Two numbers are in the ratio 4:7 and their LCM is 84. Find the two numbers.

Answer

To find the two numbers that are in the ratio 4:7 and have an LCM of 84, we can follow these steps:

### Step 1: Define the Numbers
Let the first number be \( a \) and the second number be \( b \). According to the ratio given, we can express these numbers in terms of a common variable \( x \):
- \( a = 4x \)
- \( b = 7x \)

### Step 2: Use the LCM Formula
We know that the product of two numbers is equal to the product of their LCM and GCD (or SCF). The formula is:
\[ a \times b = \text{LCM}(a, b) \times \text{GCD}(a, b) \]
Since \( x \) is the GCD of \( a \) and \( b \) (because they are expressed in terms of \( x \)), we can substitute:
\[ (4x) \times (7x) = \text{LCM}(a, b) \times x \]

### Step 3: Substitute the LCM Value
We know from the problem that the LCM is 84. So we can write:
\[ (4x) \times (7x) = 84 \times x \]
This simplifies to:
\[ 28x^2 = 84x \]

### Step 4: Simplify the Equation
We can divide both sides of the equation by \( x \) (assuming \( x \neq 0 \)):
\[ 28x = 84 \]

### Step 5: Solve for \( x \)
Now, divide both sides by 28:
\[ x = \frac{84}{28} = 3 \]

### Step 6: Find the Two Numbers
Now that we have \( x \), we can find the two numbers:
- First number \( a = 4x = 4 \times 3 = 12 \)
- Second number \( b = 7x = 7 \times 3 = 21 \)

### Conclusion
The two numbers are **12** and **21**.

---