Quick Revision — SSC CGL 2027 Quantitative Aptitude (Maths) Chapter 3 Notes

Fractions and Percentages to Know

Turn a percent into a fraction and most ‘x% of y’ steps become one division.

50% = $\frac{1}{2}$ , 25% = $\frac{1}{4}$ , 20% = $\frac{1}{5}$ , 12.5% = $\frac{1}{8}$ , 10% = $\frac{1}{10}$ , 5% = $\frac{1}{20}$ , 4% = $\frac{1}{25}$ .
$33\frac{1}{3}\% = \frac{1}{3}$ , $16\frac{2}{3}\% = \frac{1}{6}$ , $14\frac{2}{7}\% = \frac{1}{7}$ , $11\frac{1}{9}\% = \frac{1}{9}$ , $9\frac{1}{11}\% = \frac{1}{11}$ , $8\frac{1}{3}\% = \frac{1}{12}$ , 6.25% = $\frac{1}{16}$ .
Multiples: 37.5% = $\frac{3}{8}$ , 62.5% = $\frac{5}{8}$ , 87.5% = $\frac{7}{8}$ , $83\frac{1}{3}\% = \frac{5}{6}$ , $28\frac{4}{7}\% = \frac{2}{7}$ .
In general, $\frac{1}{n} = \frac{100}{n}\%$ .

a% of A = b% of B gives A : B = b : a (swap the percents). 30% of A = 45% of B gives 3 : 2.
A fraction instead of a percent: convert first, $\frac{3}{4}$ = 75%.
Three quantities: if 25% of A = 40% of B = $\frac{1}{5}$ of C = k, then A = 4k, B = 2.5k and C = 5k.
Different group sizes need a weighted average: 60% of 150 boys and 75% of 100 girls is 165 of 250 = 66%, not 67.5%.

If A is x% more than B, then B is $\frac{x}{100 + x} \times 100\%$ less than A.
If A is x% less than B, then B is $\frac{x}{100 - x} \times 100\%$ more than A.
The base is always the quantity after ‘than’. Take it as 100.
Pairs to know: 25% more and 20% less; 20% more and $16\frac{2}{3}\%$ less; 50% more and $33\frac{1}{3}\%$ less; 60% more and 37.5% less.
‘150% more than’ means 2.5 times, but ‘150% of’ means 1.5 times.

Percentage change = $\frac{\text{change}}{\text{old}} \times 100\%$ . Divide by the old value, never the new one.
After an x% rise, old = new × $\frac{100}{100 + x}$ . After a fall, use 100 − x.
Trap: a 15% rise to ₹2,300 means old = 2,300 ÷ 1.15 = ₹2,000, not ₹2,300 minus 15% of it.
To undo a rise of x%, fall by $\frac{x}{100 + x} \times 100\%$ . To undo a fall of x%, rise by $\frac{x}{100 - x} \times 100\%$ .

Changes of a% and b% equal one change of $\left(a + b + \frac{ab}{100}\right)\%$ . Put a minus sign on a fall.
Or multiply the factors: +10%, +20% and −25% give 1.1 × 1.2 × 0.75 = 0.99, a 1% fall. Best for three or more changes.
The same x% up and then down, in either order, is a loss of $\frac{x^2}{100}\%$ .
Final value given: divide by the combined factor to get the start.

[SSC Tip] Never just add the percentages.

Error % = $\frac{\text{wrong} - \text{correct}}{\text{correct}} \times 100\%$ (size only). The base is the correct answer.
Pick an easy number. Divided by $\frac{5}{4}$ instead of multiplying: with 20, it is 16 against 25, so 36%.
Multiplied by k instead of dividing: error = $(k^2 - 1) \times 100\%$ . Divided instead of multiplying: $\left(1 - \frac{1}{k^2}\right) \times 100\%$ .
Swapped digits (45% read as 54%): the 9% gap equals the given difference.
Decimal slip (7.2 for 0.72): the gap is 6.48 times the number.

Growth at r% a year: $P\left(1 + \frac{r}{100}\right)^n$ . Depreciation: $P\left(1 - \frac{r}{100}\right)^n$ .
Different rates each year: multiply the factors, as in 64,000 × 1.05 × 1.10 × 0.95 = 70,224.
Working back: divide the final value by the same factors.
Births, deaths and migration: net rate = births − deaths − people leaving + people arriving.
Each year’s change is on that year’s starting value, so two years of 10% growth is 21%, not 20%.
Compound interest works the same way and has its own chapter.

When A = B × C stays fixed:

B up x%: C falls $\frac{x}{100 + x} \times 100\%$ . B down x%: C rises $\frac{x}{100 - x} \times 100\%$ .
It works for price × consumption, speed × time and length × breadth.
Both changing by a% and b%: the product changes by $\left(a + b + \frac{ab}{100}\right)\%$ .
Price falls x% and ₹M buys k kg more: new price = (x% of M) ÷ k.
Numerator up a%, denominator down b%: new fraction = old × $\frac{100 + a}{100 - b}$ .

Income = Expenditure + Savings. Take income as 100 and fill in the rest.
After changes, new savings = new income − new expenditure. With income 100, expenditure 80 and savings 20, rises of 15% and 10% give (15 − 8) ÷ 20 = 35%.
Savings fixed: all the extra income goes into expenditure.
Two people with incomes a : b, expenditures c : d and equal savings: solve ax − cy = bx − dy.
‘Saves 15%, spends ₹25,500’ means 85% = ₹25,500.

Rectangle: sides up a% and b% change the area by $\left(a + b + \frac{ab}{100}\right)\%$ .
Square or circle: side or radius x% changes the area by $\left(2x + \frac{x^2}{100}\right)\%$ . Radius −30% gives area −51%.
Cube: side +10% gives volume × 1.331, up 33.1%.
Cylinder ( $\pi r^2 h$ ): the radius counts twice. Radius +20%, height −25% gives 1.44 × 0.75 = 1.08.
Area fixed: one side up x% means the other falls $\frac{x}{100 + x} \times 100\%$ .
Perimeter changes by the same percent as the side.

Two candidates, winner with w% of the valid votes: the margin is (2w − 100)% of them. 58% against 42% is a margin of 16%.
Valid votes = (100 − invalid%) of the votes polled.
Votes polled = (100 − absent%) of the voters on the list.
Work back one layer at a time: margin, valid votes, votes polled, list.
Three candidates: the third share is 100 minus the other two.

[SSC Tip] A percent of the valid votes is not a percent of the votes polled.

Pass mark = marks scored + marks short, or marks scored − marks above.
Maximum = pass mark × 100 ÷ pass percent.
Two students: their gap in marks equals their gap in percent. 30% and 15 short, 42% and 21 above: 12% = 36, so the maximum is 300.
Subjects: failed in at least one = A + B − both; in exactly one = A + B − 2 × both.
Overall percent = total scored ÷ total maximum, not an average of percents.

‘x% of the remaining’: multiply what is left. 30% rent, then 25% of the rest, leaves 0.7 × 0.75 = 52.5%.
Never add percents on different bases. A budget split on one base can be added straight.
Mixtures: track the part that does not change. Adding water keeps the milk fixed; boiling keeps the solute fixed.
New total = fixed part ÷ its new share: 32 litres of milk at 64% means 50 litres.
Mixing two solutions: add the actual litres of each part, then divide.