Rani , who is y yr old at present, is x yr older than Hamid 15 yr ago, Hamid's age was `(1)/(4)` of the age of Rani. Which of the following is true ?

Options

A. `3y - 4x = 45`
B. `2y - x = 15`
C. `(y)/(x) - 15 = (1)/(4)`
D. `3x - 4y = 45`

Answer

To solve the problem step by step, we will use the information given about Rani and Hamid's ages.

### Step 1: Define the variables
Let:
- Rani's present age = \( y \) years
- Hamid's present age = \( h \) years

### Step 2: Establish the relationship between Rani's and Hamid's ages
According to the problem, Rani is \( x \) years older than Hamid. Therefore, we can express Hamid's age in terms of Rani's age:
\[ h = y - x \]

### Step 3: Determine their ages 15 years ago
- Rani's age 15 years ago = \( y - 15 \)
- Hamid's age 15 years ago = \( h - 15 \)

### Step 4: Set up the equation based on the given condition
The problem states that 15 years ago, Hamid's age was \( \frac{1}{4} \) of Rani's age. Therefore, we can write:
\[ h - 15 = \frac{1}{4}(y - 15) \]

### Step 5: Substitute Hamid's age into the equation
Substituting \( h = y - x \) into the equation:
\[ (y - x) - 15 = \frac{1}{4}(y - 15) \]

### Step 6: Simplify the equation
Now, simplify the equation:
\[ y - x - 15 = \frac{1}{4}y - \frac{15}{4} \]

### Step 7: Eliminate the fraction
To eliminate the fraction, multiply the entire equation by 4:
\[ 4(y - x - 15) = y - 15 \]
This simplifies to:
\[ 4y - 4x - 60 = y - 15 \]

### Step 8: Rearrange the equation
Now, rearranging the equation gives:
\[ 4y - y - 4x = -15 + 60 \]
This simplifies to:
\[ 3y - 4x = 45 \]

### Conclusion
The derived equation \( 3y - 4x = 45 \) is the relationship that holds true based on the conditions provided in the problem.