What is the sum of two numbers whose difference is 45, and the quotient of the greater number by the lesser number is 4?

Options

A. 100
B. 90
C. 80
D. 75

Answer

To solve the problem step by step, we need to find two numbers based on the conditions provided. Let's denote the greater number as \( x \) and the lesser number as \( y \).

### Step 1: Set up the equations based on the problem statement.

We know from the problem:
1. The difference between the two numbers is 45.
\[ x - y = 45 \quad \text{(Equation 1)} \]

2. The quotient of the greater number by the lesser number is 4.
\[ \frac{x}{y} = 4 \quad \text{(Equation 2)} \]

### Step 2: Rewrite Equation 2 to express \( x \) in terms of \( y \).

From Equation 2, we can express \( x \) as:
\[ x = 4y \quad \text{(Equation 3)} \]

### Step 3: Substitute Equation 3 into Equation 1.

Now, we substitute \( x \) from Equation 3 into Equation 1:
\[ 4y - y = 45 \]

### Step 4: Simplify the equation.

This simplifies to:
\[ 3y = 45 \]

### Step 5: Solve for \( y \).

Now, divide both sides by 3:
\[ y = \frac{45}{3} = 15 \]

### Step 6: Substitute \( y \) back into Equation 3 to find \( x \).

Now that we have \( y \), we can find \( x \) using Equation 3:
\[ x = 4y = 4 \times 15 = 60 \]

### Step 7: Calculate the sum of the two numbers.

Now we need to find the sum of \( x \) and \( y \):
\[ \text{Sum} = x + y = 60 + 15 = 75 \]

### Final Answer:

Thus, the sum of the two numbers is \( \boxed{75} \).

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