If the difference of two numbers is 5 and the difference of their square is 125, then out of the two which is the smaller number?

Options

A. 15
B. 12
C. 20
D. 10

Answer

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

### Step 1: Define the Variables
Let the two numbers be \( x \) and \( y \). According to the problem, we know:
1. The difference of the two numbers is 5:
\[ x - y = 5 \quad \text{(Equation 1)} \]

### Step 2: Set Up the Second Equation
We also know that the difference of their squares is 125:
\[ x^2 - y^2 = 125 \]
Using the difference of squares formula, we can rewrite this as:
\[ (x - y)(x + y) = 125 \]
Substituting the value of \( x - y \) from Equation 1:
\[ 5(x + y) = 125 \]

### Step 3: Solve for \( x + y \)
Now, we can solve for \( x + y \):
\[ x + y = \frac{125}{5} = 25 \quad \text{(Equation 2)} \]

### Step 4: Solve the System of Equations
Now we have a system of two equations:
1. \( x - y = 5 \) (Equation 1)
2. \( x + y = 25 \) (Equation 2)

We can add these two equations together:
\[ (x - y) + (x + y) = 5 + 25 \]
This simplifies to:
\[ 2x = 30 \]
Dividing both sides by 2 gives:
\[ x = 15 \]

### Step 5: Find the Value of \( y \)
Now that we have \( x \), we can substitute it back into one of the equations to find \( y \). Using Equation 2:
\[ 15 + y = 25 \]
Subtracting 15 from both sides gives:
\[ y = 25 - 15 = 10 \]

### Step 6: Identify the Smaller Number
Now we have both numbers:
- \( x = 15 \)
- \( y = 10 \)

Since we need to identify the smaller number, we see that:
\[ \text{The smaller number is } y = 10. \]

### Final Answer
The smaller number is **10**.