fraction decreased by. The resulting fraction is then increased by 20%. What is the percentage decrease/increase in the value original fraction?

Let's solve the problem step by step.

### Step 1: Represent the Original Fraction
Let the original fraction be \( x \).

### Step 2: Apply the First Decrease
Assume the fraction is decreased by \( d \% \). The new fraction after the decrease can be represented as:
\[
x' = x \times \left(1 - \frac{d}{100}\right)
\]

### Step 3: Apply the 20% Increase
The resulting fraction from the previous step is then increased by 20%. The new fraction after the increase can be represented as:
\[
x'' = x' \times \left(1 + \frac{20}{100}\right) = x \times \left(1 - \frac{d}{100}\right) \times 1.2
\]

### Step 4: Compare the Final Value with the Original Value
We now want to find the percentage change from the original fraction \( x \) to the final fraction \( x'' \).

The percentage change is given by:
\[
\text{Percentage Change} = \frac{x'' - x}{x} \times 100
\]

Substituting the expression for \( x'' \):
\[
\text{Percentage Change} = \frac{x \times \left(1 - \frac{d}{100}\right) \times 1.2 - x}{x} \times 100
\]
\[
\text{Percentage Change} = \left[\left(1 - \frac{d}{100}\right) \times 1.2 - 1\right] \times 100
\]
\[
\text{Percentage Change} = \left[1.2 - \frac{1.2d}{100} - 1\right] \times 100
\]
\[
\text{Percentage Change} = \left[0.2 - \frac{1.2d}{100}\right] \times 100
\]
\[
\text{Percentage Change} = \left(20 - 1.2d\right)\%
\]

### Final Result:
The percentage change in the value of the original fraction is \( (20 - 1.2d)\% \).

This means the original fraction will increase by \( (20 - 1.2d)\% \) after the specified operations.

If \( d \) is such that \( 1.2d = 20 \), the net change is zero. If \( 1.2d > 20 \), the fraction decreases, and if \( 1.2d < 20 \), the fraction increases.

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Would you like more details or have any questions?

Here are 5 related questions:
1. What happens if the initial decrease \( d \) is 0%?
2. How does the percentage change vary if the decrease is 50%?
3. What is the final percentage change if the initial fraction is decreased by 10%?
4. How would the problem change if the second increase was 30% instead of 20%?
5. How can we express the final percentage change if the original fraction was given a specific value?

**Tip:** When dealing with percentage problems, it's helpful to represent operations as multiplicative factors to keep track of the cumulative effects.

Learn how to calculate the percentage change of a fraction after it is decreased and then increased by 20%. Understand the mathematical steps involved in finding the final value compared to the original fraction. Suitable for students in Grades 6-8.

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