The sum of Rs. 600 is divided in A, B and C such that Rs. 40 is more than 2/5 of A's, Rs. 20 is more than 2/7 of B's and Rs. 10 is more than 9/17 of C's share, they are equal. Then what is the share of A, B, C respectively.

Options

A. `140, 240, 190`
B. `170, 260, 170`
C. `150, 280, 170`
D. `100, 330, 170`

Answer

To solve the problem, we need to set up equations based on the information given about the shares of A, B, and C. Let's break it down step by step.

### Step 1: Set up the equations based on the conditions given.

We know from the problem statement:
1. Rs. 40 is more than \( \frac{2}{5} \) of A's share.
2. Rs. 20 is more than \( \frac{2}{7} \) of B's share.
3. Rs. 10 is more than \( \frac{9}{17} \) of C's share.

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

From the conditions, we can write the following equations:
- \( \frac{2}{5}a + 40 = k \) (for A)
- \( \frac{2}{7}b + 20 = k \) (for B)
- \( \frac{9}{17}c + 10 = k \) (for C)

Where \( k \) is a common value that all three expressions equal.

### Step 2: Rearranging the equations to express A, B, and C in terms of k.

From the equations, we can express \( a \), \( b \), and \( c \) as follows:
1. From \( \frac{2}{5}a + 40 = k \):
\[ \frac{2}{5}a = k - 40 \implies a = \frac{5}{2}(k - 40) = \frac{5k - 200}{2} \]

2. From \( \frac{2}{7}b + 20 = k \):
\[ \frac{2}{7}b = k - 20 \implies b = 7 \cdot \frac{(k - 20)}{2} = \frac{7k - 140}{2} \]

3. From \( \frac{9}{17}c + 10 = k \):
\[ \frac{9}{17}c = k - 10 \implies c = 17 \cdot (k - 10) / 9 = \frac{17k - 170}{9} \]

### Step 3: Set up the equation for the total sum of A, B, and C.

We know the total sum of A, B, and C is Rs. 600:
\[ a + b + c = 600 \]
Substituting the expressions we found:
\[ \frac{5k - 200}{2} + \frac{7k - 140}{2} + \frac{17k - 170}{9} = 600 \]

### Step 4: Finding a common denominator and solving for k.

The common denominator for the fractions is 18. Thus, we rewrite the equation:
\[ \frac{9(5k - 200) + 9(7k - 140) + 2(17k - 170)}{18} = 600 \]
Multiplying through by 18 to eliminate the denominator:
\[ 9(5k - 200) + 9(7k - 140) + 2(17k - 170) = 10800 \]
Expanding this:
\[ 45k - 1800 + 63k - 1260 + 34k - 340 = 10800 \]
Combining like terms:
\[ 142k - 3400 = 10800 \]
Adding 3400 to both sides:
\[ 142k = 14200 \]
Dividing by 142:
\[ k = 100 \]

### Step 5: Calculate A, B, and C using the value of k.

Now substituting \( k = 100 \) back into the equations for \( a \), \( b \), and \( c \):
1. For A:
\[ a = \frac{5(100) - 200}{2} = \frac{500 - 200}{2} = \frac{300}{2} = 150 \]

2. For B:
\[ b = \frac{7(100) - 140}{2} = \frac{700 - 140}{2} = \frac{560}{2} = 280 \]

3. For C:
\[ c = \frac{17(100) - 170}{9} = \frac{1700 - 170}{9} = \frac{1530}{9} = 170 \]

### Final Shares:
Thus, the shares of A, B, and C are:
- A's share = Rs. 150
- B's share = Rs. 280
- C's share = Rs. 170