If the sum of two numbers is equal to five times of their difference, then what is the ratio of the numbers?

Options

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

Answer

To solve the problem step by step, let's denote the two numbers as \( x \) and \( y \).

### Step 1: Set up the equation based on the problem statement
We are given that the sum of the two numbers is equal to five times their difference. This can be expressed mathematically as:
\[ x + y = 5(x - y) \]

### Step 2: Expand the equation
Now, let's expand the right-hand side of the equation:
\[ x + y = 5x - 5y \]

### Step 3: Rearrange the equation
Next, we will rearrange the equation to group all terms involving \( x \) on one side and all terms involving \( y \) on the other side:
\[ x + y + 5y = 5x \]
This simplifies to:
\[ x + 6y = 5x \]

### Step 4: Isolate \( x \)
Now, we will isolate \( x \) by moving \( x \) to the right side:
\[ 6y = 5x - x \]
This simplifies to:
\[ 6y = 4x \]

### Step 5: Simplify the equation
Now, we can simplify the equation further:
\[ \frac{x}{y} = \frac{6}{4} = \frac{3}{2} \]

### Step 6: Write the ratio
Thus, the ratio of the two numbers \( x \) and \( y \) is:
\[ x : y = 3 : 2 \]

### Final Answer
The ratio of the numbers is \( 3 : 2 \).

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