Numerical Ability — CUET UG General Test

Numerical Ability is the single largest topic in the Quantitative & Numerical Ability cluster, and it stays at a Class 8 level throughout — nothing here needs a method beyond what a school student already knows.

The actual difficulty is speed without a calculator: every question has to be solved by hand inside a 60-minute paper that also carries reasoning and general-knowledge questions. Automatic recall of a small set of rules — not clever technique — is what separates a fast, accurate attempt from a slow one.

1. Order of operations (BODMAS)

BODMAS fixes the sequence in which an expression's operations must be evaluated: Brackets, Of, Division, Multiplication, Addition, Subtraction, strictly in that order. Division and multiplication are evaluated left-to-right as they appear, not division-always-before-multiplication; the same left-to-right rule applies to addition and subtraction once both remain.

$15+6\times (8-3)÷2-4=26$

The bracket resolves first ($8-3=5$), then multiplication and division proceed left to right ($6\times 5=30$, then $30÷2=15$), and only then does addition and subtraction finish the expression ($15+15-4=26$). A nested bracket is resolved from the innermost pair outward, never from the outside in.

2. Fractions and decimals

Adding or subtracting fractions with different denominators requires converting every fraction to a common denominator first — the LCM of the denominators is the smallest, and therefore fastest, choice. Multiplying fractions needs no common denominator at all: multiply numerators together and denominators together, then simplify.

Decimal multiplication is ordinary integer multiplication with the decimal point placed afterward, counting the total decimal places across both numbers being multiplied. A quick sanity check — estimating the answer's rough size before multiplying — catches a decimal-point placement error immediately, since a wildly different order of magnitude signals a misplaced point rather than a genuine mistake in the digits themselves.

3. Squares, cubes and square roots

Memorising squares up to 30 and cubes up to 15 removes the single biggest speed bottleneck in this topic, since a huge share of questions either ask for one directly or use one as an intermediate step. The difference-of-squares identity turns a subtraction of two large squares into two small multiplications:

$a^2-b^2=(a-b)(a+b)$

$47^2-43^2=(47-43)(47+43)=4\times 90=360$

Finding a square root by hand is fastest through prime factorisation for a perfect square: split the number into prime factors, pair them up, and take one factor from each pair.

4. HCF and LCM

The Highest Common Factor (HCF) is the largest number dividing two or more given numbers exactly; the Lowest Common Multiple (LCM) is the smallest number that every given number divides exactly. For exactly two numbers, a fixed relationship between the two always holds and is worth using as a shortcut or a check:

$\text{HCF}\times \text{LCM}=\text{Product of the two numbers}$

For 18 and 24: $18=2\times 3^2$ and $24=2^3\times 3$. HCF takes the lowest power of every common prime ($2^1\times 3^1=6$); LCM takes the highest power of every prime present ($2^3\times 3^2=72$). Checking: $6\times 72=432=18\times 24$.

5. Ratio simplification

A ratio simplifies exactly like a fraction — divide every term by their common HCF — and a ratio with more than two terms must be simplified by the SAME single divisor applied to all terms together, never term by term with different divisors. A ratio of 18:24:30 has HCF 6 across all three terms, simplifying cleanly to 3:4:5; dividing each term by a different number would silently change the ratio's actual proportions.

6. Percentage, including reverse percentage

A straightforward percentage question asks for X% of a given quantity; a reverse-percentage question instead gives the value AFTER a percentage change and asks for the original value before that change. The trap in reverse percentage is dividing by the wrong factor — subtracting the percentage from the final value directly, instead of dividing by the correct multiplier.

If a 20% discount brings a jacket's price down to ₹960, the ORIGINAL price is not ₹960 plus 20% of ₹960 — it is found by dividing ₹960 by 0.8 (since ₹960 represents 80% of the original), giving ₹1,200.

7. Simple linear equations

A simple linear equation in one variable is solved by collecting every variable term on one side and every constant on the other, changing the sign of any term moved across the equals sign. The equation $3x-7=2x+8$ becomes $3x-2x=8+7$ once both moves are made, giving $x=15$ directly. Substituting the answer back into the original equation is a fast, reliable check that costs only a few seconds.

8. Averages

The average of a set of numbers equals their sum divided by the count, and this relationship is used in reverse just as often as it is used forward — given an average and a count, the sum is recovered by multiplying them back together.

This reverse use is exactly what solves "one value removed from a group" questions: find the original sum, find the new sum after removal, and the difference between the two sums is the removed value itself.

9. The unitary method

The unitary method finds the value of ONE unit first, then scales that single-unit value up or down to whatever quantity the question actually asks for. It applies to any proportional relationship — workers and days, cost and quantity, speed and distance — as long as the underlying rate stays constant throughout the question.

If 8 workers build a wall in 15 days, the total work equals $8\times 15=120$ worker-days; 12 workers would then take $120÷12=10$ days, assuming every worker works at the same constant rate.

Worked Examples

Example 1. Evaluate: $24-(18-(6+4))+3\times 2$.

The innermost bracket resolves first: $6+4=10$. Then $18-10=8$. Then $24-8=16$. Multiplication before addition: $3\times 2=6$. Finally $16+6=22$.

Answer: 22.

Example 2. Add: $\frac{1}{2}+\frac{1}{3}+\frac{1}{4}$.

The LCM of 2, 3 and 4 is 12. Converting: $\frac{6}{12}+\frac{4}{12}+\frac{3}{12}=\frac{13}{12}=1\frac{1}{12}$.

Answer: $1\frac{1}{12}$.

Example 3. Find $47^2-43^2$ without directly squaring either number.

Using the difference-of-squares identity: $(47-43)(47+43)=4\times 90=360$.

Answer: 360.

Example 4. Find the square root of 1,156 by prime factorisation.

$1156=2^2\times 17^2$. Pairing the primes and taking one factor from each pair: $2\times 17=34$.

Answer: 34 (check: $34^2=1156$).

Example 5. Find the HCF and LCM of 18 and 24.

$18=2\times 3^2$, $24=2^3\times 3$. HCF (lowest common powers) $=2\times 3=6$. LCM (highest powers present) $=2^3\times 3^2=72$.

Answer: HCF = 6, LCM = 72.

Example 6. Simplify the ratio 18 : 24 : 30 to its lowest terms.

The HCF of 18, 24 and 30 is 6. Dividing every term by 6: $3:4:5$.

Answer: 3 : 4 : 5.

Example 7. After a 20% discount, a jacket costs ₹960. What was its original price?

₹960 is 80% of the original price. Original price $=960÷0.8=₹1,200$.

Answer: ₹1,200.

Example 8. A shop increases a product's price by 20% and then offers a 10% discount on the new price. What is the net percentage change from the original price?

Combined multiplier $=1.20\times 0.90=1.08$, an 8% net increase — not a 10% net change, since the two percentages apply to different base values (the second discount is on the already-increased price, not the original).

Answer: 8% net increase.

Example 9. Solve for $x$: $3x-7=2x+8$.

Moving variable terms to one side and constants to the other: $3x-2x=8+7$, so $x=15$. Check: $3(15)-7=38$ and $2(15)+8=38$ — both sides match.

Answer: $x=15$.

Example 10. The average of 5 numbers is 24. If one number is removed, the average of the remaining 4 numbers becomes 22. What is the removed number?

Sum of 5 numbers $=5\times 24=120$. Sum of remaining 4 numbers $=4\times 22=88$. Removed number $=120-88=32$.

Answer: 32.

Example 11. If 8 workers can build a wall in 15 days, how many days will 12 workers take, working at the same constant rate?

Total work $=8\times 15=120$ worker-days. With 12 workers: $120÷12=10$ days.

Answer: 10 days.

Example 12. Find the cube of 13 and the square of 17, and state which is larger.

$13^3=2197$. $17^2=289$. Since $2197>289$, the cube of 13 is larger.

Answer: $13^3=2197>17^2=289$.

Summary

BODMAS fixes a strict evaluation order — brackets first, then division/multiplication left to right, then addition/subtraction left to right — and nested brackets resolve from the innermost pair outward.

Fractions need a common denominator (the LCM of the denominators) to add or subtract, but not to multiply. Memorised squares, cubes and the difference-of-squares identity $a^2-b^2=(a-b)(a+b)$ remove the biggest speed bottleneck in this topic.

HCF and LCM satisfy $\text{HCF}\times \text{LCM}=$ the product of the two numbers (for exactly two numbers) — a fast shortcut and a built-in check. A multi-term ratio simplifies by dividing every term by their single common HCF, never term by term with different divisors.

Reverse percentage questions give the value AFTER a change and ask for the value before it — divide by the correct multiplier, never subtract the percentage directly from the final value. Simple equations solve by collecting variables on one side and constants on the other; averages recover a group's sum by multiplying the average by the count; the unitary method finds one unit's value first, then scales to whatever the question asks.