Percentage for the Assam Police Constable Exam
Percentage is the single most useful idea in the whole Elementary Mathematics section of the SLPRB Assam Police Constable (AB & UB) paper, because it is the language in which the examination talks about change. Whenever the paper says that a price rose, a population fell, a student scored, a discount was offered or a profit was earned, the numbers are usually percentages. A candidate who can move quickly between a percentage, a fraction and a decimal finds that the same question can be finished in a single line instead of five. In the Assam Police test, percentage is rarely asked alone; it is woven into profit and loss, discount, simple and compound interest, data interpretation and even into questions on the number of candidates who passed or failed. This chapter therefore pays twice: once for the direct questions on percentage and once for the confidence it gives in every other chapter. The method followed here is the one in the Master Book: Maths Masti (D. Hurina): fix the base first, then convert the percentage into a fraction before multiplying, so that the arithmetic stays light.
Why Percentage Is Everywhere in the Assam Police Paper
Percentage standardises comparison. Because every percentage is measured out of one hundred, two quantities of different sizes can be compared at a glance: a village of four hundred people and a town of fifty thousand people can both be described by the same kind of figure. This is exactly why the examiners like it. A question may ask for the pass percentage of a school, the percentage of a budget spent on a scheme, the percentage of an increase in a fee, or the percentage of marks needed to reach a cut-off. The method is always the same, which means the topic is highly scoring once the standard conversions are on the fingertips.
Terms You Should Know
- Per cent: per hundred, written as a number followed by the percent sign.
- Base: the quantity with which the comparison is made, the one that follows the word "of".
- Increase and decrease: the change measured as a fraction of the original value.
- Successive change: two or more percentage changes applied one after another.
- Net change: the single percentage that has the same effect as several changes taken together.
- Marked price: the printed price before any discount is allowed.
- Selling price: the price at which an article is actually sold.
Two words decide most questions: the word "of" fixes the base, and the word "than" fixes the direction of the comparison. Reading the question once more with these two words in mind prevents the commonest of all percentage mistakes.
Converting Between Percentage, Fraction and Decimal
To change a percentage into a fraction, divide by one hundred and reduce. To change a fraction into a percentage, multiply by one hundred. To change a percentage into a decimal, divide by one hundred, which simply moves the point two places to the left. These three movements are the entire toolkit of the chapter, and a candidate who can perform them without hesitation will find that most questions collapse into one multiplication.
Standard Values Worth Memorising
Some conversions appear so often that they should be automatic: one half is fifty per cent, one quarter is twenty-five per cent, three quarters is seventy-five per cent, one fifth is twenty per cent, one eighth is twelve and a half per cent, one tenth is ten per cent, and one sixth is sixteen and two thirds per cent. The reciprocals of a rising price or a falling price are also useful. For example, a rise of twenty-five per cent must be undone by a fall of twenty per cent, because the reduction is measured on the new, larger base. Keeping this small table in memory removes most of the arithmetic from the examination.
The Three Basic Types of Percentage Questions
The first type asks for a percentage of a given number, solved by multiplying the number by the fraction. The second type asks what percentage one number is of another, solved by dividing the first by the second and multiplying by one hundred. The third type gives a percentage and a part and asks for the whole, solved by dividing the part by the fraction. Almost every percentage question is one of these three, and deciding which one is being asked is the first and most important step.
Successive Percentage Change
When two percentage changes are applied one after the other, they do not simply add. If a price rises by ten per cent and then rises again by ten per cent, the total change is not twenty per cent but twenty-one per cent, because the second rise is calculated on the already increased amount. The net change of two successive changes is found by multiplying the two multiplying factors. A rise of ten followed by a fall of ten gives a net loss of one per cent, which surprises many candidates but follows directly from the multiplication.
Increase, Decrease and the Net Effect
An increase of one quantity followed by an equal decrease of the same percentage always produces a small net loss, because the decrease is measured on the larger base. To restore the original value after a rise of a certain percentage, the necessary fall is smaller than the rise; a rise of fifty per cent needs a fall of thirty-three and one third per cent to return to the start. In income and expenditure questions, the same logic is used to find the effect of a rise in price on the amount that can be bought.
Percentage in Profit, Loss, Discount and Interest
Profit and loss percentages are always measured on the cost price, while discount is always measured on the marked price. Mixing these two bases is the classic trap. Simple interest is a fixed percentage of the principal for each year, and compound interest is a percentage applied on the amount that has already grown. Because all of these are percentages, a strong grip on this chapter makes the profit and loss and interest chapters far easier.
Percentage in Data Interpretation and Diagrams
Tables, bar diagrams and pie charts are read with percentages. In a pie chart a full circle of three hundred and sixty degrees represents one hundred per cent, so each one per cent is three point six degrees and each one degree is one hundred divided by three hundred and sixty per cent. Questions often ask for the difference between two shares or for the percentage of a total represented by one bar, and both are answered directly with the conversions above.
Shortcut Techniques for the Exam Hall
Convert the percentage into the smallest possible fraction before multiplying, so that the cancellation does the work. Use the multiplying factors for successive changes rather than the addition of percentages. When a rise or fall is given on the new value and the original is asked, work backwards with the reciprocal. For the pass or fail type questions, remember that the number of failures is the total minus the number of passes, and take care to apply each percentage to the base the question intends.
Common Traps and Errors
- Adding successive percentages instead of multiplying their factors.
- Measuring profit or loss on the selling price instead of the cost price.
- Measuring discount on the cost price instead of the marked price.
- Using the older base after an increase when the question expects the newer base.
- Forgetting the reciprocal needed to undo a rise in price or income.
- Writing a decimal as a percentage without shifting the point by two places.
A Study Plan for This Chapter
- Master the three conversions between percentage, fraction and decimal.
- Memorise the standard fractions and their percentage equivalents.
- Practise the three basic types until the choice is instant.
- Work through successive change, and income and expenditure questions.
- Apply percentage to profit, loss, discount and interest, and finish with a mixed set.
Frequently Asked Questions
Why is a ten per cent rise followed by a ten per cent fall not zero?
Because the fall is measured on the higher amount, not on the original. The two factors are one point one and zero point nine, and their product is zero point nine nine, which means a net loss of one per cent.
On which price is the profit percentage calculated?
The profit percentage is always calculated on the cost price, the amount the seller paid. The discount percentage, on the other hand, is calculated on the marked price.
How do I increase a number by a given percentage in one step?
Multiply the number by one plus the percentage as a decimal. To increase by twenty per cent, multiply by one point two; to decrease by twenty per cent, multiply by zero point eight.
What fraction is twelve and a half per cent?
Twelve and a half per cent is one eighth. This is one of the standard conversions and appears often in questions on discount and interest.
How is percentage connected with a pie chart?
In a pie chart the full circle is three hundred and sixty degrees, which represents one hundred per cent. So one per cent is three point six degrees, and a sector of a given angle can be converted into a percentage by dividing the angle by three point six.