Indicies - Explanation, Examples and Laws of Indicies

 

                     Percentages.

Percentages are one of the most practical topics in mathematics - used daily in discounts, grades, interest rates, statistics, and even health data. But beyond their everyday use, percentages teach us how to compare quantities fairly and meaningfully.

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What is a Percentage?

The word “percent” comes from the Latin per centum, meaning “by the hundred.”
A percentage is simply a fraction with a denominator of 100.

So,

$$25\mathrm{\%}=\frac{25}{100}=0.25$$

This means 25 out of every 100 parts.

Percentages are useful because they let us compare things on the same scale, even if their original values are very different.

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Converting Between Fractions, Decimals, and Percentages

To work easily with percentages, you should know how to convert between fractions, decimals, and percentages.

1. From Fraction to Percentage:
   Multiply the fraction by 100%.
   For example:

   $$\frac{3}{4}\times 100\mathrm{\%}=75\mathrm{\%}$$

   So, $\frac{3}{4}$ is equal to 75%.

2. From Decimal to Percentage:
   Multiply the decimal by 100%.
   For example:

   $$0.2 \times 100\% = 20\%$$

   So, 0.2 is equal to 20%.

3. From Percentage to Decimal:
   Divide the percentage by 100.
   For example:

   $$85\% = 0.85$$

   So, 85% is the same as 0.85.

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Finding the Percentage of a Quantity

To find x% of a number, multiply the number by $\frac{x}{100}$.

Example 1:
Find 15% of 80

$$15\% \text{ of } 80 = \frac{15}{100} \times 80 = 12$$

Example 2:
Find 7.5% of 200

$$7.5\% \text{ of } 200 = \frac{7.5}{100} \times 200 = 15$$

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Finding What Percentage One Number Is of Another

To find what percentage A is of B:

$$\text{Percentage} = \frac{A}{B} \times 100\%$$

Example 3:
What percentage of 40 is 10?

$$\frac{10}{40} \times 100\% = 25\%$$

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Increase and Decrease by a Percentage

Percentage Increase

When a value rises by a certain percent:

$$\text{New value} = \text{Original value} \times \left(1 + \frac{\text{percentage}}{100}\right)$$

Percentage Decrease

When a value drops by a certain percent:

$$\text{New value} = \text{Original value} \times \left(1 - \frac{\text{percentage}}{100}\right)$$

Example 4:
A phone costs GHS 800. Its price increases by 10%. Find the new price.

$$800 \times (1 + \frac{10}{100}) = 800 \times 1.1 = 880$$

Example 5:
A shirt originally costs GHS 150 but is sold at a 20% discount.

$$150 \times (1 - \frac{20}{100}) = 150 \times 0.8 = 120$$

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Finding Percentage Change

$$\text{Percentage Change} = \frac{\text{Change}}{\text{Original Value}} \times 100\%$$

Example 6:
A student’s score increased from 60 to 75.

$$\text{Change} = 75 - 60 = 15$$ $$\text{Percentage Change} = \frac{15}{60} \times 100\% = 25\%$$

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Reverse Percentage (Finding the Original Amount)

If the final value after increase or decrease is known, we can find the original amount.

Example 7:
After a 10% discount, a laptop costs GHS 900. Find the original price.

Let the original price = $x$.

$$900 = x \times (1 - \frac{10}{100}) = 0.9x$$ $$x = \frac{900}{0.9} = 1000$$

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Percentage Profit and Loss

$$\text{Percentage Profit or Loss} = \frac{\text{Profit or Loss}}{\text{Cost Price}} \times 100\%$$

Example 8:
A trader buys a bag for GHS 500 and sells it for GHS 600.

$$\text{Profit} = 600 - 500 = 100$$ $$\text{Percentage Profit} = \frac{100}{500} \times 100\% = 20\%$$

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Real-Life Applications of Percentages

• In finance (interest rates and profit calculations)

• In academics (test scores and averages)

• In health (body fat percentage, infection rates)

• In sales (discounts, taxes, and price increases)

Understanding percentages helps you think proportionally, a key skill in both mathematics and daily decision-making.

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                Assignment 

Try these to strengthen your understanding. Show all workings!

1. Find 18% of 250.

2. What percentage of 300 is 45?

3. A price increases from 80 to 100. Find the percentage increase.

4. A shirt costing GHS 200 is sold at a 15% discount. Find the selling price.

5. A trader makes a loss when he sells a radio for GHS 850 that cost him GHS 1000. Find the percentage loss.

6. After a 25% increase, a TV costs GHS 1250. What was its original price?

7. A student scored 72 marks out of 90. What percentage did he score?

8. The population of a town increased from 8,000 to 10,400. Find the percentage increase.

9. Find the number which is 30% more than 200.

10. Find 60% of 3/5 of 250.