SSC CGL Maths · Surds, Indices and Simplification

Fractions and Decimals

Fractions and decimals follow the same BODMAS order as whole numbers; what changes is the arithmetic: mixed numbers, reciprocals, decimal places, and knowing which form makes a step easy.

All Surds, Indices and Simplification notes

Why this matters

Forty-four questions in the bank, 38 EASY and 6 MODERATE, and 26 of them come from the 2025 shifts. The 2025 papers moved from long bracket chains to short decimal sums and small word problems, so speed with decimal places and with the common fraction and decimal pairs now decides these marks.

Concept 1 of 4: Fractions and mixed numbers in BODMAS

A fraction expression is worked in the usual BODMAS order. Make every mixed number an improper fraction, turn every division into multiplication by the reciprocal, and cancel before you multiply so the numbers stay small.

Definition

Mixed to improper: $3\frac{2}{5} = \frac{3 \times 5 + 2}{5} = \frac{17}{5}$ .
Divide by a fraction: multiply by its reciprocal, $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$ .
Add or subtract: bring both fractions to the LCM of the denominators.
Cancel first: $\frac{14}{15} \times \frac{25}{28} = \frac{1}{3} \times \frac{5}{2} = \frac{5}{6}$ .
Reduce a fraction when you see it: $\frac{38}{57} = \frac{2}{3}$ , since both are multiples of 19.
Fast route: read left to right, $x \div y \times y = x$ . Dividing and then multiplying by the same number cancels, so do not work it out.

Fraction rules

\frac{a}{b} \div \frac{c}{d} = \frac{ad}{bc} \qquad \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} \qquad p\frac{q}{r} = \frac{pr + q}{r}

Worked example

Simplify

2\frac{1}{4} \div \frac{3}{8} - \frac{5}{6} \times 1\frac{1}{5}

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 16 Sep 2025, Shift 1 · Q54Easy

Example 1 · Surds, Indices and Simplification · Fractions and Decimals

Simplify:

\left[\left(\frac{2}{3}\right) + \left(\frac{4}{5} \div \frac{2}{7}\right)\right] \div \left[\left(\frac{3}{2}\right) - \left(\frac{5}{6} \times \frac{3}{4}\right)\right]

Multiplying mixed numbers part by part

2\frac{1}{2} \times 3\frac{1}{3}

is not

6\frac{1}{6}

. Convert first:

\frac{5}{2} \times \frac{10}{3} = \frac{25}{3} = 8\frac{1}{3}

Turning the wrong fraction upside down

Only the divisor is flipped:

\frac{2}{5} \div \frac{4}{15} = \frac{2}{5} \times \frac{15}{4} = \frac{3}{2}

. Flipping the first fraction instead gives

\frac{5}{2} \times \frac{4}{15} = \frac{2}{3}

, the reciprocal of the answer.

Drill 17 more on fractions and mixed numbers in bodmas

Concept 2 of 4: Working with decimals

A decimal is a fraction with 10, 100 or 1000 below it. To multiply, count decimal places; to divide, slide the point in both numbers. And many decimals are fractions you already know, which is often the quickest way through.

Definition

Multiply: multiply as whole numbers, then give the answer as many decimal places as the two numbers have between them: $0.6 \times 0.07 = 0.042$ .
Divide: move the point the same number of places in both numbers until the divisor is whole: $1.44 \div 0.12 = 144 \div 12 = 12$ .
Add or subtract: line up the decimal points; fill gaps with zeros.
Switch to a fraction when it helps: $0.125 = \frac{1}{8}$ , $0.375 = \frac{3}{8}$ , $0.625 = \frac{5}{8}$ , $0.75 = \frac{3}{4}$ , $0.2 = \frac{1}{5}$ . Dividing by 0.25 is multiplying by 4; dividing by 0.5 is doubling.
Recurring decimals: one repeating digit goes over 9, two repeating digits over 99: $0.\overline{7} = \frac{7}{9}$ , $0.\overline{45} = \frac{45}{99} = \frac{5}{11}$ .
Estimate first: round each number and fix the sign and rough size of the answer before working. When the options differ in sign or by a lot, the estimate alone picks one. When they agree in the first figures, it is not enough.

Recurring decimals and handy divisions

0.\overline{a} = \frac{a}{9} \qquad 0.\overline{ab} = \frac{ab}{99} \qquad x \div 0.25 = 4x \qquad x \div 0.5 = 2x

athe single repeating digit
abthe two-digit number made by the repeating pair

Worked example

Simplify

3.6 \times 0.5 + 2.7 \div 0.9 - 1.25

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 17 Sep 2025, Shift 2 · Q51Easy

Example 2 · Surds, Indices and Simplification · Fractions and Decimals

Simplify:

(2.4 + 3.125) \times 0.25 - (0.75 \div 0.3)

Too few decimal places in a product

0.3 \times 0.2 = 0.06

, not 0.6. Each number has one decimal place, so the product needs two.

Expecting division to make a number smaller

Dividing by a number below 1 makes the result bigger:

4.5 \div 0.5 = 9

. An option smaller than 4.5 can be ruled out at once.

Drill 19 more on working with decimals

Concept 3 of 4: Comparing and ordering fractions

Two fractions can only be compared on one scale: a common denominator, decimals, or a cross-multiplication. Often a simpler test settles it, because only one fraction sits on the far side of one half.

Definition

Cross-multiply: for positive denominators, $\frac{a}{b} > \frac{c}{d}$ exactly when $ad > bc$ .
Benchmark first: compare each fraction with $\frac{1}{2}$ (is the top more than half the bottom?) or with 1.
Gap from 1: for fractions just below 1, the one with the smaller gap $1 - \frac{a}{b}$ is the larger.
Decimals: three decimal places is enough to separate the options SSC prints.

Cross-multiplication test

\frac{a}{b} > \frac{c}{d} \text{ exactly when } ad > bc \quad (b, d > 0)

Worked example

Arrange

\frac{3}{8}, \frac{2}{5}, \frac{5}{12}

in ascending order.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 12 Sep 2025, Shift 1 · Q51Easy

Example 3 · Surds, Indices and Simplification · Fractions and Decimals

Arrange the fractions

\frac{5}{9}

\frac{4}{7}

\frac{3}{5}

, and,

\frac{2}{3}

in ascending order.

Judging a fraction by the size of its numbers

\frac{9}{70}

has bigger numbers than

\frac{2}{3}

but is far smaller: about 0.13 against 0.67. Compare values, not the size of the top and bottom.

Drill 2 more on comparing and ordering fractions

Concept 4 of 4: Continued fractions

A continued fraction has a fraction inside the denominator of a fraction, several levels deep. You cannot start at the top. Start at the bottom, turn each level into one fraction, and climb.

Definition

Work from the lowest level: write the last $a + \frac{1}{b}$ as one fraction, flip it, add it to the level above, and repeat.
One over a fraction is that fraction upside down: $\frac{1}{\;2/7\;} = \frac{7}{2}$ .
When a letter sits inside, simplify the levels the same way, then solve the equation that is left.
Fast check: if every lower level is at least 1, the value lies between the top whole number and one more than it. For $2 + \frac{1}{3 + \dots}$ the value is between 2 and 3, which often removes two options.

Building one level

a + \frac{1}{b} = \frac{ab + 1}{b} \qquad \frac{1}{\;p/q\;} = \frac{q}{p}

Worked example

Find

3 + \frac{1}{1 + \frac{1}{4 + \frac{1}{3}}}

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 15 Sep 2025, Shift 3 · Q53Easy

Example 4 · Surds, Indices and Simplification · Fractions and Decimals

Which of the following represents the correct simplified value of the continued fraction:

x = 1 + \frac{1}{2 + \frac{1}{2 + \frac{1}{2}}}

Starting a continued fraction from the top

2 + \frac{1}{3 + \frac{1}{2}}

you cannot add 2 and 1 first. The bottom gives

3 + \frac{1}{2} = \frac{7}{2}

, its reciprocal is

\frac{2}{7}

, and the value is

2 + \frac{2}{7} = \frac{16}{7}

Drill 2 more on continued fractions

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (4)

Fractions and mixed numbers in BODMAS
Fraction rules
$\frac{a}{b} \div \frac{c}{d} = \frac{ad}{bc} \qquad \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} \qquad p\frac{q}{r} = \frac{pr + q}{r}$
Working with decimals
Recurring decimals and handy divisions
$0.\overline{a} = \frac{a}{9} \qquad 0.\overline{ab} = \frac{ab}{99} \qquad x \div 0.25 = 4x \qquad x \div 0.5 = 2x$
Comparing and ordering fractions
Cross-multiplication test
$\frac{a}{b} > \frac{c}{d} \text{ exactly when } ad > bc \quad (b, d > 0)$
Continued fractions
Building one level
$a + \frac{1}{b} = \frac{ab + 1}{b} \qquad \frac{1}{\;p/q\;} = \frac{q}{p}$

Watch out for (6)

Multiplying mixed numbers part by part
→ Fractions and mixed numbers in BODMAS
Turning the wrong fraction upside down
→ Fractions and mixed numbers in BODMAS
Too few decimal places in a product
→ Working with decimals
Expecting division to make a number smaller
→ Working with decimals
Judging a fraction by the size of its numbers
→ Comparing and ordering fractions
Starting a continued fraction from the top
→ Continued fractions

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