Hello, and welcome to today's mathematics lesson. Today, we are going to explore percent and percentage. By the end of this lesson, you will understand what percentages mean, how to convert between fractions, decimals and percentages, how to express one quantity as a percentage of another, how to find a percentage of a given quantity, and how to calculate percentage change.
Let us begin with the fundamental idea behind percentages. The word cent means hundred. Therefore, percent literally means per hundred or out of hundred. We use the symbol % to represent percent. So when we write 5%, we read it as five percent, which means five out of every hundred.
Now, how do we express an ordinary statement as a percentage? Here is the method. First, express the given statement as a fraction. Second, convert this fraction into an equivalent fraction with denominator 100.
Let me illustrate with an example. Suppose 7 out of 35 children in a class are absent. We write this as the fraction 7/35, which simplifies to 1/5. To convert to a percentage, we multiply by 100%: 1/5 × 100% equals 20%. Alternatively, you can simply calculate 7/35 × 100% which directly gives 20%. Therefore, 20% of the children are absent.
Here is a useful rule to remember. To express any fraction or decimal as a percent, multiply it by 100 and write the percent sign. For instance, 4/10 becomes 4/10 × 100% which equals 40%. Similarly, 0.3 becomes 0.3 × 100% which equals 30%.
Conversely, to change a percent to a fraction or decimal, divide by 100 and remove the percent sign. For example, 3/4% becomes 3/400 as a fraction, or 0.0075 as a decimal. And 12.5% becomes 12.5/100 which equals 1/8 or 0.125.
Let us move on to expressing one quantity as a percentage of another. The steps are straightforward. First, if necessary, convert both quantities to the same units. Second, form a fraction with the quantity to be compared as the numerator and the reference quantity as the denominator. Third, multiply this fraction by 100 and attach the percent sign.
Consider this example. Express 40 paise as a percent of 6 rupees. Since 1 rupee equals 100 paise, 6 rupees equals 600 paise. Our fraction is 40/600 which simplifies to 1/15. Multiplying by 100% gives us 100/15%, which equals 6 2/3%.
Here is the general formula. If two quantities x and y are in the same unit, then x as percent of y equals x/y × 100%, and y as percent of x equals y/x × 100%.
Next, we learn how to find a percentage of a given quantity. This is one of the most common calculations in daily life. To find 20% of 60, we calculate 20/100 × 60, which equals 12. Similarly, 40% of 7.5 equals 40/100 × 7.5, which gives us 3.
Let us work through a practical example. In a class of 50 children, 10% are taking part in dramatics. How many children are not taking part? First, 10% of 50 equals 10/100 × 50, which is 5 children. Therefore, 50 minus 5 equals 45 children are not taking part.
Alternatively, we could reason that if 10% are participating, then 90% are not. So 90% of 50 equals 90/100 × 50, which again gives us 45.
Now we come to an important application: percentage change. This helps us measure how much a quantity has increased or decreased relative to its original value.
The formula for percentage change is: the change in value divided by the original value, multiplied by 100%. Crucially, we always calculate the change percent on the original value, not the new value.
Here is an example. A bicycle costs 800 rupees. After six months, its value becomes 650 rupees. The decrease in price is 800 − 650 equals 150 rupees. The percentage decrease equals 150/800 × 100%, which simplifies to 75/4% or 18 3/4%.
We can also find the new value after a percentage increase or decrease. To increase 80 by 25%, we first find 25% of 80, which is 20. The new value is 80 plus 20, giving us 100. To decrease 60 by 10%, we find 10% of 60, which is 6. The new value is 60 minus 6, giving us 54.
Sometimes we need to work backwards. What number, when increased by 25%, becomes 150? Let the original number be x. Then x plus 25% of x equals 150. This gives us x plus x/4 equals 150, so 5x/4 equals 150, which means 5x = 600. Solving, x equals 600 divided by 5, which is 120.
One final important concept: successive percentage changes. When a number changes by percentages in sequence, each change applies to the current value, not the original. Suppose a number first decreases by 60%, then decreases by 80%. Starting with 100, after the first decrease we have 40. The second decrease is 80% of 40, which is 32. So we end with 8. The total decrease is 60 plus 32 equals 92, which is 92% of the original.
Let us now recap the key takeaways from this lesson.
First, percent means per hundred, and we use the symbol % to denote it.
Second, to convert a fraction or decimal to a percent, multiply by 100 and add the percent sign. To convert a percent to a fraction or decimal, divide by 100 and remove the percent sign.
Third, to express one quantity as a percent of another, ensure both are in the same units, form a fraction, and multiply by 100%.
Fourth, to find a percentage of a quantity, multiply the quantity by the percentage divided by 100.
Fifth, percentage change is always calculated on the original value using the formula: change divided by original value, times 100%.
Sixth, for successive percentage changes, apply each change to the current value, not the starting value.
That brings us to the end of our lesson on Percent and Percentage. I hope you now feel confident working with percentages in all their forms. Remember, percentages are simply a way of expressing parts per hundred, and with practice, these calculations will become second nature. Keep practicing, stay curious, and I look forward to seeing you in the next lesson.