There are two numbers. When first number is increased by `125%`, the obtained number is 300 less than second number and if the first number is increased by 150% then the obtained number is 200 less than the second number. Find out the sum of the two numbers.

Answer

To solve the problem step by step, let's denote the first number as \( x \) and the second number as \( y \).

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

1. When the first number \( x \) is increased by \( 125\% \):
\[ x + 1.25x = 2.25x \]
According to the problem, this result is \( 300 \) less than the second number \( y \):
\[ 2.25x = y - 300 \quad \text{(Equation 1)} \]

2. When the first number \( x \) is increased by \( 150\% \):
\[ x + 1.5x = 2.5x \]
This result is \( 200 \) less than the second number \( y \):
\[ 2.5x = y - 200 \quad \text{(Equation 2)} \]

### Step 2: Solve the equations simultaneously.

From Equation 1:
\[ y = 2.25x + 300 \]

From Equation 2:
\[ y = 2.5x + 200 \]

Now, we can set the two expressions for \( y \) equal to each other:
\[ 2.25x + 300 = 2.5x + 200 \]

### Step 3: Simplify the equation.

Rearranging gives:
\[ 300 - 200 = 2.5x - 2.25x \]
\[ 100 = 0.25x \]

### Step 4: Solve for \( x \).

Dividing both sides by \( 0.25 \):
\[ x = \frac{100}{0.25} = 400 \]

### Step 5: Substitute \( x \) back to find \( y \).

Using \( x = 400 \) in Equation 1:
\[ y = 2.25(400) + 300 \]
\[ y = 900 + 300 = 1200 \]

### Step 6: Find the sum of the two numbers.

Now we can find the sum of \( x \) and \( y \):
\[ x + y = 400 + 1200 = 1600 \]

### Final Answer:
The sum of the two numbers is \( 1600 \).

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