The ratio of the amounts with Mr Umar and Mr Gumar is ` 3 : 4`. If Mr Gumar gives Rs. 5 to Mr Umar, then the ratio of the amounts with Uma and Gumar is `4 : 3`. Mr Umar gives Rs. 5 to Mr Gumar. Find the ratio of the amounts with them.

Options

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

Explanation

To solve the problem step by step, we need to establish the amounts Mr. Umar and Mr. Gumar have based on the given ratios and conditions.

### Step 1: Define the amounts based on the initial ratio
Let the amount with Mr. Umar be \( 3x \) and the amount with Mr. Gumar be \( 4x \), where \( x \) is a common multiplier.

### Step 2: Set up the equation after Mr. Gumar gives Rs. 5 to Mr. Umar
After Mr. Gumar gives Rs. 5 to Mr. Umar, the amounts become:
- Mr. Umar: \( 3x + 5 \)
- Mr. Gumar: \( 4x - 5 \)

According to the problem, the new ratio of their amounts becomes \( 4:3 \). Therefore, we can set up the equation:
\[ \frac{3x + 5}{4x - 5} = \frac{4}{3} \]

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

### Step 4: Expand both sides
Expanding both sides results in:
\[ 9x + 15 = 16x - 20 \]

### Step 5: Rearrange the equation to solve for \( x \)
Rearranging the equation gives:
\[ 15 + 20 = 16x - 9x \]
\[ 35 = 7x \]
\[ x = 5 \]

### Step 6: Calculate the original amounts
Now that we have \( x \), we can find the original amounts:
- Amount with Mr. Umar:
\[ 3x = 3 \times 5 = 15 \]
- Amount with Mr. Gumar:
\[ 4x = 4 \times 5 = 20 \]

### Step 7: Find the amounts after Mr. Umar gives Rs. 5 to Mr. Gumar
After Mr. Umar gives Rs. 5 to Mr. Gumar, the amounts become:
- Mr. Umar: \( 15 - 5 = 10 \)
- Mr. Gumar: \( 20 + 5 = 25 \)

### Step 8: Find the final ratio of their amounts
The final ratio of the amounts with Mr. Umar and Mr. Gumar is:
\[ \frac{10}{25} = \frac{2}{5} \]

### Final Answer
The ratio of the amounts with Mr. Umar and Mr. Gumar after the transactions is \( 2:5 \).

---