The sum of two positive numbers x and y is 2.5 times their difference. If the product of numbers is 84, then what is the sum of those two numbers?

Options

A. 26
B. 24
C. 22
D. 20

Answer

To solve the problem step by step, we will define the two positive numbers as \( x \) and \( y \).

### Step 1: Set up the equations based on the problem statement.
The problem states that the sum of the two numbers \( x \) and \( y \) is 2.5 times their difference. We can express this mathematically as:

\[ x + y = 2.5(x - y) \]

Additionally, we know that the product of the two numbers is 84:

\[ xy = 84 \]

### Step 2: Simplify the first equation.
Let's simplify the first equation:

\[ x + y = 2.5x - 2.5y \]

Rearranging gives:

\[ x + y + 2.5y = 2.5x \]

This simplifies to:

\[ x + 3.5y = 2.5x \]

Now, rearranging further:

\[ 3.5y = 2.5x - x \]

\[ 3.5y = 1.5x \]

Dividing both sides by 1.5 gives:

\[ y = \frac{1.5}{3.5}x = \frac{3}{7}x \]

### Step 3: Substitute \( y \) in the product equation.
Now we will substitute \( y \) in the product equation \( xy = 84 \):

\[ x \left(\frac{3}{7}x\right) = 84 \]

This simplifies to:

\[ \frac{3}{7}x^2 = 84 \]

### Step 4: Solve for \( x^2 \).
To eliminate the fraction, multiply both sides by 7:

\[ 3x^2 = 588 \]

Now, divide both sides by 3:

\[ x^2 = 196 \]

Taking the square root of both sides gives:

\[ x = 14 \]

### Step 5: Find \( y \) using the value of \( x \).
Now that we have \( x \), we can find \( y \):

\[ y = \frac{3}{7} \times 14 = 6 \]

### Step 6: Find the sum of \( x \) and \( y \).
Now we can find the sum of \( x \) and \( y \):

\[ x + y = 14 + 6 = 20 \]

### Final Answer:
The sum of the two numbers is \( \boxed{20} \).

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