The product of two positive numbers is 11520 and their quotient is `(9)/(5)`. Find the difference of two numbers.

Options

A. 60
B. 64
C. 74
D. 70

Answer

To solve the problem, we need to find two positive numbers \( X \) and \( Y \) such that:

1. The product of the two numbers is \( XY = 11520 \).
2. The quotient of the two numbers is \( \frac{X}{Y} = \frac{9}{5} \).

Let's denote the two numbers as \( X \) and \( Y \).

### Step 1: Express \( X \) and \( Y \) in terms of a variable
From the quotient, we can express \( X \) and \( Y \) in terms of a variable \( a \):
\[ X = 9a \quad \text{and} \quad Y = 5a \]

### Step 2: Substitute into the product equation
Now, we can substitute \( X \) and \( Y \) into the product equation:
\[ XY = (9a)(5a) = 45a^2 \]
Setting this equal to 11520 gives us:
\[ 45a^2 = 11520 \]

### Step 3: Solve for \( a^2 \)
To find \( a^2 \), we divide both sides by 45:
\[ a^2 = \frac{11520}{45} \]
Calculating the right side:
\[ a^2 = 256 \]

### Step 4: Solve for \( a \)
Taking the square root of both sides gives us:
\[ a = \sqrt{256} = 16 \]

### Step 5: Find \( X \) and \( Y \)
Now we can find the values of \( X \) and \( Y \):
\[ X = 9a = 9 \times 16 = 144 \]
\[ Y = 5a = 5 \times 16 = 80 \]

### Step 6: Find the difference of \( X \) and \( Y \)
Finally, we find the difference between the two numbers:
\[ \text{Difference} = X - Y = 144 - 80 = 64 \]

Thus, the difference of the two numbers is \( \boxed{64} \).