The ratio of the present age of A to B is. `4:5` and that of the present age of C to D is `7 : 5`. The average of the present ages of A, B, C and D is 27 years and D’s age 3 years hence will be 14 years less than A’s present age. What is B’s present age? (in years)

Options

A. 30
B. 25
C. 40
D. 50

Answer

To solve the problem step by step, we will use the information given in the question.

### Step 1: Set up the ratios
Let the present ages of A and B be represented as:
- A's age = 4x
- B's age = 5x

Let the present ages of C and D be represented as:
- C's age = 7y
- D's age = 5y

### Step 2: Use the average age information
We know that the average age of A, B, C, and D is 27 years. Therefore, we can write the equation:
\[ \frac{(4x + 5x + 7y + 5y)}{4} = 27 \]
This simplifies to:
\[ \frac{(9x + 12y)}{4} = 27 \]
Multiplying both sides by 4 gives:
\[ 9x + 12y = 108 \quad \text{(Equation 1)} \]

### Step 3: Use the information about D's age in 3 years
According to the problem, D’s age in 3 years will be 14 years less than A’s present age. Therefore, we can write:
\[ 5y + 3 = 4x - 14 \]
Rearranging this gives:
\[ 4x - 5y = 17 \quad \text{(Equation 2)} \]

### Step 4: Solve the equations simultaneously
Now we have two equations:
1. \(9x + 12y = 108\) (Equation 1)
2. \(4x - 5y = 17\) (Equation 2)

To eliminate one variable, we can multiply Equation 2 by 3:
\[ 12x - 15y = 51 \quad \text{(Equation 3)} \]

Now we can multiply Equation 1 by 4:
\[ 36x + 48y = 432 \quad \text{(Equation 4)} \]

### Step 5: Solve for x and y
Now we will subtract Equation 3 from Equation 4:
\[ (36x + 48y) - (12x - 15y) = 432 - 51 \]
This simplifies to:
\[ 24x + 63y = 381 \]
Now we can solve for y:
From Equation 2, we can express \(x\) in terms of \(y\):
\[ 4x = 5y + 17 \implies x = \frac{5y + 17}{4} \]

Substituting this into Equation 1:
\[ 9\left(\frac{5y + 17}{4}\right) + 12y = 108 \]
Multiplying through by 4 to eliminate the fraction:
\[ 9(5y + 17) + 48y = 432 \]
Expanding gives:
\[ 45y + 153 + 48y = 432 \]
Combining like terms:
\[ 93y = 432 - 153 \]
\[ 93y = 279 \implies y = 3 \]

### Step 6: Find x
Now substituting \(y = 3\) back into the equation for \(x\):
\[ 4x = 5(3) + 17 = 15 + 17 = 32 \implies x = 8 \]

### Step 7: Find B's present age
Now we can find B's present age:
\[ B's \, age = 5x = 5(8) = 40 \, \text{years} \]

### Final Answer
B's present age is **40 years**.