\section*{Task 1 - 011041}

As announced at the XXII Congress of the CPSU, the production of means of production (i.e., raw materials, machines, equipment for industry, agriculture, and transport, etc.) in the Soviet Union will increase to 6.8 times from 1960 to 1980.

However, the production of consumer goods (goods intended for the needs of the population) is also expected to grow significantly, increasing to five times. The total industrial production will increase to 6.2 times.

a) What percentage of the total industrial production did the production of means of production account for in 1960?

b) What percentage would it account for in 1980?
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Let \(a_{0}, b_{0}\) and \(c_{0}\) be the production of means of production, consumer goods, and the total industrial production in 1960, respectively, and \(a_{1}, b_{1}\) and \(c_{1}\) be the planned quantities in 1980.

Then, \(a_{1}=6.8 a_{0}, b_{1}=5.0 b_{0}\) and \(c_{1}=6.2 c_{0}\) and further \(a_{0}+b_{0}=c_{0}\) and \(a_{1}+b_{1}=c_{1}\) or \(6.8 a_{0}+5.0 b_{0}=6.2 c_{0}\). The first equation is divided by \(c_{0}\), and the second by \(6.2 c_{0}\). This yields the linear system of equations

\[
\frac{a_{0}}{b_{0}}+\frac{b_{0}}{c_{0}}=1 \quad ; \quad \frac{34}{31} \frac{a_{0}}{c_{0}}+\frac{25}{31} \frac{b_{0}}{c_{0}}=1
\]

which has the solution \(\frac{a_{0}}{c_{0}}=\frac{2}{3}\) and \(\frac{b_{0}}{c_{0}}=\frac{1}{3}\).

a) In 1960, the share of the production of means of production was thus \(\frac{a_{0}}{c_{0}}=\frac{2}{3}=66.7 \%\).

b) In 1980, this share would have been \(\frac{34}{31} \frac{a_{0}}{c_{0}}=\frac{68}{93}=73.1 \%\).