Let's denote the shares of Annie, Bola, and Charles as [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex] respectively.
We are given the following relationships:
1. Annie's share ([tex]\( A \)[/tex]) is [tex]\(\frac{2}{5}\)[/tex] of Bola's share ([tex]\( B \)[/tex]).
2. Bola's share ([tex]\( B \)[/tex]) is [tex]\(\frac{3}{4}\)[/tex] of Charles's share ([tex]\( C \)[/tex]).
We need to find out Charles's share, [tex]\( C \)[/tex], given that the total amount distributed among them is ₹8200.
First, let's express [tex]\( A \)[/tex] and [tex]\( B \)[/tex] in terms of [tex]\( C \)[/tex].
Since [tex]\( B = \frac{3}{4} \times C \)[/tex], we can write:
[tex]\[ B = \frac{3}{4}C \][/tex]
Since [tex]\( A = \frac{2}{5} \times B \)[/tex], substituting [tex]\( B = \frac{3}{4}C \)[/tex] into the equation for [tex]\( A \)[/tex]:
[tex]\[ A = \frac{2}{5} \times \left( \frac{3}{4}C \right) = \frac{2 \times 3}{5 \times 4}C = \frac{6}{20}C = \frac{3}{10}C \][/tex]
Next, we need to sum the shares of Annie, Bola, and Charles and set the sum equal to the total amount ₹8200:
[tex]\[ A + B + C = 8200 \][/tex]
Substituting the expressions for [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ \frac{3}{10}C + \frac{3}{4}C + C = 8200 \][/tex]
To combine these fractions, we need a common denominator. The least common multiple of 10 and 4 is 20. Rewriting each fraction with a denominator of 20:
[tex]\[ \frac{3}{10}C = \frac{3 \times 2}{10 \times 2}C = \frac{6}{20}C \][/tex]
[tex]\[ \frac{3}{4}C = \frac{3 \times 5}{4 \times 5}C = \frac{15}{20}C \][/tex]
Now we add these up:
[tex]\[ \frac{6}{20}C + \frac{15}{20}C + \frac{20}{20}C = 8200 \][/tex]
Combining the fractions:
[tex]\[ \left( \frac{6 + 15 + 20}{20} \right)C = 8200 \][/tex]
[tex]\[ \left( \frac{41}{20} \right)C = 8200 \][/tex]
To solve for [tex]\( C \)[/tex], multiply both sides of the equation by [tex]\(\frac{20}{41}\)[/tex]:
[tex]\[ C = 8200 \times \frac{20}{41} \][/tex]
Calculating the value:
[tex]\[ C = \frac{8200 \times 20}{41} = \frac{164000}{41} = 4000 \][/tex]
So, Charles's share is ₹4000.
