Numerical Ability for GATE

Numerical Ability

GATE GA typically has 5 marks from Numerical Ability (3-4 questions). These are solvable in 2-3 minutes each with the right formulas.

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Percentages — "Out of 100"

The Word "Percent" literally means "per hundred."

x% of y = (x/100) × y = xy/100

Example: 30% of 200 = (30/100) × 200 = 60

Percentage Change:

% change = (new - old) / old × 100

Example: Price goes from ₹80 to ₹100
Change = (100 - 80)/80 × 100 = 20/80 × 100 = 25% increase

Successive changes (very common in GATE): If price first increases by a%, then decreases by b%:

Net % = a - b - (a×b)/100

Example: 20% up then 20% down
Net = 20 - 20 - (20×20)/100 = 0 - 4 = -4%  ← You LOSE 4%!
(This surprises people — equal up and down = net loss)

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Ratio & Proportion — Scaling Recipes

Analogy: A recipe for 4 people needs 2 cups of flour. For 6 people, how much? That's ratio and proportion!

a:b = c:d  means  a/b = c/d  means  ad = bc (cross multiply)

Example: A:B = 2:3 and B:C = 4:5. Find A:B:C.
Make B common: A:B = 2:3 = 8:12 and B:C = 4:5 = 12:15
∴ A:B:C = 8:12:15

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Simple & Compound Interest — The Magic of Compounding

Analogy: Simple Interest = you earn interest only on the original money (Principal). Compound Interest = you earn interest on interest — like a snowball rolling downhill!

Type	Formula
Simple Interest	SI = (P × N × R) / 100
Total Amount (SI)	A = P + SI = P(1 + NR/100)
Compound Interest	A = P(1 + R/100)^N
CI =	A - P

Key trick: When compounded half-yearly → replace R with R/2, N with 2N.

GATE Example: ₹1000 at 10% for 2 years. Find SI and CI.

SI = 1000 × 2 × 10 / 100 = ₹200
CI: A = 1000(1 + 10/100)² = 1000 × 1.21 = 1210; CI = ₹210
Difference = CI - SI = ₹10  ← This difference = P(R/100)² for 2 years

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Time & Work — The Rate Method

Key Insight: Always convert to RATES (work done per day), then add or subtract.

If A completes work in 'a' days → A's rate = 1/a per day
If B completes work in 'b' days → B's rate = 1/b per day

Working together: combined rate = 1/a + 1/b
Time together = 1 / (1/a + 1/b) = ab / (a+b)

Example: A does work in 6 days, B in 12 days. Together?

Rate of A = 1/6, Rate of B = 1/12
Together = 1/6 + 1/12 = 2/12 + 1/12 = 3/12 = 1/4 work per day
Time = 4 days

Pipes & Cisterns — same concept! Inlet pipe = positive rate, outlet pipe = negative rate.

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Time, Speed & Distance — D = S × T

The Golden Triangle:

        Distance
       /         \
Speed × Time

Cover any one letter to get the formula: D = S×T, S = D/T, T = D/S.

Average Speed TRAP (most common mistake):

Equal DISTANCES at different speeds:  use HARMONIC MEAN
Average speed = 2v₁v₂ / (v₁ + v₂)

Equal TIME at different speeds:  use ARITHMETIC MEAN
Average speed = (v₁ + v₂) / 2

Example: Drive 100km at 40 km/h, return 100km at 60 km/h.
Average speed = 2 × 40 × 60 / (40 + 60) = 4800/100 = 48 km/h  (NOT 50!)

Relative Speed:

Moving in SAME direction:    |v₁ - v₂|
Moving in OPPOSITE direction: v₁ + v₂

Train crossing a pole: time = Length of train / Speed
Train crossing a platform: time = (Length of train + Length of platform) / Speed

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Profit & Loss — The Shopkeeper Problem

CP = Cost Price (what you paid)
SP = Selling Price (what you sold for)
MP = Marked Price (price on tag)

Profit = SP - CP  (if positive)
Loss   = CP - SP  (if negative)

Profit% = (Profit / CP) × 100
SP      = CP × (100 + Profit%) / 100
Discount% = (MP - SP) / MP × 100

Example: CP = ₹200, SP = ₹250. Find Profit%.

Profit = 250 - 200 = ₹50
Profit% = 50/200 × 100 = 25%

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Mixtures — The Alligation Cross

When to use: You mix two things of different values and want to find the ratio.

The Cross Method (draw this):

Value of A (a)          Value of B (b)
          \              /
           \            /
            (target m)
           /            \
          /              \
   (b - m)           (a - m)  ← these are the ratio parts

Example: Mix milk at ₹20/litre and ₹30/litre to get a mixture worth ₹24/litre.

a = 20, b = 30, m = 24
Ratio of milk₂₀ : milk₃₀ = (30-24) : (24-20) = 6 : 4 = 3 : 2

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Data Interpretation — Read Before You Calculate

Tables, bar charts, pie charts, line graphs appear in GATE.

Smart approach: 1. Read the title and units carefully (lakhs? millions? percentage?) 2. Read all the questions first before calculating 3. Use approximations (±5% error is usually fine for MCQs) 4. Check if values are absolute numbers or percentages

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Quick Check Questions

Q1. A product's price increases by 10% then decreases by 10%. Net change? Answer: -1% (lose 1% — use successive change formula: 10 - 10 - 10×10/100 = -1)

Q2. A can do work in 8 days, B in 12 days. Working together for 2 days, A leaves. How many more days for B to finish?

Together rate = 1/8 + 1/12 = 5/24 per day
Work done in 2 days = 2 × 5/24 = 10/24 = 5/12
Remaining = 1 - 5/12 = 7/12
B alone takes: 7/12 ÷ 1/12 = 7 days

Q3. A train 150m long passes a pole in 15 seconds. Time to pass a 300m platform?

Speed = 150/15 = 10 m/s
Time for platform = (150 + 300)/10 = 450/10 = 45 seconds