The LCM of two numbers is 224 and the numbers are in the ratio 4:7. Find the numbers.

Answer

To solve the problem step by step, we will follow the given information about the LCM and the ratio of the two numbers.

### Step 1: Define the Numbers
Let the two numbers be represented as \(4x\) and \(7x\), based on the given ratio of 4:7.

### Step 2: Write the LCM Formula
The LCM of two numbers \(a\) and \(b\) can be expressed as:
\[ \text{LCM}(a, b) = \frac{a \times b}{\text{GCD}(a, b)} \]
In our case, we can also use the property that for two numbers in the ratio \(m:n\), the LCM can be calculated as:
\[ \text{LCM}(4x, 7x) = \frac{4x \times 7x}{\text{GCD}(4x, 7x)} \]

### Step 3: Calculate the GCD
Since 4 and 7 are co-prime (they have no common factors other than 1), the GCD of \(4x\) and \(7x\) is \(x\). Thus:
\[ \text{GCD}(4x, 7x) = x \]

### Step 4: Substitute into the LCM Formula
Now substituting into the LCM formula:
\[ \text{LCM}(4x, 7x) = \frac{(4x)(7x)}{x} = 28x \]

### Step 5: Set the LCM Equal to 224
We know from the problem that the LCM is 224:
\[ 28x = 224 \]

### Step 6: Solve for \(x\)
To find \(x\), divide both sides by 28:
\[ x = \frac{224}{28} = 8 \]

### Step 7: Find the Numbers
Now substitute \(x\) back to find the two numbers:
- First number:
\[ 4x = 4 \times 8 = 32 \]
- Second number:
\[ 7x = 7 \times 8 = 56 \]

### Final Answer
The two numbers are **32 and 56**.

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