different lengths.

The total length of the two middle-length escalators is $136 \mathrm{~m}$, with one being $8 \mathrm{~m}$ longer than the other. The length of the longest escalator is 310, and the length of the shortest is 314 of the total length of all four escalators.

Calculate the length of each individual escalator!
Let $a$ be the measure of the longest, $b$ the measure of the second longest, $c$ the measure of the third longest, and $d$ the measure of the shortest escalator, and let $g$ be the measure of the total length of all escalators.

Then it holds: $b+c=136$ and $b=c+8$. From this, it follows: $b=72$ and $c=64$.

For the total length of the longest and the shortest escalator, it holds:

$$
a+d=\frac{3}{10} g+\frac{3}{14} g=\frac{36}{70} g
$$

then it follows:

$$
136=b+c=g-(a+d)=\frac{34}{70} g
$$

From this, it follows: $\frac{1}{70} g=4$ and thus $\frac{3}{10} g=84$ and $\frac{3}{14} g=60$.

The lengths of the escalators are $84 \mathrm{~m}, 72 \mathrm{~m}, 64 \mathrm{~m}$, and $60 \mathrm{~m}$.

Indeed, as a check, $72+64=136,72=64+8,84=\frac{3}{10}(84+72+64+60)$ and $60=$ $\frac{3}{14}(84+72+64+60)$.