Quantitative Reasoning (Competitive Exams) Exam Survival Guide

Window: CAT/GRE/GMAT/SAT/CUET/BITSAT & similar | Components: Arithmetic • Algebra • Number Theory • Geometry/Mensuration • Data Interpretation • Counting/Probability/Sets

Must-do topics (80/20 focus)

• Arithmetic (Core):
  Percent & percentage change; ratio & proportion; averages/weighted mean & mixture (alligation); profit–loss–discount; SI/CI; time–speed–distance (incl. relative/avg speed); time–work/rates; unit conversions.
• Algebra:
  Linear/quad equations & inequalities (domains, roots, discriminant); factorising & identities; functions (inverses, composition); absolute value; sequences (AP/GP).
• Number Theory:
  Divisibility, gcd–lcm; remainders & modular arithmetic (incl. negatives); last digit/cycles; primes/composites/parity; place value.
• Geometry/Mensuration & Coordinate:
  Triangles (similarity, Pythagorean triples), circles (angles/arcs), polygons; area/CSA/TSA/volume (rect/cyl/cone/sphere/frustum); slope, distance, midpoint.
• Data Interpretation (DI)/Statistics:
  Tables, bar/line/pie, stacked charts; “rate vs count”; mean/median/mode; SD intuition.
• Counting/Probability/Set Theory:
  nPrnPrnPr, nCrnCrnCr, complement rule; independence vs mutual exclusivity; Venn (2/3-set) & inclusion–exclusion; simple Bayes.

Top traps (avoid)

• Percent: mixing change vs “percentage points”; wrong base; chaining changes additively.
• Ratio/Mixture: scaling parts inconsistently; using plain average instead of weighted; alligation cross mistaken.
• Algebra/Inequalities: forgetting to flip inequality when ×/÷ by negative; extraneous roots after squaring; domain ignored (logs/roots/denominators).
• Quadratics: copying formula wrong; misreading discriminant sign.
• TSD: average speed ≠ mean of speeds; wrong frame for relative speed (same/opposite directions).
• Work/Rates: adding times instead of rates; not LCM’ing task sizes.
• Geometry: trusting not-to-scale diagrams; missing similar triangles cue; mixing r, d, h, l (slant).
• Number Theory: mod with negatives mishandled; remainder vs last digit; gcd–lcm properties forgotten.
• DI: ignoring scale/units; comparing totals when chart shows rates.
• Probability/Counting: double-counting arrangements; assuming independence; not using complement when easier.

Time split (tune to exam)

• Harvest first: Arithmetic/Algebra single-step (~40–50% of Qs/time).
• Then: Geometry/Mensuration & DI (~30–35%).
• Finally: Number theory & PnC/Probability/sets (~15–25%).
  Per-item caps (typical): 60–90 s MCQ; 90–120 s DI/calc-heavy. Use two-pass: sweep sure → mark 50–50s → one revisit.

Last-48h checklist (do this exactly)

• Formula ring (1 page): percent/ratio, TSD/work, SI/CI, quad formula & D, area/volume set, combinatorics & probability basics, logs/indices, common primes/squares/cubes.
• Drills (90–120 min/day):
  • 40 Qs Arithmetic mix (percent, ratio/mixture, TSD, work).
  • 20 Qs Algebra (ineq/quad/abs).
  • 2 DI sets (mixed charts).
  • 10 Qs Geo/mensuration.
  • 10 Qs PnC/Prob/Set-Venn.
• Mental math sprints (10 min): squares (to 30), cubes (to 10), reciprocals (1/2..1/12), %–fraction pairs (12.5%=1/8, 16.66%=1/6…).
• Error log: tag each miss by trap above; redo just those patterns.

Quick facts & frames (carry card)

Arithmetic

• Percent change: %Δ=new−oldold×100%\%\Delta = \frac{\text{new}-\text{old}}{\text{old}}\times100\%%Δ=oldnew−old​×100% ; successive a%, b% ⇒ net (1+a)(1+b)−1(1+a)(1+b)-1(1+a)(1+b)−1.
• Weighted avg: xˉ=∑wixi∑wi\bar x=\frac{\sum w_ix_i}{\sum w_i}xˉ=∑wi​∑wi​xi​​. Alligation: cross-difference gives ratio of quantities.
• TSD: d=std=std=st; avg speed on equal distances =2v1v2v1+v2=\frac{2v_1v_2}{v_1+v_2}=v1​+v2​2v1​v2​​. Relative speed: same dir =∣v1−v2∣=|v_1-v_2|=∣v1​−v2​∣, opposite =v1+v2=v_1+v_2=v1​+v2​.
• Work: Rate =1/time=1/\text{time}=1/time; combined rates add; use LCM of jobs.
• SI/CI: SI=Prt \text{SI}=PrtSI=Prt; CI=P(1+rn)nt−P \text{CI}=P(1+\frac{r}{n})^{nt}-PCI=P(1+nr​)nt−P.

Algebra/Numbers

• Quadratic roots: x=−b±b2−4ac2ax=\frac{-b\pm\sqrt{b^2-4ac}}{2a}x=2a−b±b2−4ac​​, nature by D=b2−4acD=b^2-4acD=b2−4ac.
• Inequalities: multiply/divide by negative ⇒ flip sign.
• Mod: if a≡b(modn)a\equiv b\pmod na≡b(modn), then a±c≡b±ca\pm c \equiv b\pm ca±c≡b±c, ka≡kbka\equiv kbka≡kb. For negatives: −7≡n−7 (mod n)-7\equiv n-7 \ (\text{mod }n)−7≡n−7 (mod n).
• Logs/indices: log⁡ab=log⁡a+log⁡b\log ab=\log a+\log blogab=loga+logb; never split log⁡(a+b)\log(a+b)log(a+b).

Geometry/Mensuration

• Similar triangles: equal angles ⇒ sides in proportion.
• Right-triangle triples: (3,4,5), (5,12,13), (8,15,17).
• Circles: A=πr2, C=2πrA=\pi r^2,\ C=2\pi rA=πr2, C=2πr; sector A=θ360∘πr2A=\frac{\theta}{360^\circ}\pi r^2A=360∘θ​πr2.
• 3D: Cylinder V=πr2hV=\pi r^2hV=πr2h; Cone V=13πr2hV=\frac13\pi r^2hV=31​πr2h, CSA=πrl \text{CSA}=\pi rlCSA=πrl; Sphere V=43πr3, SA=4πr2V=\frac43\pi r^3, \ \text{SA}=4\pi r^2V=34​πr3, SA=4πr2.

DI/Stats/Counting/Prob

• TAUSL: Title • Axes • Units • Scale • Legend (say aloud before answering).
• Sets (2/3): ∣A∪B∣=∣A∣+∣B∣−∣A∩B∣ |A\cup B|=|A|+|B|-|A\cap B|∣A∪B∣=∣A∣+∣B∣−∣A∩B∣; extend to 3 with +++ then −−− then + A∩B∩CA\cap B\cap CA∩B∩C.
• Combinatorics: nPr=n!(n−r)!nPr=\frac{n!}{(n-r)!}nPr=(n−r)!n!​, nCr=n!r!(n−r)!nCr=\frac{n!}{r!(n-r)!}nCr=r!(n−r)!n!​. Order? use PPP; no order? use CCC.
• Probability: P(A)=favourabletotalP(A)=\frac{\text{favourable}}{\text{total}}P(A)=totalfavourable​; use complement when easier; independence: P(A∩B)=P(A)P(B)P(A\cap B)=P(A)P(B)P(A∩B)=P(A)P(B).

Speed tactics (during test)

• Pick smart numbers for percent/ratio/variables (e.g., base 100).
• Back-solve from options on linear/quad & mixture.
• Bound & eliminate: compute rough upper/lower bounds; kill options outside.
• Units & dimension check to spot absurd answers.
• Sketch first (geo/DI) then compute.
• When stuck @ 60–90 s: switch method (plug, draw, pick numbers), else mark & move.

Exam-day mini-plan

• 3-min warm-up: 6 mental-math items + 1 ratio quickie.
• Pass 1: Harvest singles (Arithmetic/Algebra, easy DI) — no Q > 90 s.
• Pass 2: Geo/mensuration & set-Venn/PnC you flagged.
• Final 8–10 min: Revisit 50–50s; do DI TAUSL and unit audits; bubble check.