Hello, and welcome to today's mathematics lesson! We are going to explore a very useful and practical topic: percent. By the end of this lesson, you will understand what percent means, how to convert between fractions, decimals, and percentages, and how to solve real-life problems involving percentages.
Let us begin with the basic idea of percent. The word percent comes from two Latin words: per, meaning "for every" or "out of," and cent, meaning "hundred." So, percent literally means "for every hundred" or "out of a hundred."
Here is a simple way to think about it. Imagine you score 83 marks out of 100 in an examination. We say you have scored 83 percent. The symbol for percent is the percent sign: %.
Mathematically, percent is the numerator of a fraction whose denominator is 100. For example, 60 out of 100 can be written as 60/100, which equals 60 percent, or 60%. When we express a fraction with denominator 100, we call the numerator the percentage.
Let us look at some examples. 3/100 equals 3%. What about 7/50? We multiply both numerator and denominator by 2 to get 14/100, which equals 14%.
Now, how do we convert any fraction or decimal into a percentage? The rule is simple: multiply the given fraction or decimal by 100, and write the percent sign.
For example, 3/4 becomes 3/4 × 100%, which equals 75%. Similarly, the decimal 0.225 becomes 0.225 × 100%, which equals 22.5%.
Let us try a practical example. Suppose 5 out of 20 eggs are bad. What percentage is this? We write 5/20, then multiply by 100 percent. This gives us 25% bad eggs.
Another example: 3 children are absent in a class of 30. We calculate 3/30 × 100%, which equals 10% absent.
Now, let us learn the reverse process: converting a percentage back into a fraction or decimal.
The rule is: remove the percent sign and divide by 100. Then simplify the fraction or write it as a decimal.
For example, 25% becomes 25/100, which simplifies to 1/4, or as a decimal, 0.25.
What about 37.5%? We write 37.5/100. To eliminate the decimal, multiply numerator and denominator by 10 to get 375/1000, which simplifies to 3/8. As a decimal, this is 0.375.
Here is a useful tip. When you divide a whole number by 100, place the decimal point two places from the right. When you divide a decimal by 100, shift the decimal point two places to the left.
Next, we learn how to express one quantity as a percentage of another.
The method is: divide the first quantity by the second, then multiply by 100 percent.
For example, what is 20 kilograms as a percentage of 200 kilograms? We calculate 20/200 × 100%, which equals 10%.
Here is an important point: the two quantities must have the same unit. For instance, to find 60 paise as a percentage of 3 rupees, first convert 3 rupees to 300 paise. Then calculate 60/300 × 100%, which equals 20%.
Also remember, percentage has no unit. It is a pure number representing a ratio.
Now, let us learn how to find a percentage of a given quantity.
Express the percentage as a fraction and multiply by the given number.
For example, 25% of 500 becomes 25/100 × 500, which equals 125. Similarly, 30% of 400 equals 30/100 × 400, which is 120.
Let us work through a complete example. In a class of 50 students, 40 percent are girls. How many girls and boys are there?
Number of girls equals 40% of 50, which is 40/100 × 50, giving us 20 girls. Therefore, the number of boys is 50 minus 20, which equals 30.
Here is another way to think about it. If girls are 40 percent, then boys must be 100 minus 40, which is 60 percent. So boys equal 60% of 50, which is 30.
Now we come to a very practical application: finding increase or decrease in percent.
When a quantity changes, we often want to know by what percentage it has increased or decreased.
The formula for percentage increase is: the increase divided by the original quantity, multiplied by 100 percent. Similarly, for percentage decrease: the decrease divided by the original quantity, multiplied by 100 percent.
Let us see an example. The price of milk rises from 24 rupees per litre to 32 rupees and 40 paise per litre. The increase is 8 rupees and 40 paise. The percentage increase is 8.40/24 × 100%, which equals 35%.
For decrease: if sugar price drops from 40 per kilogram to 32, the decrease is 8. The percentage decrease is 8/40 × 100%, which equals 20%.
We can also find the new value after a percentage change. If 70 is increased by 40%, first find 40% of 70, which is 28. The increased number is 70 + 28, which equals 98.
Similarly, if a cost of 80 is decreased by 15%, the decrease is 12. The decreased cost is 80 − 12, which equals 68.
Percentages appear everywhere in our world, especially in environmental science.
Did you know that only 2.7% of all water on Earth is fresh and fit for drinking? This means (100 − 2.7)%, which is 97.3%, is salty and unfit for drinking.
If we take 5,00,000 cubic metres of water from various parts of Earth and mix it together, how much is drinkable? We calculate 2.7% of 5,00,000, which equals 2.7/100 × 5,00,000, giving us 13,500 cubic metres of fresh water fit for drinking.
Another example: air composition. Nitrogen makes up 78 percent of air, oxygen 21 percent, and other gases just 1 percent. In 800 cubic metres of air, nitrogen would be 78% of 800, which is 624 cubic metres. Oxygen would be 21% of 800, or 168 cubic metres. Other gases would be 8 cubic metres.
Our own bodies are about 70% water by weight. In a person whose body weight is 56 kilograms, the weight of water equals 70% of 56, which is 39.2 kilograms.
Let us recap the key points from today's lesson.
First, percent means "for every hundred" or "out of a hundred" and is represented by the symbol %. To convert a fraction or decimal to percentage, multiply by 100 and add the percent sign.
Second, to convert a percentage to a fraction or decimal, divide by 100 and simplify.
Third, to express one quantity as a percentage of another, divide the first by the second and multiply by 100 percent. Remember, both quantities must have the same unit.
Fourth, to find a percentage of a given quantity, convert the percentage to a fraction and multiply.
Fifth, percentage increase or decrease equals the change divided by the original amount, multiplied by 100%. Sixth, remember that percentage has no unit , it is a pure number.
And finally, percentages are used everywhere: in examinations, shopping discounts, environmental studies, and understanding our own bodies.
That brings us to the end of our lesson on percent. I hope you now feel confident working with percentages and can see how useful they are in everyday life. Keep practicing, and you will master these skills in no time. Thank you for listening, and see you in the next lesson!