Ex.13.2 Q2 Direct and Inverse Proportions Solution - NCERT Maths Class 8

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Question

 In a television game show, the prize money \(₹ 1,00,000\) is to be divided equally amongst the winners. Complete the table and find whether the prize money given to an individual winner is directly or inversely proportional to the number of winners?

Number of winners

\(1\)

\(2\)

\(4\)

\(5\)

\(8\)

\(10\)

\(20\)

Prize for each winner (in ₹)

\(1,00,000\)

\(50,000\)

\( \ldots \)

\( \ldots \)

\( \ldots \)

\( \ldots \)

\( \ldots \)

Text Solution

What is known?

Amount for \(1\) winner and \(2\) winners.

What is unknown?

Prize amount for \(4, 5, 8, 10,\) and \(20\) winners.

Reasoning:

Two numbers \(x\) and \(y\) are said to vary in inverse proportion if

\[\begin{align}xy = {\rm{ }}k,{\rm{ }}\,\, x{\rm{ }} = {\rm{ }}\frac{1}{y}k\end{align}\]

Where, \(k\) is a constant.

\[\begin{align}{x_1}\;{y_1} = {x_2}\;{y_2}\end{align}\]

Steps:

If winners increase the prize amount will decrease.

\[\begin{align}{x_1}\;{y_1} &= {x_3}\;{y_3}\\1 \times 100000 &= \,4 \times {y_3}\\{y_3} &= \frac{{1 \times 100000}}{4} \quad \therefore {y_3} = 25000\\1 \times 100000 &= 5 \times {y_4}\\{y_4} &= \frac{{1 \times 100000}}{5} \quad \therefore {y_4} = 20000\\1 \times 100000 &= 8 \times {y_5}\\{y_5} &= \frac{{1 \times 100000}}{8} \quad \therefore {y_5} = 12,500\\1 \times 100000 &= 10 \times {y_6}\\{y_6} &= \frac{{1 \times 100000}}{{10}} \quad \therefore {y_6} = 10000\\1 \times 100000 &= 20 \times {y_7}\\{y_7} &= \frac{{1 \times 100000}}{{20}} \quad \therefore {y_7} = 5000\end{align}\]

No. of winners

\(4\)

\(5\)

\(8\)

\(10\)

\(20\)

Prize for winners

\(25,000\)

\(20,000\)

\(12,500 \)

\(10,000\)

\(5,000\)