The ages of sonal and manoj are in the ratio `7:5` ten years hence, the ratio of their ages will be `9:7`. Find their present ages

Options

A. `"40 years 34 years"`
B. `"37 years 27 years"`
C. `"35 years 25 years"`
D. `"36 years 26 years"`

Answer

To find the present ages of Sonal and Manoj, we can follow these steps:

### Step 1: Define the present ages
Let the present age of Sonal be \( 7x \) and the present age of Manoj be \( 5x \), where \( x \) is a common multiplier.

### Step 2: Set up the equation for ages after 10 years
After 10 years, Sonal's age will be \( 7x + 10 \) and Manoj's age will be \( 5x + 10 \). According to the problem, the ratio of their ages after 10 years will be \( 9:7 \).

### Step 3: Write the equation based on the given ratio
We can set up the equation based on the ratio:
\[ \frac{7x + 10}{5x + 10} = \frac{9}{7} \]

### Step 4: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 7(7x + 10) = 9(5x + 10) \]

### Step 5: Expand both sides
Expanding both sides results in:
\[ 49x + 70 = 45x + 90 \]

### Step 6: Rearrange the equation
Now, we will rearrange the equation to isolate \( x \):
\[ 49x - 45x = 90 - 70 \]
\[ 4x = 20 \]

### Step 7: Solve for \( x \)
Dividing both sides by 4 gives:
\[ x = 5 \]

### Step 8: Find the present ages
Now we can find the present ages of Sonal and Manoj:
- Sonal's age: \( 7x = 7 \times 5 = 35 \) years
- Manoj's age: \( 5x = 5 \times 5 = 25 \) years

### Final Answer
- Sonal's present age is **35 years**.
- Manoj's present age is **25 years**.

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