Percentage
Percentage is the language in which the ADRE Grade IV examination compares quantities of different sizes, and it is one of the most heavily used ideas in the whole paper. A percentage simply expresses a number as a part of one hundred, so fifty per cent means fifty out of every hundred, and the whole of anything is one hundred per cent. The chapter looks small but it reaches into profit and loss, simple and compound interest, ratio, average and data interpretation, which is why a candidate who is fluent here gains marks in several sections at once. The method followed in the Master Book: Maths Masti (D. Hurina) is to convert every problem into one clean equation, and the questions below are solved in that spirit. Because the paper sets these as short questions, speed matters, so the common conversions and the standard formulas should be known by heart.
Basic Terms
- Per cent: the word means "for every hundred", and it is written with the sign %.
- Base: the whole quantity on which the percentage is calculated.
- Rate: the number of parts taken for every hundred.
- Percentage value: the actual quantity that the rate represents.
- Increase per cent: the rise expressed as a percentage of the original value.
- Decrease per cent: the fall expressed as a percentage of the original value.
Every percentage question can be reduced to the same relationship: the percentage value is equal to the base multiplied by the rate and divided by one hundred. Once this one line is clear, the formulas below no longer have to be remembered separately.
Percentage to Fraction and Decimal
To change a percentage into a fraction, drop the per cent sign and write the number over one hundred, then reduce it to its lowest terms. Thus twenty-five per cent becomes one quarter, forty per cent becomes two over five, and twelve and a half per cent becomes one eighth. To change a percentage into a decimal, simply shift the decimal point two places to the left. Fifty per cent becomes zero point five, seven point five per cent becomes zero point zero seven five, and one hundred and twenty per cent becomes one point two. The direction of the shift is the only thing that must never be confused.
Fraction and Decimal to Percentage
The reverse change is just as useful. Multiply the fraction by one hundred and write the per cent sign, so three quarters becomes seventy-five per cent and one fifth becomes twenty per cent. For a decimal, shift the decimal point two places to the right. A useful check is that a value greater than one is always more than one hundred per cent, and a value less than one is always less than one hundred per cent.
Common Conversions to Remember
- One half is fifty per cent, and one quarter is twenty-five per cent.
- Three quarters is seventy-five per cent, and one fifth is twenty per cent.
- One eighth is twelve and a half per cent, and one tenth is ten per cent.
- One third is thirty-three and one third per cent, and two thirds is sixty-six and two thirds per cent.
- One sixth is sixteen and two thirds per cent, and one twelfth is eight and one third per cent.
Main Formulas
- Percentage value equals base multiplied by rate divided by one hundred.
- Rate equals percentage value multiplied by one hundred divided by base.
- Base equals percentage value multiplied by one hundred divided by rate.
- Increase per cent equals increase divided by original value, multiplied by one hundred.
- Decrease per cent equals decrease divided by original value, multiplied by one hundred.
In every one of these the base is the value over which the percentage is taken. Identifying the base correctly before doing any calculation removes most of the confusion in this chapter.
Percentage Increase and Decrease
A percentage increase is calculated always on the original value, never on the new value. If a salary of eight thousand rupees rises to ten thousand rupees, the increase is two thousand rupees, and the increase per cent is two thousand divided by eight thousand and multiplied by one hundred, which is twenty-five per cent. A fall is treated in exactly the same way. A common trap is to add and subtract percentages as if they were ordinary numbers; a ten per cent rise followed by a ten per cent fall does not return to the starting value, and the reason is that the two percentages are taken on different bases.
Successive Percentage Change
When a quantity is changed twice in succession, the two changes cannot simply be added. The short result for two successive changes of a per cent and b per cent is that the net change is the sum of the two changes plus or minus their product divided by one hundred. Using this small formula saves a full calculation and is especially helpful in questions on successive discount and on compound interest. For example, a rise of ten per cent followed by a fall of ten per cent gives a net fall of one per cent, not a return to the original value.
Percentage in Profit, Loss and Interest
Profit per cent and loss per cent are always percentages of the cost price, while discount is always a percentage of the marked price. Simple interest and compound interest are both percentages of the principal, applied over the given time. Once the candidate sees that all of these are the same percentage relation applied to a different base, the arithmetic becomes routine and the chapter stops looking like several separate topics.
Shortcut Methods for Rapid Calculation
Because the examination is timed, a few shortcuts are worth memorising. To find ten per cent of a number, move the decimal point one place to the left, so ten per cent of four hundred and fifty is forty-five. Five per cent is half of ten per cent, and one per cent is ten per cent divided by ten. To find fifteen per cent, add ten per cent and five per cent; to find twenty-five per cent, take a quarter; and to find fifty per cent, take a half. A percentage such as twelve and a half per cent is an eighth, so dividing by eight gives the value at once. These steps let most options be checked mentally before the full sum is written.
Percentage in Data Interpretation
Charts and tables in the reasoning and quantitative sections almost always report figures as percentages, and the questions ask the candidate to compare two percentages, to find the percentage share of one item in a total, or to find the percentage rise from one year to the next. The same three formulas are used throughout, so the skill learned in this chapter transfers directly. When a table gives actual values and asks for a percentage, the base is the total in the same row or column, and reading the table carefully is often the only real difficulty.
How This Chapter Appears in the Examination
Percentage questions in ADRE Grade IV, APSC and Assam Police papers usually ask the candidate to find a percentage of a given number, to express one quantity as a percentage of another, to find the original value when the changed value and the rate are known, or to work out a percentage increase or decrease. Data interpretation sets lean heavily on percentage, and many profit, loss and interest questions are percentage questions in disguise. The section is dependable for marks because the method is fixed and the numbers are usually simple.
Common Mistakes to Avoid
- Taking the base as the new value instead of the original value.
- Adding or subtracting two percentages directly in a successive change.
- Forgetting to reduce a fraction to its lowest terms before writing the percentage.
- Shifting the decimal point in the wrong direction while converting.
- Reading a value above one hundred per cent as impossible, when it only means an increase.
How to Prepare This Topic
- Learn the meaning of base and rate, and write every problem in the form of one equation.
- Memorise the common fraction and percentage conversions given above.
- Practise finding the percentage value, the rate and the base in turn.
- Work on increase, decrease and successive change until the shortcut is automatic.
- Finish with a mixed set from profit, loss and interest to see percentage used in context.
Frequently Asked Questions
How do I express one number as a percentage of another?
Divide the first number by the second and multiply the result by one hundred. For example, ten as a percentage of fifty is ten divided by fifty, multiplied by one hundred, which is twenty per cent. Always check that both numbers are in the same unit before dividing.
Why does a ten per cent rise followed by a ten per cent fall not return to the original value?
Because the two percentages are taken on different bases. The rise is calculated on the original value, but the fall is calculated on the increased value. The net result is a fall of one per cent, so the quantity ends below where it started.
How do I find the original value when the change and the new value are known?
If the new value and the increase per cent are known, the original value equals the new value multiplied by one hundred and divided by one hundred plus the increase per cent. For a decrease, subtract the decrease per cent from one hundred in the denominator.
What is the quickest way to remember the common conversions?
Keep a short list of the standard fractions and their percentages: one half is fifty, one quarter is twenty-five, one fifth is twenty, and one eighth is twelve and a half. Reading the list once a day fixes it in memory, and it saves valuable time in the examination.
How much weight does percentage carry in ADRE Grade IV?
Percentage is often asked directly, and many questions on profit and loss, interest, ratio and data interpretation depend on it indirectly. It is a dependable scoring area, so a thorough preparation of this chapter helps across the whole paper.