The sum of Rs. 500 is divided among A, B and C such that Rs. 16 is more than 2/5 of A's share, Rs. 70 is less than 3/4 of B's share and Rs. 4 is less than 3/5 of C's share they are equal. Then what are the share of A, B, C.

Options

A. `160, 200, 140`
B. `150, 200, 150`
C. `140, 190, 170`
D. `100, 150, 250`

Answer

To solve the problem, we need to find the shares of A, B, and C from a total of Rs. 500, given the conditions about their shares.

### Step-by-Step Solution:

1. **Set Up the Equations:**
We are given three conditions:
- Rs. 16 is more than \( \frac{2}{5} \) of A's share.
- Rs. 70 is less than \( \frac{3}{4} \) of B's share.
- Rs. 4 is less than \( \frac{3}{5} \) of C's share.

Let's denote A's share as \( a \), B's share as \( b \), and C's share as \( c \).

From the conditions, we can write the following equations:
\[ k = \frac{2}{5}a + 16 \]
\[ k = \frac{3}{4}b - 70 \]
\[ k = \frac{3}{5}c - 4 \]

2. **Express Each Share in Terms of k:**
Rearranging the equations gives us:
- For A:
\[ \frac{2}{5}a = k - 16 \implies a = \frac{5}{2}(k - 16) = \frac{5k - 80}{2} \]
- For B:
\[ \frac{3}{4}b = k + 70 \implies b = \frac{4}{3}(k + 70) = \frac{4k + 280}{3} \]
- For C:
\[ \frac{3}{5}c = k + 4 \implies c = \frac{5}{3}(k + 4) = \frac{5k + 20}{3} \]

3. **Sum of Shares:**
We know that the total sum of A, B, and C is Rs. 500:
\[ a + b + c = 500 \]
Substituting the expressions for \( a \), \( b \), and \( c \):
\[ \frac{5k - 80}{2} + \frac{4k + 280}{3} + \frac{5k + 20}{3} = 500 \]

4. **Finding a Common Denominator:**
The common denominator for the fractions is 6. Multiplying the entire equation by 6 to eliminate the fractions:
\[ 6 \left(\frac{5k - 80}{2}\right) + 6 \left(\frac{4k + 280}{3}\right) + 6 \left(\frac{5k + 20}{3}\right) = 3000 \]
This simplifies to:
\[ 3(5k - 80) + 2(4k + 280) + 2(5k + 20) = 3000 \]

5. **Expanding and Combining Like Terms:**
Expanding the left side:
\[ 15k - 240 + 8k + 560 + 10k + 40 = 3000 \]
Combine like terms:
\[ 33k + 360 = 3000 \]

6. **Solving for k:**
Subtract 360 from both sides:
\[ 33k = 2640 \implies k = \frac{2640}{33} = 80 \]

7. **Finding Shares A, B, and C:**
Now substitute \( k = 80 \) back into the equations for \( a \), \( b \), and \( c \):
- For A:
\[ a = \frac{5(80) - 80}{2} = \frac{400 - 80}{2} = \frac{320}{2} = 160 \]
- For B:
\[ b = \frac{4(80) + 280}{3} = \frac{320 + 280}{3} = \frac{600}{3} = 200 \]
- For C:
\[ c = \frac{5(80) + 20}{3} = \frac{400 + 20}{3} = \frac{420}{3} = 140 \]

8. **Final Shares:**
Thus, the shares are:
- A's share = Rs. 160
- B's share = Rs. 200
- C's share = Rs. 140