To find the percentage change from 40 to 48, we need to first calculate the difference between the two numbers.

Let's break down the problem step by step and address the students in the second person to explain the solution in detail.

**Step 1: Determine the initial value.**

First, let's identify the initial value. In this case, the initial value is 40.

Step 2: Determine the final value.

Now, let's find the final value, which is 48.

**Step 3: Calculate the increase in value.**

To find the increase, subtract the initial value from the final value:

Increase = Final Value - Initial Value

Increase = 48 - 40

**Step 4: Calculate the percent increase.**

To calculate the percent increase, divide the increase by the initial value and multiply by 100:

Percent Increase = (Increase / Initial Value) * 100

Percent Increase = (8 / 40) * 100

**Step 5: Simplify the fraction (if necessary).**

Let's simplify the fraction 8/40. Both the numerator and denominator are divisible by 8, so we get:

Percent Increase = (1 / 5) * 100

**Step 6: Calculate the percent.**

Now, let's calculate the percent increase using a calculator or by dividing and multiplying:

Percent Increase = (0.2) * 100

Percent Increase = 20%

So, the percent increase from 40 to 48 is 20%.

It's important to note that the same step-by-step approach can be used to solve any percent increase problem.

Simply substitute the appropriate values for the initial and final amounts, and follow the steps accordingly.

**Example 1:** Imagine you had 60 marbles, and your collection has now grown to 84 marbles. What is the percent increase in the number of marbles?

Step 1: Determine the initial value. The initial value is 60 marbles.

Step 2: Determine the final value. The final value is 84 marbles.

Step 3: Calculate the increase in value. Increase = Final Value - Initial Value Increase = 84 - 60 Increase = 24

Step 4: Calculate the percent increase. Percent Increase = (Increase / Initial Value) * 100 Percent Increase = (24 / 60) * 100 Percent Increase = 0.4 * 100 Percent Increase = 40%

Therefore, the percent increase in the number of marbles is 40%.

**Example 2:** Suppose the population of a city was 500,000 in 2020 and has grown to 600,000 in 2023. What is the percent increase in the population?

Step 1: Determine the initial value. The initial value is 500,000.

Step 2: Determine the final value. The final value is 600,000.

Step 3: Calculate the increase in value. Increase = Final Value - Initial Value Increase = 600,000 - 500,000 Increase = 100,000

Step 4: Calculate the percent increase. Percent Increase = (Increase / Initial Value) * 100 Percent Increase = (100,000 / 500,000) * 100 Percent Increase = 0.2 * 100 Percent Increase = 20%

Therefore, the percent increase in the population of the city is 20%.

These examples demonstrate how percent increase can be applied in various situations. By following the step-by-step approach, you can easily calculate the percent increase between two values.