The present ratio of ages of A and B is `4 : 5`. 18 years ago, this ratio was `11 : 16`. Find the sum total of their present ages.

Options

A. 90 years
B. 105 years
C. 110 years
D. 80 years

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the present ages of A and B
Let the present age of A be \(4x\) and the present age of B be \(5x\), based on the given ratio of their ages \(4:5\).

### Step 2: Set up the equation for their ages 18 years ago
18 years ago, the age of A would be \(4x - 18\) and the age of B would be \(5x - 18\). According to the problem, the ratio of their ages 18 years ago was \(11:16\). Therefore, we can set up the equation:
\[ \frac{4x - 18}{5x - 18} = \frac{11}{16} \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 16(4x - 18) = 11(5x - 18) \]

### Step 4: Expand both sides of the equation
Expanding both sides results in:
\[ 64x - 288 = 55x - 198 \]

### Step 5: Rearrange the equation to isolate x
Now, we will move all terms involving \(x\) to one side and constant terms to the other side:
\[ 64x - 55x = 288 - 198 \]
This simplifies to:
\[ 9x = 90 \]

### Step 6: Solve for x
Dividing both sides by 9 gives:
\[ x = 10 \]

### Step 7: Find the present ages of A and B
Now, substituting \(x\) back into the expressions for the ages:
- Present age of A: \(4x = 4 \times 10 = 40\)
- Present age of B: \(5x = 5 \times 10 = 50\)

### Step 8: Calculate the sum of their present ages
Now, we can find the sum of their present ages:
\[ 40 + 50 = 90 \]

### Final Answer
The sum total of their present ages is \(90\).

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