The sum of two numbers is 4500. If `10%` of one number is `12.5%` of the other, find the numbers.

Answer

To solve the problem step by step, we will define the two numbers and set up 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 First Equation
According to the problem, the sum of the two numbers is 4500. Therefore, we can write the first equation as:
\[ x + y = 4500 \quad \text{(1)} \]

### Step 3: Set Up the Second Equation
The problem states that 10% of one number is equal to 12.5% of the other number. We can express this mathematically:
\[ 0.1x = 0.125y \]
To eliminate the decimals, we can multiply the entire equation by 100:
\[ 10x = 12.5y \]
Now, to make it easier to work with, we can multiply through by 8 to eliminate the decimal in 12.5:
\[ 80x = 100y \quad \text{(2)} \]

### Step 4: Simplify the Second Equation
We can simplify equation (2) by dividing both sides by 20:
\[ 4x = 5y \quad \text{(3)} \]

### Step 5: Substitute Equation (3) into Equation (1)
From equation (3), we can express \( x \) in terms of \( y \):
\[ x = \frac{5y}{4} \]
Now we will substitute this expression for \( x \) into equation (1):
\[ \frac{5y}{4} + y = 4500 \]

### Step 6: Combine Like Terms
To combine the terms on the left side, we need a common denominator:
\[ \frac{5y}{4} + \frac{4y}{4} = 4500 \]
This simplifies to:
\[ \frac{9y}{4} = 4500 \]

### Step 7: Solve for \( y \)
To solve for \( y \), multiply both sides by 4:
\[ 9y = 18000 \]
Now, divide both sides by 9:
\[ y = 2000 \]

### Step 8: Substitute Back to Find \( x \)
Now that we have \( y \), we can substitute \( y = 2000 \) back into equation (1) to find \( x \):
\[ x + 2000 = 4500 \]
Subtracting 2000 from both sides gives:
\[ x = 2500 \]

### Final Answer
The two numbers are:
\[ x = 2500 \quad \text{and} \quad y = 2000 \]