There are two numbers such that the sum of twice the first number and thrice the second number is 150 and the sum of thrice of first number and twice the second number is 170. What is the greater number ?

Options

A. 42
B. 22
C. 32
D. 52

Answer

To solve the problem step-by-step, let's define the two numbers as \( a \) and \( b \).

### Step 1: Set up the equations
From the problem statement, we can derive the following two equations based on the information given:

1. \( 2a + 3b = 150 \) (Equation 1)
2. \( 3a + 2b = 170 \) (Equation 2)

### Step 2: Solve the equations
We can solve these equations simultaneously. First, let's manipulate the equations to eliminate one variable.

#### Step 2.1: Multiply both equations to align coefficients
To eliminate \( b \), we can multiply Equation 1 by 2 and Equation 2 by 3:

- \( 2(2a + 3b) = 2(150) \) → \( 4a + 6b = 300 \) (Equation 3)
- \( 3(3a + 2b) = 3(170) \) → \( 9a + 6b = 510 \) (Equation 4)

#### Step 2.2: Subtract Equation 3 from Equation 4
Now, we can subtract Equation 3 from Equation 4 to eliminate \( b \):

\[ (9a + 6b) - (4a + 6b) = 510 - 300 \]

This simplifies to:

\[ 5a = 210 \]

#### Step 2.3: Solve for \( a \)
Now, divide both sides by 5:

\[ a = \frac{210}{5} = 42 \]

### Step 3: Substitute \( a \) back to find \( b \)
Now that we have \( a \), we can substitute \( a = 42 \) back into one of the original equations to find \( b \). Let's use Equation 1:

\[ 2(42) + 3b = 150 \]

This simplifies to:

\[ 84 + 3b = 150 \]

Subtract 84 from both sides:

\[ 3b = 150 - 84 \]
\[ 3b = 66 \]

Now, divide by 3:

\[ b = \frac{66}{3} = 22 \]

### Step 4: Identify the greater number
Now we have both numbers: \( a = 42 \) and \( b = 22 \). The greater number is:

\[ \text{Greater number} = 42 \]

### Final Answer
The greater number is **42**.

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