Each question has four options(A), (B), (C) and (D) for answers. Select the right answer and write in English letters in the box against each question in the enclosed answer sheet.
If the product of two numbers is 21 and their difference is 4, then the ratio of the sum of their cubes to the difference of their cubes is

Options

A. 185 : 165
B. 165 : 158
C. 185 : 158
D. 158 : 145

Answer

To solve the problem step by step, we will first define the two numbers and set up the equations based on the information given in the question.

### Step 1: Define the Variables
Let the two numbers be \( x \) and \( y \).

### Step 2: Set Up the Equations
From the problem, we know:
1. The product of the two numbers is 21:
\[ x \cdot y = 21 \] (Equation 1)
2. The difference of the two numbers is 4:
\[ x - y = 4 \] (Equation 2)

### Step 3: Solve for One Variable
From Equation 2, we can express \( x \) in terms of \( y \):
\[ x = y + 4 \]

### Step 4: Substitute into the First Equation
Now, substitute \( x \) from Equation 2 into Equation 1:
\[ (y + 4) \cdot y = 21 \]
Expanding this gives:
\[ y^2 + 4y = 21 \]

### Step 5: Rearrange the Equation
Rearranging the equation to set it to zero:
\[ y^2 + 4y - 21 = 0 \]

### Step 6: Factor the Quadratic Equation
Now, we need to factor the quadratic equation:
\[ (y + 7)(y - 3) = 0 \]
This gives us two possible values for \( y \):
1. \( y = 3 \)
2. \( y = -7 \)

### Step 7: Find Corresponding Values for \( x \)
Using \( y = 3 \):
\[ x = 3 + 4 = 7 \]

Using \( y = -7 \):
\[ x = -7 + 4 = -3 \]

### Step 8: Choose the Positive Pair
Since we are looking for positive numbers, we take \( x = 7 \) and \( y = 3 \).

### Step 9: Calculate the Sum of Their Cubes
Now we calculate the sum of their cubes:
\[ x^3 + y^3 = 7^3 + 3^3 = 343 + 27 = 370 \]

### Step 10: Calculate the Difference of Their Cubes
Next, we calculate the difference of their cubes:
\[ x^3 - y^3 = 7^3 - 3^3 = 343 - 27 = 316 \]

### Step 11: Find the Ratio
Now we find the ratio of the sum of their cubes to the difference of their cubes:
\[ \text{Ratio} = \frac{x^3 + y^3}{x^3 - y^3} = \frac{370}{316} \]

### Step 12: Simplify the Ratio
To simplify the ratio, we divide both the numerator and denominator by 2:
\[ \frac{370 \div 2}{316 \div 2} = \frac{185}{158} \]

### Final Answer
Thus, the ratio of the sum of their cubes to the difference of their cubes is:
\[ \boxed{185 \text{ ratio } 158} \]