Simplification and Approximation for CUET General Aptitude

Q: How do you compare or order fractions?

Cross-multiply. To compare a/b with c/d, compare a × d with b × c; the fraction whose numerator is in the bigger product is bigger. Is 5/7 bigger than 7/10 ? 5 × 10 = 50 and 7 × 7 = 49, so yes, just.

Q: How do you simplify surds and rationalise a denominator?

Pull out the largest perfect square: √(72) = √(36 × 2) = 6√(2). Only like surds add: 6√(2) + √(8) = 6√(2) + 2√(2) = 8√(2). To rationalise 1/(√(a) - √(b)), multiply top and bottom by √(a) + √(b); the denominator becomes a - b.

Q: What identities make mental arithmetic faster?

Three identities do most of the work: (a + b)^2 = a^2 + 2ab + b^2, (a - b)^2 = a^2 - 2ab + b^2 and (a - b)(a + b) = a^2 - b^2. Take a round number for a and a small one for b.

About 10 min to learn · 71 questions

CUET's simplification questions ask you to turn a long expression into one number without a slip, and its approximation questions ask you to round sensibly and still land on the right option. Both reward a fixed routine: BODMAS and brackets, fractions and recurring decimals, laws of indices, surds, quick-square identities and friendly rounding targets. This lesson gives the routine, the rules behind it and the traps that cost marks.

Syllabus area: General Mental Ability, Numerical Ability

On this page · 9 sections

01The core idea
02Order of operations
03Fractions and decimals
04Powers and roots
05Approximation
06Exam shortcuts
07Common mistakes
08Quick revision
09Check your understanding

The core idea

Simplification tests whether you can turn a long expression into a single number quickly and without a slip. Approximation tests the opposite skill: rounding sensibly and still landing on the right option. Both reward a fixed routine more than cleverness.

Before you start working, look at the options. If they are far apart, round and estimate. If they are close, work exactly. And when an expression looks messy, check whether it is a standard identity in disguise; in this chapter it very often is.

Order of operations

What is the correct order of operations (BODMAS)?

Work in this order: brackets, then "of", then division and multiplication, then addition and subtraction. Division and multiplication have equal rank and are done left to right, and so are addition and subtraction.

Step	What you do	Example
B	Brackets, innermost first	$(3 + 5)$ becomes 8
O	"Of", which means multiply	$\tfrac{1}{2}$ of 40 is 20
D, M	Divide and multiply, left to right	$30 \div 5 \times 2 = 6 \times 2 = 12$
A, S	Add and subtract, left to right	$20 - 8 + 3 = 15$

Example 1 · BODMAS

Evaluate $90 - 6 \times 8 \div 4 + 5$ .

Multiply and divide, left to right: $6 \times 8 = 48$ , then $48 \div 4 = 12$ .
Add and subtract, left to right: $90 - 12 + 5 = 83$ .

Answer

Working strictly left to right gives $(90 - 6) \times 8 \div 4 + 5 = 173$ . That wrong answer is almost always an option.

Watch out

Division does not come before multiplication, and addition does not come before subtraction. In $30 \div 5 \times 2$ , left to right gives 12, not $30 \div 10 = 3$ .

How do you remove nested brackets?

Remove brackets from the inside out: round brackets first, then curly, then square. Inside every bracket the full BODMAS order still applies.

Example 2 · Nested brackets

Simplify $5 - [8 - (6 - 9)]$ .

Innermost: $6 - 9 = -3$ .
Next: $8 - (-3) = 8 + 3 = 11$ .
Finally: $5 - 11 = -6$ .

Answer

−6

Watch out

A minus sign in front of a bracket flips every sign inside when you open it: $10 - (4 - 7) = 10 - 4 + 7 = 13$ . Most sign errors in this chapter happen here.

Why does "of" come before division?

"Of" is a multiplication tied to the number right after it, so it binds more tightly than an ordinary ÷ or ×. Treat the fraction and the number after "of" as one package.

$360 \div$ ( $\tfrac{1}{4}$ of 48) $= 360 \div 12 = 30$ , with "of" first.
$360 \div \tfrac{1}{4} \times 48 = 1440 \times 48 = 69{,}120$ is a different expression, because × is not "of".

How do you find a missing number in an equation?

Simplify everything you can on both sides, then undo the operations around the unknown in reverse order. Addition is undone by subtraction, multiplication by division, and so on.

Example 3 · Missing number

Find $\square$ if $\square \times 7 + 36 \div 4 = 72$ .

Simplify the known part: $36 \div 4 = 9$ , so $\square \times 7 + 9 = 72$ .
Undo the $+9$ : $\square \times 7 = 63$ .
Undo the $\times 7$ : $\square = 9$ .

Answer

Check: $9 \times 7 + 9 = 72$ . Putting the answer back takes five seconds and catches most slips.

Fractions and decimals

How do you add, subtract, multiply and divide fractions quickly?

To add or subtract, write every fraction over the LCM of the denominators. To multiply, cancel first, then multiply tops and bottoms. To divide, multiply by the reciprocal of the second fraction.

$\tfrac{2}{3} + \tfrac{1}{4} = \tfrac{8}{12} + \tfrac{3}{12} = \tfrac{11}{12}$ .
$\tfrac{4}{9} \times \tfrac{27}{8} = \tfrac{3}{2}$ , after cancelling 4 with 8 and 9 with 27.
$\tfrac{7}{10} \div \tfrac{14}{15} = \tfrac{7}{10} \times \tfrac{15}{14} = \tfrac{3}{4}$ .

Watch out

Turn mixed numbers into improper fractions before you multiply or divide: $2\tfrac{1}{2} \times 1\tfrac{1}{5} = \tfrac{5}{2} \times \tfrac{6}{5} = 3$ , not $2 \times 1 + \tfrac{1}{2} \times \tfrac{1}{5}$ . And never add fractions straight across: $\tfrac{1}{2} + \tfrac{1}{3} = \tfrac{5}{6}$ , not $\tfrac{2}{5}$ .

Example 4 · A fraction of what is left

Asha has ₹1,200. She spends $\tfrac{1}{3}$ of it, and then $\tfrac{1}{4}$ of what is left. How much remains?

After the first spend, $\tfrac{2}{3}$ remains: $\tfrac{2}{3} \times 1{,}200 = 800$ .
She then spends $\tfrac{1}{4}$ of 800, which is 200, so 600 remains.
In one step: $\tfrac{2}{3} \times \tfrac{3}{4} = \tfrac{1}{2}$ of the original remains.

Answer

₹600

How do you compare or order fractions?

Cross-multiply. To compare $\tfrac{a}{b}$ with $\tfrac{c}{d}$ , compare $a \times d$ with $b \times c$ ; the fraction whose numerator is in the bigger product is bigger. Is $\tfrac{5}{7}$ bigger than $\tfrac{7}{10}$ ? $5 \times 10 = 50$ and $7 \times 7 = 49$ , so yes, just.

Exam shortcut

Equal numerators: the smaller denominator gives the bigger fraction, so $\tfrac{3}{7}$ beats $\tfrac{3}{8}$ .
Fractions one short of 1, like $\tfrac{4}{5}$ , $\tfrac{7}{8}$ and $\tfrac{10}{11}$ : the one with the larger numbers is closer to 1, because each is 1 minus 1 over its denominator.
Very close values: three decimal places settle it. $\tfrac{4}{7} \approx 0.571$ and $\tfrac{5}{9} \approx 0.556$ .

How do you convert a recurring decimal into a fraction?

For a pure recurring decimal, put the repeating block over as many 9s as it has digits. For a mixed one, subtract the non-repeating part from the whole digit string and divide by one 9 for each repeating digit followed by one 0 for each non-repeating digit.

Decimal	Rule	Fraction
0.777…	7 over 9	$\tfrac{7}{9}$
0.3636…	36 over 99	$\tfrac{4}{11}$
0.4666…, only the 6 repeating	$46 - 4$ over 90	$\tfrac{42}{90} = \tfrac{7}{15}$
0.12525…, 25 repeating	$125 - 1$ over 990	$\tfrac{124}{990} = \tfrac{62}{495}$

Example 5 · Mixed recurring decimal

Write 0.1888…, where only the 8 repeats, as a fraction.

All the digits: 18. The non-repeating part: 1. Difference: $18 - 1 = 17$ .
One repeating digit and one non-repeating digit give 90.
So $0.1888\ldots = \tfrac{17}{90}$ .

Answer

\tfrac{17}{90}

Check: $17 \div 90 = 0.1888\ldots$

Watch out

A mixed recurring decimal never goes over plain 9s: $0.3888\ldots = \tfrac{35}{90}$ , not $\tfrac{38}{99}$ . And convert before adding: $0.444\ldots + 0.555\ldots = \tfrac{4}{9} + \tfrac{5}{9} = 1$ , not 0.99.

Powers and roots

What are the laws of indices?

For the same base, multiply by adding powers and divide by subtracting them. A power of a power multiplies the powers.

Law	In words	Example
$a^m \times a^n = a^{m + n}$	Multiply: add powers	$3^4 \times 3^2 = 3^6$
$a^m \div a^n = a^{m - n}$	Divide: subtract powers	$5^7 \div 5^5 = 5^2 = 25$
$(a^m)^n = a^{mn}$	Power of a power	$(2^3)^2 = 2^6 = 64$
$a^0 = 1$	Any non-zero base	$7^0 = 1$
$a^{-n} = \tfrac{1}{a^n}$	Negative power: reciprocal	$2^{-3} = \tfrac{1}{8}$
$a^{p/q} = (\sqrt[q]{a})^p$	Root, then power	$16^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8$

Example 6 · Same-prime bases

Simplify $25^{1.5} \times 8^{2/3} \div 2^4$ .

$25 = 5^2$ , so $25^{1.5} = 5^3 = 125$ .
$8 = 2^3$ , so $8^{2/3} = 2^2 = 4$ .
$125 \times 4 \div 16 = 500 \div 16 = 31.25$ .

Answer

31.25

Exam shortcut

Rewrite every base as a power of the same prime: $4 = 2^2$ , $8 = 2^3$ , $9 = 3^2$ , $27 = 3^3$ , $125 = 5^3$ , $243 = 3^5$ . A negative fractional power then does three jobs: the minus flips the fraction, the denominator takes a root, and the numerator raises a power. $(\tfrac{16}{81})^{-3/4} = (\tfrac{81}{16})^{3/4} = (\tfrac{3}{2})^3 = \tfrac{27}{8}$ .

How do you simplify surds and rationalise a denominator?

Pull out the largest perfect square: $\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$ . Only like surds add: $6\sqrt{2} + \sqrt{8} = 6\sqrt{2} + 2\sqrt{2} = 8\sqrt{2}$ . To rationalise $\tfrac{1}{\sqrt{a} - \sqrt{b}}$ , multiply top and bottom by $\sqrt{a} + \sqrt{b}$ ; the denominator becomes $a - b$ .

Example 7 · Rationalising

Simplify $\tfrac{4}{\sqrt{7} + \sqrt{3}}$ .

Multiply top and bottom by $\sqrt{7} - \sqrt{3}$ .
The denominator becomes $7 - 3 = 4$ .
So the fraction is $\tfrac{4(\sqrt{7} - \sqrt{3})}{4} = \sqrt{7} - \sqrt{3}$ .

Answer

\sqrt{7} - \sqrt{3}

Watch out

$\sqrt{a} + \sqrt{b}$ is never $\sqrt{a + b}$ : $\sqrt{9} + \sqrt{16} = 3 + 4 = 7$ , not $\sqrt{25} = 5$ . And the conjugate denominator is $a - b$ : $(\sqrt{5})^2 - (\sqrt{3})^2 = 2$ , not 8.

Telescoping sums and nested roots

Since $\tfrac{1}{\sqrt{n + 1} + \sqrt{n}} = \sqrt{n + 1} - \sqrt{n}$ , a long sum of such terms collapses: almost everything cancels, and only the first and last roots survive.

To find $\sqrt{11 + 2\sqrt{30}}$ , use $(\sqrt{a} + \sqrt{b})^2 = a + b + 2\sqrt{ab}$ . You need $a + b = 11$ and $ab = 30$ , so $a = 6$ and $b = 5$ , and the root is $\sqrt{6} + \sqrt{5}$ .

What identities make mental arithmetic faster?

Three identities do most of the work: $(a + b)^2 = a^2 + 2ab + b^2$ , $(a - b)^2 = a^2 - 2ab + b^2$ and $(a - b)(a + b) = a^2 - b^2$ . Take a round number for $a$ and a small one for $b$ .

Question	Identity	Working	Answer
$104^2$	$(a + b)^2$	$10000 + 800 + 16$	10,816
$97^2$	$(a - b)^2$	$10000 - 600 + 9$	9,409
$995 \times 1005$	$(a - b)(a + b)$	$1000000 - 25$	9,99,975
$63^2 - 37^2$	$a^2 - b^2$	$26 \times 100$	2,600

Identities in disguise

$\tfrac{a^3 + b^3}{a^2 - ab + b^2} = a + b$ and $\tfrac{a^3 - b^3}{a^2 + ab + b^2} = a - b$ .
$(a + b)^2 - (a - b)^2 = 4ab$ and $(a + b)^2 + (a - b)^2 = 2(a^2 + b^2)$ .
$\tfrac{0.7 \times 0.7 - 0.2 \times 0.2}{0.7 - 0.2}$ is $a^2 - b^2$ over $a - b$ , so it is $0.7 + 0.2 = 0.9$ .

Example 8 · A square root by pattern

Find $\sqrt{1369}$ .

It ends in 9, so the root ends in 3 or 7.
$35^2 = 1225$ and $40^2 = 1600$ , and 1369 is nearer 1225, so try 37.
$37^2 = 1369$ .

Answer

Key idea

Squares up to 30 and cubes up to 15 are worth knowing by heart. They turn roots like this into a two-second job.

Approximation

How do you approximate an answer quickly?

Round every number to a value that is easy to work with, simplify, and pick the option closest to your estimate. Easy means a round number, a perfect square or cube, or a percentage that is a simple fraction.

You see	Think
24.9% or 25.1%	$\tfrac{1}{4}$
33.2%	$\tfrac{1}{3}$
12.48%	$\tfrac{1}{8}$
19.97%	$\tfrac{1}{5}$
$\sqrt{399}$	20, since $20^2 = 400$
$\sqrt[3]{1330}$	11, since $11^3 = 1331$

Example 9 · Rounding to friendly numbers

What is 19.96% of 1,504 plus $\sqrt{3598}$ , approximately?

19.96% of 1,504 is close to 20% of 1,500, which is 300.
3,598 is close to $3600 = 60^2$ , so the root is about 60.
Estimate: $300 + 60 = 360$ .

Answer

About 360

The exact value is about 360.2.

Watch out

BODMAS still applies to the rounded numbers. And in a product, rounding one number up and the other down lets the errors partly cancel; rounding both the same way makes the estimate drift, so look at how far apart the options are before trusting a close call.

Exam shortcuts

Look at the options first. Far apart: estimate. Close: work exactly.
"Of" before ÷. Treat the fraction and the number after "of" as one package.
Same prime base. Rewrite 4, 8, 9, 27 and 125 as powers of 2, 3 and 5.
Spot the identity. $a^2 - b^2$ , $(a \pm b)^2$ and the cube identities hide in messy decimals.
Put the answer back. A five-second check catches most slips.

Common mistakes

Multiplying before a division that comes first. $48 \div 6 \times 2 = 16$ , not 4.
Ignoring "of". $200 \div$ ( $\tfrac{1}{2}$ of 20) $= 200 \div 10 = 20$ .
Losing a sign when opening a bracket. $7 - (3 - 5) = 7 - 3 + 5 = 9$ .
Adding fractions straight across.
99 for a mixed recurring decimal.
Adding surds under one root.
$a + b$ in the conjugate denominator.
Dropping the middle term. $(x + 5)^2 = x^2 + 10x + 25$ , not $x^2 + 25$ .
$a^0$ and $a^{-1}$ . $6^0 = 1$ , not 0 or 6. $5^{-1} = \tfrac{1}{5}$ , not −5.
Stopping one step early. Many options answer an intermediate step. Re-read what is asked.

Quick revision

Order of operations

Brackets, of, ÷ and × left to right, + and − left to right
Brackets inside out. A minus before a bracket flips every sign

Fractions and decimals

Add over the LCM. Divide by multiplying by the reciprocal
Compare by cross-multiplying
$0.\overline{ab} = \tfrac{ab}{99}$ . Mixed: (all − non-repeating) over 9s then 0s

Indices and surds

$a^m a^n = a^{m + n}$ , $(a^m)^n = a^{mn}$ , $a^0 = 1$ , $a^{-n} = \tfrac{1}{a^n}$
$a^{p/q}$ : root first, then power
Rationalise with the conjugate: denominator $a - b$

Identities and estimates

$(a \pm b)^2 = a^2 \pm 2ab + b^2$ , $a^2 - b^2 = (a - b)(a + b)$
25% is $\tfrac{1}{4}$ , 12.5% is $\tfrac{1}{8}$ , 20% is $\tfrac{1}{5}$
Round to squares and cubes

Percentages inside approximation questions are worked in percentages, and prime factors and divisibility in number system.

Next · 08Quick revision

Check your understanding

71 questions, grouped by type, easy to hard. Pick an answer and check it. The worked solution opens when you ask for it.

Order of operations (BODMAS)Q1 of 71Q1

Order of operations (BODMAS)

21 questions

Check your understanding

Order of operations (BODMAS)

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Fractions and decimals

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Powers, roots and surds

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Approximation

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