Sometimes you are given the final value after a percentage increase or decrease and need to find the original quantity. These problems are best solved using multipliers and inverse operations.

First, identify the multiplier that was applied.

• For a percentage increase of \(p\%\):
  $$
  \text{multiplier} = 1 + \frac{p}{100}
  $$
• For a percentage decrease of \(p\%\):
  $$
  \text{multiplier} = 1 - \frac{p}{100}
  $$

To find the original value, divide the final value by the multiplier.

$$
\text{original value} = \frac{\text{final value}}{\text{multiplier}}
$$

For example, a price increases by 20% to give a final price of \(72\).

$$
\text{multiplier} = 1.2
$$

$$
\text{original value} = \frac{72}{1.2} = 60
$$

For a decrease example, a value is reduced by 15% and the final value is \(85\).

$$
\text{multiplier} = 0.85
$$

$$
\text{original value} = \frac{85}{0.85} = 100
$$

If the change happens more than once, divide by the multiplier raised to the number of changes.

$$
\text{original value} = \frac{\text{final value}}{\text{multiplier}^{\text{number of changes}}}
$$

Always check your answer. After finding the original value, apply the percentage change again to confirm that it produces the given final value.

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