Why percentages first
Roughly a third of a typical aptitude paper is percentages wearing a costume. Profit and loss is percentage of cost price. Discount is percentage of marked price. Data interpretation is percentage of a total. Get fast here and four other topics get faster.
Aptitude rounds usually give you about 45 to 60 seconds per question. That budget is the reason for everything below. The method has to be short, not just correct.
The idea
Per cent means per hundred. x% of N is N x / 100.
Two properties do most of the work.
Commutativity. x% of y equals y% of x. So 16% of 25 is hard, but 25% of 16 is 4 in your head. Same answer.
Percentage is always of something. Write down what the base is before you compute. Half the wrong answers in this topic are right arithmetic on the wrong base.
The fraction table — memorise this
Converting a percentage to a fraction turns multiplication into division by a small number.
| Fraction | Percentage | Fraction | Percentage |
|---|---|---|---|
| 1/2 | 50% | 1/9 | 11.11% |
| 1/3 | 33.33% | 1/10 | 10% |
| 1/4 | 25% | 1/11 | 9.09% |
| 1/5 | 20% | 1/12 | 8.33% |
| 1/6 | 16.66% | 1/16 | 6.25% |
| 1/7 | 14.28% | 1/20 | 5% |
| 1/8 | 12.5% | 1/25 | 4% |
Combinations follow at once: 37.5% is 3/8, 62.5% is 5/8, 66.66% is 2/3, 83.33% is 5/6, 87.5% is 7/8.
Increase and decrease as multipliers
An increase of 20% means multiply by 1.20. A decrease of 20% means multiply by 0.80. Chain them by multiplying, never by adding.
Successive change formula. For two changes of a% and b%, the net change is
a + b + (a*b)/100
Use signs: a decrease is negative.
A 20% rise followed by a 20% fall is not zero. It is 20 - 20 - 400/100, which is -4%, a 4% loss. This one question appears in some form in almost every paper. The reason is that the fall is taken on the larger amount.
The reverse case. If a price increases by 25%, by what percentage must it be reduced to come back to the original? Not 25%. If the increase is a%, the required decrease is 100a/(100+a). Here 100*25/125 = 20%.
Worked problem 1
Find 37.5% of 896.
37.5% is 3/8.
896 / 8 = 112
112 * 3 = 336
The long route is 896 * 37.5 / 100. The fraction route takes about four seconds.
Worked problem 2
Ravi's salary is increased by 20% and then decreased by 20%. What is the net change?
By formula: a = +20, b = -20.
Net = 20 + (-20) + (20 * -20)/100 = 0 - 400/100 = -4%, a 4% decrease.
Check with a number. Take the salary as 25,000.
After the rise: 25000 * 1.20 = 30,000
After the fall: 30000 * 0.80 = 24,000
Loss = 25000 - 24000 = 1,000
1000 / 25000 = 0.04 = 4% decrease. The formula agrees.
Whenever a percentage problem has no numbers in it, put in 100 or a number that divides cleanly. It is not cheating and it is often faster than the formula.
Worked problem 3
In an exam a student scores 35% and fails by 40 marks. Another student scores 50% and gets 20 marks more than the pass mark. Find the total marks and the pass mark.
Let the total be T.
The pass mark, written two ways:
From the first student: pass = 0.35T + 40
From the second student: pass = 0.50T - 20
Set them equal:
0.35T + 40 = 0.50T - 20
40 + 20 = 0.50T - 0.35T
60 = 0.15T
T = 60 / 0.15 = 400 marks
Pass mark = 0.35 * 400 + 40 = 140 + 40 = 180 marks
Check with the second student: 0.50 * 400 = 200, and 200 - 180 = 20. Correct.
What to practise
- The fraction table until it is automatic. Test yourself on 12.5% of 648. (81)
- The successive change formula with mixed signs.
- Reading the base correctly: "20% more than" versus "20% of".
Next: ratio and proportion, which is percentages with the base removed.