If p and q are in the ratio `4:3` and their LCM is `36, p + q = ?`

Options

A. 18
B. 21
C. 24
D. 12

Answer

To solve the problem step by step, we will follow the given information about the ratio of \( p \) and \( q \), and their least common multiple (LCM).

### Step 1: Understand the ratio
We are given that \( p \) and \( q \) are in the ratio \( 4:3 \). This means we can express \( p \) and \( q \) in terms of a common variable \( x \):
\[ p = 4x \quad \text{and} \quad q = 3x \]

### Step 2: Use the LCM
We are also given that the LCM of \( p \) and \( q \) is \( 36 \). The LCM of two numbers \( a \) and \( b \) can be calculated using the formula:
\[ \text{LCM}(a, b) = \frac{a \times b}{\text{GCD}(a, b)} \]
In our case, since \( p = 4x \) and \( q = 3x \), we can find the LCM:
\[ \text{LCM}(4x, 3x) = \frac{(4x) \times (3x)}{\text{GCD}(4x, 3x)} \]
Since \( 4 \) and \( 3 \) are coprime (their GCD is \( 1 \)), we have:
\[ \text{GCD}(4x, 3x) = x \]
Thus, the LCM simplifies to:
\[ \text{LCM}(4x, 3x) = \frac{(4x) \times (3x)}{x} = 12x \]
We know that this LCM equals \( 36 \):
\[ 12x = 36 \]

### Step 3: Solve for \( x \)
To find \( x \), we divide both sides by \( 12 \):
\[ x = \frac{36}{12} = 3 \]

### Step 4: Calculate \( p \) and \( q \)
Now that we have \( x \), we can find \( p \) and \( q \):
\[ p = 4x = 4 \times 3 = 12 \]
\[ q = 3x = 3 \times 3 = 9 \]

### Step 5: Find \( p + q \)
Finally, we can find the sum \( p + q \):
\[ p + q = 12 + 9 = 21 \]

### Final Answer
Thus, the value of \( p + q \) is \( 21 \).