A sum of ₹ 1,710 is divided in A, B and C such that 4 times of A, 6 times of B and 9 times of C are equal. What is the difference between A and C?

Options

A. ₹ 360
B. ₹ 450
C. ₹ 480
D. ₹ 540

Answer

To solve the problem, we need to find the values of A, B, and C based on the given conditions. Let's break it down step by step.

### Step 1: Set up the equations based on the given conditions.
We know that:
- \( 4A = 6B = 9C \)

Let’s denote this common value as \( k \). Therefore, we can express A, B, and C in terms of \( k \):
- \( A = \frac{k}{4} \)
- \( B = \frac{k}{6} \)
- \( C = \frac{k}{9} \)

### Step 2: Express the total sum in terms of k.
The total sum of A, B, and C is given as ₹ 1,710. Thus, we can write:
\[ A + B + C = \frac{k}{4} + \frac{k}{6} + \frac{k}{9} = 1710 \]

### Step 3: Find a common denominator and simplify.
The least common multiple (LCM) of 4, 6, and 9 is 36. We can rewrite the fractions:
\[ A + B + C = \frac{9k}{36} + \frac{6k}{36} + \frac{4k}{36} = \frac{(9k + 6k + 4k)}{36} = \frac{19k}{36} \]
Setting this equal to ₹ 1,710 gives us:
\[ \frac{19k}{36} = 1710 \]

### Step 4: Solve for k.
To solve for \( k \), we multiply both sides by 36:
\[ 19k = 1710 \times 36 \]
Calculating the right side:
\[ 1710 \times 36 = 61,560 \]
Now divide by 19:
\[ k = \frac{61,560}{19} = 3,240 \]

### Step 5: Calculate A, B, and C.
Now that we have \( k \), we can find A, B, and C:
- \( A = \frac{3,240}{4} = 810 \)
- \( B = \frac{3,240}{6} = 540 \)
- \( C = \frac{3,240}{9} = 360 \)

### Step 6: Find the difference between A and C.
Now we need to find the difference between A and C:
\[ \text{Difference} = A - C = 810 - 360 = 450 \]

### Final Answer:
The difference between A and C is ₹ 450.

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