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 for each winner (in Rs.)	1,00,000	50,000	-	-	-	-	-

Answer 1

We use a unitary method to calculate the amount of prize money which will be equally divided into the number of winners by dividing the total prize money by number of winners.

• Unitary method helps us to calculate the value of a single unit by dividing the value of multiple units by the number of units given.
• Proportionality is a concept that gives relation between two values or elements. Either a value is directly proportional to other value or inversely proportional to other value.
• If a value ‘a’ is directly proportional to a value ‘b’ then we can write $a \propto b$which converts into an equation $a = kb$using constant of proportionality i.e. k.
  We can also define directly proportional as increase in a value of one item increases the value of second item ( same for decrease) and inversely proportional as increase in a value of one item decreases the value of second item.

Complete step by step answer:
We are given total prize money as Rs.1,00,000
When number of winners is 1
$\Rightarrow$Prize money for each winner $=$Rs.1,00,000
When number of winners is 2
$\Rightarrow$Prize money for each winner $= \dfrac{{1,00,000}}{2}$
$\Rightarrow$Prize money for each winner $=$Rs.50,000
When number of winners is 4
$\Rightarrow$Prize money for each winner $= \dfrac{{1,00,000}}{4}$
$\Rightarrow$Prize money for each winner $=$Rs.25,000
When number of winners is 5
$\Rightarrow$Prize money for each winner $= \dfrac{{1,00,000}}{5}$
$\Rightarrow$Prize money for each winner $=$Rs.20,000
When number of winners is 8
$\Rightarrow$Prize money for each winner $= \dfrac{{1,00,000}}{8}$
$\Rightarrow$Prize money for each winner $=$Rs.12,500
When number of winners is 10
$\Rightarrow$Prize money for each winner $= \dfrac{{1,00,000}}{{10}}$
$\Rightarrow$Prize money for each winner $=$Rs.10,000
When number of winners is 20
$\Rightarrow$Prize money for each winner $= \dfrac{{1,00,000}}{{20}}$
$\Rightarrow$Prize money for each winner $=$Rs.5,000
Now we substitute the values in the table

Number of winners	1	2	4	5	8	10	20
Prize for each winner (in Rs.)	1,00,000	50,000	25,000	20,000	12,500	10,000	5,000

Since we see that as the number of winners increases, prize money for each winner decreases.
We can say that the prize money given to an individual winner is inversely proportional to the number of winners.

$\therefore$ The prize money given to an individual winner is inversely proportional to the number of winners.

Note: Many students make mistake of writing the prize money given to an individual winner as directly proportional to the number of winners as they use the concept of proportionality and write $a \propto b$ which converts into an equation $a = kb$ where k is in inverse form, keep in mind the change in value of denominator is bringing change in the total value so that means it is inversely proportional.