Problem Analysis

We are given a total amount of ₹1200 to be divided among three individuals: X, Y, and Z. The condition is that specific expressions related to their shares are equal.

• Let the shares of X, Y, and Z be represented by $X$, $Y$, and $Z$ respectively.
• The total sum is $X + Y + Z = 1200$.
• The problem states that:
  • $ \frac{3}{5}X + 30 $
  • $ \frac{2}{5}Y + 10 $
  • $ \frac{5}{7}Z + 20 $
  are all equal. Let this common value be $k$.

Setting Up Equations

We can express the individual shares ($X$, $Y$, $Z$) in terms of $k$:

• From $ \frac{3}{5}X + 30 = k $: $ \frac{3}{5}X = k - 30 $ $ X = \frac{5}{3}(k - 30) $
• From $ \frac{2}{5}Y + 10 = k $: $ \frac{2}{5}Y = k - 10 $ $ Y = \frac{5}{2}(k - 10) $
• From $ \frac{5}{7}Z + 20 = k $: $ \frac{5}{7}Z = k - 20 $ $ Z = \frac{7}{5}(k - 20) $

Calculating the Common Value (k)

Substitute these expressions into the total sum equation $X + Y + Z = 1200$:

$ \frac{5}{3}(k - 30) + \frac{5}{2}(k - 10) + \frac{7}{5}(k - 20) = 1200 $

Expand the terms:

$ \frac{5}{3}k - 50 + \frac{5}{2}k - 25 + \frac{7}{5}k - 28 = 1200 $

Group the $k$ terms and the constant terms:

$ k \left( \frac{5}{3} + \frac{5}{2} + \frac{7}{5} \right) = 1200 + 50 + 25 + 28 $

Find a common denominator (30) for the fractions:

$ k \left( \frac{5 \times 10}{30} + \frac{5 \times 15}{30} + \frac{7 \times 6}{30} \right) = 1303 $

$ k \left( \frac{50 + 75 + 42}{30} \right) = 1303 $

$ k \left( \frac{167}{30} \right) = 1303 $

Solve for $k$:

$ k = 1303 \times \frac{30}{167} $

$ k = \frac{39090}{167} $

Calculating Z's Share

Now, use the value of $k$ to find Z's share using the formula $ Z = \frac{7}{5}(k - 20) $:

$ Z = \frac{7}{5} \left( \frac{39090}{167} - 20 \right) $

Convert 20 to a fraction with denominator 167:

$ Z = \frac{7}{5} \left( \frac{39090}{167} - \frac{20 \times 167}{167} \right) $

$ Z = \frac{7}{5} \left( \frac{39090 - 3340}{167} \right) $

$ Z = \frac{7}{5} \left( \frac{35750}{167} \right) $

Simplify the expression:

$ Z = \frac{7 \times 35750}{5 \times 167} $

$ Z = \frac{250250}{835} $

Divide both numerator and denominator by 5:

$ Z = \frac{50050}{167} $

Calculate the final value:

$ Z \approx 299.698199 $

Rounding to three decimal places, Z's share is ₹299.698.