Hello, and welcome to today's mathematics lesson. We are going to explore Chapter Seven: Percent and Percentage. By the end of this session, you will understand what percentages really mean, how to convert between fractions, decimals and percentages, and how to solve practical problems involving increase, decrease, and successive percentage changes.
Let us begin with the fundamental idea. The word percent literally means for every hundred. We abbreviate it as p.c., and we represent it with this symbol: %. So when we say fifty percent, we mean fifty parts out of every hundred parts.
Now, here is the precise definition you need to remember.
A fraction whose denominator is one hundred is called a percentage, and the numerator is called the rate percent. For example, 8/100 equals eight percent, which means eight out of one hundred. In everyday use, the words percent and percentage mean essentially the same thing.
Let us learn how to convert. To express any number as a percent, you multiply by one hundred and attach the percentage sign. Take three-fifths: 3/5 × 100% gives us sixty percent. Or consider zero point four five: multiply by one hundred percent and you get forty-five percent.
To go the other way, from percent to fraction, you divide by one hundred and remove the percent sign. Forty-five percent becomes 45/100, which simplifies to nine-twentieths as a common fraction, or zero point four five as a decimal.
Here are two essential formulas you must master.
First: x as the percent of y equals x/y × 100%. Second: x percent of y equals x/100 × y.
Notice the difference. Five as the percent of twenty means 5/20 × 100%, which equals twenty-five percent. But five percent of twenty means 5/100 × 20, which equals just one. The wording changes everything.
Now let us talk about increase and decrease. The formula for increase percent is: (Increase)/(Original) × 100%. Similarly, decrease percent equals (Decrease)/(Original) × 100%. Always remember: the original quantity is your base, your starting point.
Let me walk you through some worked examples. First, what percent of one hundred forty-four is thirty-six? Using our formula: 36/144 × 100%. Thirty-six divided by one hundred forty-four equals one-fourth, and one-fourth of one hundred equals twenty-five. So the answer is twenty-five percent.
Here is another type: eighty is thirty-two percent of a certain number, find that number. We write: thirty-two percent of the number equals eighty. So 32/100 × the number equals eighty. To find the number, we rearrange: eighty multiplied by one hundred, divided by thirty-two. That gives us two hundred fifty.
Now consider this practical situation. A man spends sixty-five percent of his salary and saves five hundred twenty-five rupees per month. What is his monthly salary?
If he spends sixty-five percent, he saves the remaining thirty-five percent. So thirty-five percent of his salary equals five hundred twenty-five rupees. Therefore, his salary equals five hundred twenty-five multiplied by one hundred, divided by thirty-five. That comes to one thousand five hundred rupees.
Let us look at percentage error. A number four point zero is wrongly read as four point four eight. The error is four point four eight minus four point zero, which equals zero point four eight. Percentage error equals Error/Original number × 100%, so 0.48/4.0 × 100% equals twelve percent.
In a consignment of five hundred articles, seventy are broken. The remaining articles are five hundred minus seventy, which is four hundred thirty. The percentage of remaining articles is 430/500 × 100%, which equals eighty-six percent.
Election problems are common in percentage questions. One candidate secured forty-three percent of total votes and lost by four thousand nine hundred votes. The winner therefore secured fifty-seven percent. The difference between fifty-seven percent and forty-three percent is fourteen percent, and this fourteen percent equals four thousand nine hundred votes. So total votes polled equals four thousand nine hundred multiplied by one hundred, divided by fourteen, giving thirty-five thousand votes.
Now we come to depreciation and successive percentage changes. A machine costs ten thousand rupees and depreciates by ten percent each year.
After one year: ten percent of ten thousand is one thousand, so the value becomes nine thousand rupees. After two years: ten percent of nine thousand is nine hundred, so the value becomes eight thousand one hundred rupees.
There is a direct formula for this. Value after one year equals present value multiplied by (1 - x/100). Value after two years equals present value multiplied by (1 - x/100)². With ten thousand rupees and ten percent depreciation: ten thousand multiplied by ninety over one hundred, then again by ninety over one hundred, gives eight thousand one hundred rupees.
Successive increases and decreases need careful handling. Take five thousand, decrease by ten percent, then increase by twenty percent. The result equals five thousand multiplied by (1 - 10/100) × (1 + 20/100). This becomes five thousand multiplied by ninety over one hundred multiplied by one hundred twenty over one hundred, which equals five thousand four hundred.
Notice something important: a twenty percent increase followed by a twenty percent decrease does not bring you back to the start. Start with one hundred, increase by twenty percent to get one hundred twenty, then decrease by twenty percent of one hundred twenty, which is twenty-four, giving ninety-six. You end with a net decrease of four percent.
Here is a more complex situation. Five percent of a town's population was killed in a bombardment, and seven percent of the remaining died in panic. The final population is forty-four thousand one hundred seventy-five. What was the original population?
Assume original population is one hundred. After bombardment: ninety-five remain. Seven percent of ninety-five die in panic: 7/100 × 95 equals six point six five.
Using unitary method: if eighty-eight point three five corresponds to one hundred original, then forty-four thousand one hundred seventy-five corresponds to 100/88.35 × 44175, which equals fifty thousand.
Let us examine problems with overlapping percentages. In an examination, thirty percent failed in English, thirty-five percent failed in Mathematics, and twenty-seven percent failed in both.
Failed only in English: thirty percent minus twenty-seven percent equals three percent. Failed only in Mathematics: thirty-five percent minus twenty-seven percent equals eight percent. Failed in both: twenty-seven percent. Total failed: three plus eight plus twenty-seven equals thirty-eight percent. Therefore, total passed equals one hundred minus thirty-eight, which is sixty-two percent.
If two hundred forty-eight candidates passed, and this represents sixty-two percent, then total candidates equals two hundred forty-eight multiplied by one hundred, divided by sixty-two, giving four hundred.
Comparative percentages require careful thought. If A's income is ten percent more than B's, by what percent is B's income less than A's?
Let B's income be one hundred rupees. Then A's income is one hundred ten rupees. The difference is ten rupees. But this ten rupees is being compared to A's one hundred ten, not to B's one hundred. So the percentage is 10/110 × 100, which equals 100/11 percent, or nine and one-eleventh percent.
Finally, reversing percentage changes. If wheat price increases by twenty percent today, by what percent must it decrease tomorrow to return to original?
Original price: one hundred rupees. New price: one hundred twenty rupees. To return to one hundred, we need to decrease by twenty rupees. But this twenty rupees is compared to one hundred twenty, not one hundred. So percentage decrease equals 20/120 × 100, which equals sixteen and two-thirds percent.
Here is a direct method for finding original numbers. If a number decreased by eighteen percent becomes four hundred ten, find the original number. We can write: (100 - 18)/100 × original equals four hundred ten. So eighty-two over one hundred multiplied by original equals four hundred ten. Therefore, original number equals four hundred ten multiplied by one hundred, divided by eighty-two, which equals five hundred.
Let us recap the key takeaways from today's lesson.
First: Percent means for every hundred, and a percentage is a fraction with denominator one hundred.
Second: To convert to percent, multiply by one hundred; to convert from percent, divide by one hundred.
Third: x as percent of y uses division; x percent of y uses multiplication—distinguish these carefully.
Fourth: For increase or decrease percent, always divide by the original quantity.
Fifth: Successive percentage changes multiply as factors: (1 ± x/100) for each change.
Sixth: When comparing percentages across different bases, the percentage change depends on what you choose as your reference point.
That brings us to the end of our lesson on Percent and Percentage. Practice converting between forms, identify your base quantity carefully in each problem, and remember that percentages are powerful tools for comparing quantities of different sizes. Keep practicing, stay curious, and I look forward to seeing you in the next lesson.