If '1066 is divided among A, B, C and D such that A : B = 3 : 4, B : C = 5 : 6 and C : D = 7 : 5, who will get the maximum?

Options

A. B
B. A
C. C
D. D

Answer

To solve the problem of dividing 1066 among A, B, C, and D based on the given ratios, we can follow these steps:

### Step 1: Set up the ratios
We have the following ratios:
1. A : B = 3 : 4
2. B : C = 5 : 6
3. C : D = 7 : 5

Let's express A, B, C, and D in terms of a common variable \( x \).

### Step 2: Express B in terms of A
From the ratio A : B = 3 : 4, we can write:
- \( A = 3x \)
- \( B = 4x \)

### Step 3: Express C in terms of B
From the ratio B : C = 5 : 6, we can express C as:
- \( C = \frac{6}{5}B = \frac{6}{5}(4x) = \frac{24x}{5} \)

### Step 4: Express D in terms of C
From the ratio C : D = 7 : 5, we can express D as:
- \( D = \frac{5}{7}C = \frac{5}{7}\left(\frac{24x}{5}\right) = \frac{24x}{7} \)

### Step 5: Write the equation for the total
Now, we can write the equation for the total amount:
\[ A + B + C + D = 1066 \]
Substituting the expressions we found:
\[ 3x + 4x + \frac{24x}{5} + \frac{24x}{7} = 1066 \]

### Step 6: Find a common denominator
To simplify the equation, we need a common denominator for the fractions. The least common multiple of 5 and 7 is 35. Therefore, we can rewrite the equation as:
\[ 3x + 4x + \frac{24x \cdot 7}{35} + \frac{24x \cdot 5}{35} = 1066 \]
This simplifies to:
\[ 3x + 4x + \frac{168x + 120x}{35} = 1066 \]
\[ 7x + \frac{288x}{35} = 1066 \]

### Step 7: Combine the terms
To combine the terms, convert \( 7x \) to a fraction with a denominator of 35:
\[ \frac{245x}{35} + \frac{288x}{35} = 1066 \]
Combine the fractions:
\[ \frac{533x}{35} = 1066 \]

### Step 8: Solve for x
Now, we can solve for \( x \):
\[ 533x = 1066 \cdot 35 \]
\[ 533x = 37210 \]
\[ x = \frac{37210}{533} \]
\[ x = 70 \]

### Step 9: Calculate the amounts for A, B, C, and D
Now that we have \( x \), we can find the amounts:
- \( A = 3x = 3 \cdot 70 = 210 \)
- \( B = 4x = 4 \cdot 70 = 280 \)
- \( C = \frac{24x}{5} = \frac{24 \cdot 70}{5} = 336 \)
- \( D = \frac{24x}{7} = \frac{24 \cdot 70}{7} = 240 \)

### Step 10: Identify the maximum
Now we can compare the amounts:
- A = 210
- B = 280
- C = 336
- D = 240

The maximum amount is \( C = 336 \).

### Conclusion
C will get the maximum amount.

---