There are two numbers such that sum of twice the first number and thrice the second number is 100 and the sum of thrice the first number and twice the second number is 120. Which is the larger number?

Options

A. 32
B. 12
C. 14
D. 35

Answer

To solve the problem, we need to set up a system of linear equations based on the information given. Let's denote the first number as \( x \) and the second number as \( y \).

### Step 1: Set up the equations
From the problem, we have two statements that can be translated into equations:
1. The sum of twice the first number and thrice the second number is 100:
\[ 2x + 3y = 100 \quad \text{(Equation 1)} \]
2. The sum of thrice the first number and twice the second number is 120:
\[ 3x + 2y = 120 \quad \text{(Equation 2)} \]

### Step 2: Solve the equations simultaneously
We can solve these equations using the method of substitution or elimination. Here, we will use the elimination method.

First, let's multiply Equation 1 by 3 and Equation 2 by 2 to make the coefficients of \( x \) the same:
\[ 3(2x + 3y) = 3(100) \implies 6x + 9y = 300 \quad \text{(Equation 3)} \]
\[ 2(3x + 2y) = 2(120) \implies 6x + 4y = 240 \quad \text{(Equation 4)} \]

### Step 3: Subtract Equation 4 from Equation 3
Now, we will subtract Equation 4 from Equation 3 to eliminate \( x \):
\[ (6x + 9y) - (6x + 4y) = 300 - 240 \]
This simplifies to:
\[ 5y = 60 \]
Now, divide both sides by 5:
\[ y = 12 \]

### Step 4: Substitute \( y \) back into one of the original equations
Now that we have \( y \), we can substitute it back into Equation 1 to find \( x \):
\[ 2x + 3(12) = 100 \]
This simplifies to:
\[ 2x + 36 = 100 \]
Subtract 36 from both sides:
\[ 2x = 64 \]
Now, divide by 2:
\[ x = 32 \]

### Step 5: Identify the larger number
Now we have both numbers:
- First number \( x = 32 \)
- Second number \( y = 12 \)

Thus, the larger number is:
\[ \text{Larger number} = 32 \]

### Summary of the solution:
The larger number is \( 32 \).