A sum of ₹14460 is divided among A, B, C and D such that the ratio of share of A and B is 3 : 5, that of B and C is 6 : 7 and that of C and D is 14 : 15. What is the difference between the shares of A and C?

Options

A. ₹1440
B. ₹1500
C. ₹2100
D. ₹2040

Answer

To solve the problem step by step, we will follow the ratios given and find the shares of A, B, C, and D.

### Step 1: Set up the ratios
We are given the following ratios:
- A : B = 3 : 5
- B : C = 6 : 7
- C : D = 14 : 15

### Step 2: Find a common representation for the ratios
To combine these ratios, we need to express them in a way that allows us to compare all four individuals. We will express B in terms of a common variable.

Let:
- A = 3x
- B = 5x (from A : B)
- From B : C, since B = 5x, we can find C:
\[ \frac{B}{C} = \frac{6}{7} \implies C = \frac{7}{6}B = \frac{7}{6}(5x) = \frac{35}{6}x \]
- From C : D, since C = \(\frac{35}{6}x\), we can find D:
\[ \frac{C}{D} = \frac{14}{15} \implies D = \frac{15}{14}C = \frac{15}{14}\left(\frac{35}{6}x\right) = \frac{75}{12}x = \frac{25}{4}x \]

### Step 3: Write the combined ratio
Now we have:
- A = 3x
- B = 5x
- C = \(\frac{35}{6}x\)
- D = \(\frac{25}{4}x\)

To combine these ratios, we will convert them to a common denominator. The least common multiple of the denominators (1, 1, 6, 4) is 12.

Now, we convert each term:
- A = \(3x = \frac{36}{12}x\)
- B = \(5x = \frac{60}{12}x\)
- C = \(\frac{35}{6}x = \frac{70}{12}x\)
- D = \(\frac{25}{4}x = \frac{75}{12}x\)

Thus, the combined ratio of A : B : C : D is:
\[ 36 : 60 : 70 : 75 \]

### Step 4: Calculate the total parts
Now, we sum the parts of the ratio:
\[ 36 + 60 + 70 + 75 = 241 \]

### Step 5: Find the value of x
We know the total amount is ₹14460. Therefore, we can set up the equation:
\[ \text{Total} = 241x = 14460 \]
To find x, we divide:
\[ x = \frac{14460}{241} = 60 \]

### Step 6: Calculate the shares of A and C
Now we can find the shares:
- A's share = \(3x = 3 \times 60 = 180\)
- C's share = \(\frac{35}{6}x = \frac{35}{6} \times 60 = 350\)

### Step 7: Find the difference between the shares of A and C
Now we find the difference:
\[ \text{Difference} = C - A = 350 - 180 = 170 \]

### Final Answer
The difference between the shares of A and C is ₹170.

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