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Number - Reverse Percentages

Grade 8IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Reverse percentages are used to find the original amount (the 100%100\%) after a percentage increase or decrease has been applied.

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The fundamental principle is that the new value is a specific percentage of the original value. For an increase of r%r\%, the new value is (100+r)%(100 + r)\% of the original. For a decrease of r%r\%, the new value is (100−r)%(100 - r)\% of the original.

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A common error is calculating the percentage of the new value and adding or subtracting it. This is incorrect because the percentage was originally applied to the old value.

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The most efficient way to solve these is using the multiplier method. Divide the new amount by the decimal multiplier to find the original amount.

📐Formulae

Original Value=New ValueMultiplier\text{Original Value} = \frac{\text{New Value}}{\text{Multiplier}}

Multiplier (Increase)=1+r100\text{Multiplier (Increase)} = 1 + \frac{r}{100}

Multiplier (Decrease)=1−r100\text{Multiplier (Decrease)} = 1 - \frac{r}{100}

Original Value=New ValuePercentage Remaining×100\text{Original Value} = \frac{\text{New Value}}{\text{Percentage Remaining}} \times 100

💡Examples

Problem 1:

A pair of shoes is sold for Rs 2550 after a 15%15\% discount. Calculate the original price of the shoes.

Solution:

  1. Identify the percentage remaining: 100−1585\begin{array}{r} 100 \\ - 15 \\ \hline 85 \end{array} So, Rs 2550 represents 85%85\% of the original price.
  2. Find the decimal multiplier: Multiplier=85100=0.85\text{Multiplier} = \frac{85}{100} = 0.85
  3. Calculate the original price: Original Price=25500.85=3000\text{Original Price} = \frac{2550}{0.85} = 3000

Explanation:

Since there was a 15%15\% discount, the customer paid 85%85\% of the original price. Dividing the sale price by 0.850.85 reverses the percentage change to find the 100%100\% value.

Problem 2:

The population of a town increased by 8%8\% to 54,00054,000. What was the population before the increase?

Solution:

  1. Express the new population as a percentage of the original: New Percentage=100%+8%=108%\text{New Percentage} = 100\% + 8\% = 108\%
  2. Find the multiplier: Multiplier=1.08\text{Multiplier} = 1.08
  3. Solve for original value: Original Population=540001.08=50000\text{Original Population} = \frac{54000}{1.08} = 50000

Explanation:

The current population represents 108%108\% of the original. By dividing the current population by 1.081.08, we determine the base value before the 8%8\% growth occurred.

Problem 3:

A jeweler sells a ring for Rs 1260, which includes a profit of 20%20\%. Find the cost price for the jeweler.

Solution:

  1. The selling price is 120%120\% of the cost price.
  2. Multiplier =1.20= 1.20.
  3. Cost Price: Cost Price=12601.20\text{Cost Price} = \frac{1260}{1.20} Cost Price=1050\text{Cost Price} = 1050

Explanation:

Profit is added to the cost price. If the profit is 20%20\%, the selling price is 120%120\% of the cost price (100%+20%100\% + 20\%). We divide the selling price by the multiplier 1.21.2 to find the original cost.