The ratio of present ages of two brothers is 1 : 2 and 5 years back. the ratio was 1: 3. What will be the ratio of their ages after 5 years ?

Options

A. `1:4`
B. `2:3`
C. `3:5`
D. `5:6`

Answer

To solve the problem step by step, we will denote the present ages of the two brothers as follows:

Let the present age of Brother 1 be \( x \) years.
Let the present age of Brother 2 be \( 2x \) years (since the ratio of their ages is 1:2).

### Step 1: Set up the equation based on the age ratio 5 years ago
Five years ago, Brother 1's age would have been \( x - 5 \) and Brother 2's age would have been \( 2x - 5 \). According to the problem, the ratio of their ages 5 years ago was 1:3. Therefore, we can set up the equation:

\[ \frac{x - 5}{2x - 5} = \frac{1}{3} \]

### Step 2: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:

\[ 3(x - 5) = 1(2x - 5) \]

### Step 3: Simplify the equation
Expanding both sides:

\[ 3x - 15 = 2x - 5 \]

### Step 4: Solve for \( x \)
Now, we will isolate \( x \):

\[ 3x - 2x = -5 + 15 \]
\[ x = 10 \]

### Step 5: Find the present ages of both brothers
Now that we have \( x \), we can find the present ages:

- Brother 1's present age = \( x = 10 \) years
- Brother 2's present age = \( 2x = 20 \) years

### Step 6: Calculate their ages after 5 years
After 5 years, their ages will be:

- Brother 1's age = \( 10 + 5 = 15 \) years
- Brother 2's age = \( 20 + 5 = 25 \) years

### Step 7: Find the ratio of their ages after 5 years
Now we can find the ratio of their ages after 5 years:

\[ \text{Ratio} = \frac{15}{25} = \frac{3}{5} \]

### Final Answer
The ratio of their ages after 5 years will be \( 3:5 \).

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