Question

In a Television game show, the prize money of Rs 1, 00,000 is to be divided equally amongst the winners. Complete the following 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 of each winner in rupees(Rs)	1,00,000	50,000

Answer 1

Hint: We recall the definition of directly proportional quantities and inversely proportional quantities. We find the amount of prize an individual winner will get when there are 4, 5, 8, 10, and 20 winners. We find the quantities number of winner denoted as $ a $ and prize of each winner denoted as $ b $ increase or decrease with each other. If the ratio $ \dfrac{a}{b} $ remains constant then $ a $ is directly proportional to $ b $ and if the product $ ab $ remains constant then $ a $ is inversely proportional to $ b $ .\[\]

Complete step by step answer:
We know that when any measurable quantity $ a $ increases with increases in another quantity $ b $ and $ a $ decreases with the decreases in $ b $ then we say $ a $ varies directly with $ b $ . If their ratio $ \dfrac{a}{b} $ remains constant then we say $ a $ is directly proportional to $ b $ . \[\]

We also know that when quantity $ a $ increases with a decrease in another quantity $ b $ and $ a $ decreases with the increases in $ b $ then we say $ a $ varies inversely with $ b $ . If their product $ ab $ remains constant then we say $ a $ is inversely proportional to $ b $ . \[\]

We are given in the question that the prize money of Rs 1, 00,000 is to be divided equally amongst the winners. Let us denote the number of winners as $ a $ and $ b $ as the amount of prize money an individual will win when there are $ a $ winners. So we have ;
\[\begin{align}
  & \text{Prize of each winner=}\dfrac{\text{Total prize money}}{\text{Number of winners}} \\
 & \Rightarrow b=\dfrac{\text{Rs}.100000}{a} \\
\end{align}\]
If there are $ a=4 $ winners , the prize money for each winner is
\[b=\dfrac{100000}{a}=\dfrac{100000}{4}=\text{Rs}.25000\]
If there are $ a=5 $ winners , the prize money for each winner is
\[b=\dfrac{100000}{a}=\dfrac{100000}{5}=\text{Rs}.20000\]
If there are $ a=8 $ winners , the prize money for each winner is
\[b=\dfrac{100000}{a}=\dfrac{100000}{8}=\text{Rs}\text{.125}000\]
If there are $ a=10 $ winners , the prize money for each winner is
\[b=\dfrac{100000}{a}=\dfrac{100000}{10}=\text{Rs}\text{.10}000\]
If there are $ a=20 $ winners , the prize money for each winner is
\[b=\dfrac{100000}{a}=\dfrac{100000}{20}=\text{Rs}\text{.5}000\]
Now we fill the second row of the table and have; \[\]

Number of winners	1	2	4	5	8	10	20
Prize of each winner in rupees(Rs)	100000	50000	25,000	20000	12500	10000	5000

We see that as of the number of winners $ \left( a \right) $ increases from left to right in the table, the prize of each winner $ \left( b \right) $ decreases. So $ a $ varies indirectly with $ b $ . We also see in each column the product number of winners $ \left( a \right) $ and the prize of each winner $ \left( b \right) $ is constant which Rs.100000. Mathematically we have, $ ab=\text{Rs}.100000 $ . So the number of winners and the prize of each winner are inversely proportional. \[\]
Note:
We must be careful that proportional requires either constant ratio $ \dfrac{a}{b}=k $ in case of direct proportionality and or constant product $ ab=k $ in case of indirect proportionality. The real number $ k $ is called the proportionality constant. The increase or decrease with each other does not satisfy the condition for proportionality. The graph of two quantities in direct proportional will be a straight line and the graph of two indirectly proportional will be a rectangular hyperbola.