There are two number such that the sum of twice the first number and thrice the second number is 300 and the sum of thrice the first number and twice the second number is 265. What is the larger number?

Options

A. 24
B. 39
C. 85
D. 74

Answer

To solve the problem, we need to set up a system of linear equations based on the information provided.

### Step 1: Define the Variables
Let:
- \( a \) = the first number
- \( b \) = the second number

### Step 2: Set Up the Equations
From the problem statement, we can derive two equations:
1. The sum of twice the first number and thrice the second number is 300:
\[ 2a + 3b = 300 \quad \text{(Equation 1)} \]
2. The sum of thrice the first number and twice the second number is 265:
\[ 3a + 2b = 265 \quad \text{(Equation 2)} \]

### Step 3: Solve the Equations
We can solve these equations using the elimination or substitution method. Here, we will use the elimination method.

First, we can multiply both equations to align the coefficients of one of the variables. Let's multiply Equation 1 by 3 and Equation 2 by 2:

- Multiply Equation 1 by 3:
\[ 6a + 9b = 900 \quad \text{(Equation 3)} \]

- Multiply Equation 2 by 2:
\[ 6a + 4b = 530 \quad \text{(Equation 4)} \]

### Step 4: Subtract the Equations
Now, subtract Equation 4 from Equation 3:
\[ (6a + 9b) - (6a + 4b) = 900 - 530 \]
This simplifies to:
\[ 5b = 370 \]

### Step 5: Solve for \( b \)
Now, divide both sides by 5:
\[ b = \frac{370}{5} = 74 \]

### Step 6: Substitute \( b \) Back to Find \( a \)
Now that we have \( b \), we can substitute it back into one of the original equations to find \( a \). We can use Equation 1:
\[ 2a + 3(74) = 300 \]
This simplifies to:
\[ 2a + 222 = 300 \]
Now, subtract 222 from both sides:
\[ 2a = 300 - 222 \]
\[ 2a = 78 \]
Now, divide by 2:
\[ a = \frac{78}{2} = 39 \]

### Step 7: Identify the Larger Number
Now we have both numbers:
- \( a = 39 \)
- \( b = 74 \)

The larger number is:
\[ \text{Larger number} = b = 74 \]

### Final Answer
The larger number is **74**.

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