One positive number is one-fifth of another number. The difference between the two numbers is 216, find the numbers.

Let's solve this step by step.

### Step 1: Define the Variables
Let \( x \) represent the larger number. Then the smaller number is one-fifth of the larger number, so it can be represented as \( \frac{1}{5}x \).

### Step 2: Set Up the Equation
The problem states that the difference between the two numbers is 216. Therefore, the equation is:

\[
x - \frac{1}{5}x = 216
\]

### Step 3: Solve the Equation
First, combine the terms on the left side of the equation:

\[
\frac{5}{5}x - \frac{1}{5}x = 216
\]

\[
\frac{4}{5}x = 216
\]

Now, solve for \( x \) by multiplying both sides of the equation by \( \frac{5}{4} \):

\[
x = 216 \times \frac{5}{4}
\]

\[
x = 216 \times 1.25 = 270
\]

So, the larger number is **270**.

### Step 4: Find the Smaller Number
The smaller number is one-fifth of the larger number:

\[
\frac{1}{5} \times 270 = 54
\]

So, the smaller number is **54**.

### Final Answer:
- The larger number is **270**.
- The smaller number is **54**.

Would you like more details, or do you have any other questions?

Here are 5 related questions to explore:
1. What if the difference between the two numbers was 300 instead of 216?
2. How would the problem change if the smaller number was one-third of the larger number?
3. If the larger number is reduced by 10%, what will be the new difference between the two numbers?
4. What is the ratio of the smaller number to the larger number?
5. How would the solution change if the difference between the two numbers was given as a percentage?

**Tip:** When dealing with ratios or fractional relationships, it's helpful to express the relationship in a clear equation before attempting to solve for the unknowns.

Discover how to find two numbers related by one being one-fifth of the other and a given difference of 216. Follow a step-by-step solution including defining variables, setting up equations, and solving for the unknowns. Suitable for Grades 7-9 mathematics students.

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