Arithmetic — XAT Quantitative Ability & Data Interpretation

No calculator, five options, and options close enough together that a rushed approximation lands on the wrong one as often as the right one. XAT arithmetic rewards a specific discipline: decide how much precision the question actually needs before you start calculating, not after.

1. What XAT actually asks

Arithmetic is estimated at roughly 7-9 of QA&DI's 28 questions across recent papers (see docs/exam-briefs/xat-2026-brief.md) — the single heaviest topic in the section, reflecting XAT's stated lean toward arithmetic and mensuration over the algebra-heavy style CAT favours. Every question carries five options and the standard +1/-0.25 marking.

The topic spans a recognisable set of sub-areas: percentages, ratio and proportion, averages and mixtures, time-speed-distance, time and work, and simple/compound interest. None of these are individually hard in isolation — the difficulty is entirely in execution speed and in resisting the five-option format's tightly-spaced traps.

2. Percentages — the base always matters

The single most common arithmetic trap across all MBA entrances, XAT included: percentage change calculated on the wrong base.

$\text{percentage change}=\frac{\text{new value}-\text{old value}}{\text{old value}}\times 100$

If a price rises 20% then falls 20%, the net change is not zero — the second 20% is calculated on the already-increased value. Starting at 100: rises to 120, then falls by 20% of 120 (24), landing at 96 — a net 4% decrease. This asymmetry, and the specific "successive percentage change" formula it generalises to, is tested constantly:

$\text{net change}=a+b+\frac{ab}{100}\%,\text{where }a,b\text{ are signed percentage changes}$

For the 20%-then-20% example: $20+(-20)+\frac{20\times (-20)}{100}=0-4=-4\%$, confirming the 96 result without recomputing from scratch.

3. Ratio, proportion and mixtures

Ratio compares quantities of the same kind; a ratio $a:b$ can always be scaled to $ka:kb$ without changing its meaning. Proportion equates two ratios: $a:b=c:d$ means $ad=bc$ (cross-multiplication), the fastest way to solve for an unknown in a stated proportion.

Mixture problems (combining two quantities of different concentration, price, or speed) resolve fastest through the alligation shortcut:

$\frac{\text{quantity of cheaper}}{\text{quantity of dearer}}=\frac{\text{dearer price}-\text{mean price}}{\text{mean price}-\text{cheaper price}}$

This single formula replaces setting up and solving a full weighted-average equation, and is worth memorising cold — XAT's five-option format punishes the extra time a full algebraic setup costs relative to a direct alligation.

4. Time, speed and distance

$\text{speed}=\frac{\text{distance}}{\text{time}}$

Relative speed governs almost every XAT question in this sub-area: for two bodies moving toward each other, relative speed is the sum of their speeds; moving in the same direction, it is the difference. A boat's speed downstream is (boat speed + current speed); upstream is (boat speed − current speed) — the current always helps one direction and hinders the other by the same amount.

Unit conversion is the most common careless error here: $1\text{ km/h}=\frac{5}{18}\text{ m/s}$, and forgetting this conversion (or inverting it) silently produces an answer off by a factor of $3.6$, which is exactly the kind of error a five-option question is built to catch — a wrong-conversion answer often sits among the five options as a deliberate trap.

5. Time and work

If a person can complete a task in $n$ days, their work rate is $\frac{1}{n}$ of the task per day. Combined work rates simply add:

$\text{combined rate}=\frac{1}{n_1}+\frac{1}{n_2}+\cdots ,\text{combined time}=\frac{1}{\text{combined rate}}$

The trap to avoid: averaging the individual times directly (e.g., treating two workers who take 12 and 18 days as "would take 15 days together") is never correct — rates add, times do not. Work backward from a stated efficiency ratio when the problem gives one (e.g., "A is twice as efficient as B") by assigning A and B work rates in that ratio directly, rather than solving for absolute days first.

6. Simple and compound interest

$\text{Simple Interest}=\frac{P\times R\times T}{100},\text{Amount}=P\left(1+\frac{RT}{100}\right)$

$\text{Compound Interest, Amount}=P{\left(1+\frac{R}{100}\right)}^T$

For short time periods (2-3 years) and round rates, compound interest can be computed directly without the exponent, by applying the rate successively year over year — often faster under no-calculator conditions than evaluating a power. The difference between CI and SI over 2 years at rate $R\%$ has a direct shortcut worth memorising:

$\text{CI}-\text{SI (2 years)}=P{\left(\frac{R}{100}\right)}^2$

This shortcut alone resolves a recurring XAT question type — "find the difference between CI and SI on a sum over 2 years" — in one step instead of computing both amounts separately.

8. Common traps

• Applying a percentage change to the wrong base, especially in successive changes (a discount after a markup, or two consecutive price changes) — always identify what 100% refers to at each step.
• Averaging times instead of adding rates in time-and-work problems — this single confusion produces a wrong answer that often appears as a deliberate five-option trap.
• Forgetting the km/h to m/s conversion factor ($\frac{5}{18}$) or inverting it, in time-speed-distance problems — a factor-of-3.6 error is common and deliberately trapped.
• Confusing relative speed's sum-vs-difference rule — moving toward each other adds speeds, moving in the same direction subtracts them; downstream/upstream follows the same logic with current speed.
• Computing compound interest via the full exponential formula when a shortcut applies — for 2-year CI-vs-SI differences or short, round-rate periods, a direct shortcut is faster and less error-prone under no-calculator conditions.
• Not using alligation for mixture problems, instead setting up a full weighted-average equation that costs more time than the five-option format allows.

9. When to guess, and why

Within your first 8 skips across all of Part 1, an arithmetic question requiring a long, error-prone calculation under time pressure is a reasonable one to skip, since QA&DI is XAT's most time-hungry section and a rushed calculation risks landing on a trap option rather than genuinely saving time. Beyond your 8th skip, a blind 1-in-5 guess has an expected value of exactly 0, while a blank costs -0.10 — mark your best remaining guess.

Because arithmetic options are often numerically close, even a rough estimate (checking magnitude, or the last digit of a multiplication) frequently eliminates two or three options before you need to guess blind, turning a late guess meaningfully better than chance.

Summary

• Arithmetic is roughly 7-9 of QA&DI's 28 questions — the section's single heaviest topic, reflecting XAT's lean toward arithmetic over CAT's algebra-heavier style.
• Percentage change must always be computed on the correct base; successive changes use $a+b+\frac{ab}{100}$, never simple addition.
• Alligation solves mixture problems in one step, faster than a full weighted-average setup.
• Relative speed adds when moving toward each other, subtracts when moving the same direction; remember the $\frac{5}{18}$ km/h-to-m/s conversion.
• Time-and-work rates add; times never average directly — this is the chapter's most common single error.
• The CI-vs-SI 2-year difference shortcut ($P(R/100{)}^2$) and the consecutive-year-amount-difference trick both bypass slower full formula evaluation.
• Past your 8th free skip in Part 1, even a rough magnitude estimate on a hard arithmetic question usually narrows five close options enough to make a guess meaningfully better than chance.