Two positive numbers x and y are such that `x gt y`. If the difference of these numbers is 5 and their product is 24, find:
sum of these numbers

Answer

To solve the problem step by step, we will define the two positive numbers \( x \) and \( y \) based on the information given in the question.

### Step 1: Set up the equations
We know from the problem:
1. The difference between the two numbers is 5:
\[ x - y = 5 \]
(Equation 1)

2. The product of the two numbers is 24:
\[ xy = 24 \]
(Equation 2)

### Step 2: Express \( x \) in terms of \( y \)
From Equation 1, we can express \( x \) in terms of \( y \):
\[ x = y + 5 \]
(Equation 3)

### Step 3: Substitute \( x \) in Equation 2
Now, we will substitute Equation 3 into Equation 2:
\[ (y + 5)y = 24 \]
Expanding this gives:
\[ y^2 + 5y = 24 \]

### Step 4: Rearrange the equation
Rearranging the equation to set it to zero:
\[ y^2 + 5y - 24 = 0 \]
(Equation 4)

### Step 5: Factor the quadratic equation
Now we will factor Equation 4. We need two numbers that multiply to \(-24\) and add to \(5\). The numbers \(8\) and \(-3\) work:
\[ (y + 8)(y - 3) = 0 \]

### Step 6: Solve for \( y \)
Setting each factor to zero gives us:
1. \( y + 8 = 0 \) → \( y = -8 \) (not a valid solution since \( y \) must be positive)
2. \( y - 3 = 0 \) → \( y = 3 \)

### Step 7: Find \( x \)
Now we can find \( x \) using Equation 3:
\[ x = y + 5 = 3 + 5 = 8 \]

### Step 8: Calculate the sum of \( x \) and \( y \)
Finally, we calculate the sum of \( x \) and \( y \):
\[ x + y = 8 + 3 = 11 \]

### Final Answer
The sum of the two numbers \( x \) and \( y \) is \( 11 \).

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