Quantitative Reasoning — CUET UG General Test

Quantitative Reasoning differs from plain Numerical Ability in what it actually tests: the numbers involved are simple, but each question requires translating a word description into the right relationship before any calculation begins. The skill is setting the problem up correctly — an age comparison, a proportion, a pattern — not performing a harder calculation once it is set up.

1. Direct and inverse proportion

In direct proportion, two quantities increase or decrease together at a constant ratio — speed and distance covered in a fixed time, or cost and quantity at a fixed rate. In inverse proportion, one quantity increases as the other decreases, with their PRODUCT staying constant — more workers finishing a fixed job faster, or more pipes filling a tank sooner.

The tell is the underlying constant: if it's a constant RATE (like speed or unit price), the relationship is direct; if it's a fixed TOTAL amount of work being shared (like a job or a tank's capacity), the relationship is inverse.

2. Age problems

Age problems always compare the SAME two people's ages at different points in time, and the correct setup states each condition as its own equation before solving. A common structure gives a ratio of current ages and a second condition after a stated number of years — writing both conditions in terms of a single variable resolves the problem directly.

If B's current age is $x$ and A is three times as old, A's age is $3x$. Five years from now, A's age is $3x+5$ and B's age is $x+5$; a condition like "A will then be twice as old as B" becomes the equation $3x+5=2(x+5)$.

3. Clock angles

The minute hand moves 6° per minute and the hour hand moves 0.5° per minute (30° per hour), so the angle between them at any H hours and M minutes is found from one formula:

$\text{Angle}=|30H-5.5M|\text{ degrees (take the smaller of this value and }360°\text{ minus it)}$

At 3:40, $|30(3)-5.5(40)|=|90-220|=130$, and since 130 is already the smaller of 130° and 230°, the angle between the hands is 130°.

4. Number patterns: triangular and geometric

A triangular number is the sum of the first $n$ natural numbers, following the formula $\frac{n(n+1)}{2}$, and the sequence 1, 3, 6, 10, 15, 21… grows by one more than the previous gap each time. A geometric pattern instead grows by a constant RATIO between consecutive terms — 3, 6, 12, 24… doubles at every step, so the next term is always the previous term multiplied by that fixed ratio.

Recognising which type of pattern a sequence follows — constant added gap (arithmetic), constant ratio (geometric), or a growing gap matching a known formula (triangular, square) — is the entire skill, since once identified, extending or filling the pattern is direct substitution.

5. Sum-and-difference problems

Given the sum and the difference of two numbers, the larger number equals half the sum plus half the difference, and the smaller number equals half the sum minus half the difference. This shortcut avoids setting up and solving two simultaneous equations from scratch every time the same structure appears.

$\text{Larger}=\frac{\text{Sum}+\text{Difference}}{2},\text{Smaller}=\frac{\text{Sum}-\text{Difference}}{2}$

6. Consecutive numbers

For a set of consecutive integers, the AVERAGE of the set always equals its middle value (for an odd count) or the average of the two middle values (for an even count), which makes the sum-divided-by-count shortcut faster than naming each number with a separate variable. Three consecutive integers summing to 72 must average $72÷3=24$, so the middle number is 24 and the three numbers are 23, 24 and 25.

7. Rate reasoning and combined work

When two agents work at different but constant individual rates, their COMBINED rate is the sum of their individual rates, expressed as "fraction of the job done per unit time," not the sum of the times they individually take. A worker who finishes a job in 10 days completes $\frac{1}{10}$ of it per day; combining two such rates and inverting the sum gives the time taken working together.

8. Ratio sharing

Dividing a total quantity in a given ratio means treating the ratio's terms as the number of equal "parts" the total is split into — find the value of one part first by dividing the total by the sum of the ratio terms, then scale up. This is the unitary method applied specifically to proportional division rather than to rates.

Worked Examples

Example 1. A car travels 180 km in 3 hours at a constant speed. How far will it travel in 5 hours at the same speed?

Speed $=180÷3=60$ km/h (direct proportion — a constant rate). Distance in 5 hours $=60\times 5=300$ km.

Answer: 300 km.

Example 2. 6 identical pipes fill a tank in 4 hours. How long will 8 identical pipes take, working at the same individual rate?

Total pipe-hours (a fixed quantity of work) $=6\times 4=24$. With 8 pipes: $24÷8=3$ hours.

Answer: 3 hours.

Example 3. B's current age is $x$ years and A is three times as old as B. In 5 years, A will be exactly twice as old as B. Find both their current ages.

$3x+5=2(x+5)\Rightarrow 3x+5=2x+10\Rightarrow x=5$. So B is 5 and A is 15. Check: in 5 years, A is 20 and B is 10, and $20=2\times 10$.

Answer: B = 5 years, A = 15 years.

Example 4. Find the angle between the hour and minute hands at 3:40.

$|30(3)-5.5(40)|=|90-220|=130$. Since 130 is already less than $360-130=230$, the angle is 130°.

Answer: 130°.

Example 5. Find the angle between the hour and minute hands at 7:20.

$|30(7)-5.5(20)|=|210-110|=100$. Since 100 is less than $360-100=260$, the angle is 100°.

Answer: 100°.

Example 6. Find the 10th triangular number, and list the first 5 terms of the sequence.

Using $\frac{n(n+1)}{2}$ with $n=10$: $\frac{10\times 11}{2}=55$. The first 5 terms are $1,3,6,10,15$ (each adding one more than the previous gap: $+2,+3,+4,+5$).

Answer: 10th triangular number = 55.

Example 7. A sequence begins $3,6,12,24,\dots$. Find the next two terms.

Each term is double the previous one (a constant ratio of 2, so the pattern is geometric): $24\times 2=48$, then $48\times 2=96$.

Answer: 48, 96.

Example 8. Two numbers have a sum of 84 and a difference of 12. Find both numbers.

Larger $=\frac{84+12}{2}=48$. Smaller $=\frac{84-12}{2}=36$. Check: $48+36=84$ and $48-36=12$.

Answer: 48 and 36.

Example 9. The sum of three consecutive integers is 72. Find the three integers.

The average of three consecutive integers equals the middle one: $72÷3=24$. The three integers are $23,24,25$.

Answer: 23, 24, 25.

Example 10. A can complete a job in 10 days and B can complete the same job in 15 days. Working together at their own constant rates, how many days will they take?

A's rate $=\frac{1}{10}$ of the job per day; B's rate $=\frac{1}{15}$ per day. Combined rate $=\frac{1}{10}+\frac{1}{15}=\frac{3}{30}+\frac{2}{30}=\frac{5}{30}=\frac{1}{6}$ per day. Time together $=6$ days.

Answer: 6 days.

Example 11. ₹1,200 is divided among A, B and C in the ratio 2 : 3 : 5. Find each person's share.

Total parts $=2+3+5=10$. One part $=1200÷10=₹120$. A gets $2\times 120=₹240$, B gets $3\times 120=₹360$, C gets $5\times 120=₹600$.

Answer: A = ₹240, B = ₹360, C = ₹600.

Summary

Quantitative Reasoning tests correct problem SETUP over harder calculation — identify whether a relationship is direct proportion (constant rate) or inverse proportion (fixed total work, more agents means less time each) before calculating.

Age problems compare the same two people's ages at different times; writing every condition as its own equation in one variable, rather than guessing, solves them directly. Clock angle questions use one formula, $|30H-5.5M|$ degrees, taking the smaller of that value and 360° minus it.

Recognise a number pattern's type — constant added gap, constant ratio, or a growing-gap formula like triangular numbers — before extending it. Sum-and-difference problems solve instantly via (Sum+Difference)/2 and (Sum-Difference)/2; consecutive-integer sums use the average-equals-middle-value shortcut; combined work-rate problems add individual RATES, never individual times; and ratio sharing finds one "part's" value first, then scales up to every share.