General Aptitude

Sheet 1

Percentages, Profit/Loss & Ratios

5 formulas

Key formulas

Percentage Change ☆

\[ \%\Delta = \frac{\text{New}-\text{Old}}{\text{Old}} \times 100 \]

Successive Percentage Change ☆

\[ \text{net} = a+b+\frac{ab}{100} \]

a% then b%

Profit % & Discount ☆

\[ \text{Profit}\% = \frac{SP-CP}{CP}\times 100 \]

Discount% is on Marked Price

Combining Ratios ☆

\[ A{:}B{=}p{:}q,\ B{:}C{=}r{:}s \Rightarrow A{:}B{:}C = pr:qr:qs \]

Compound & Simple Interest ☆

\[ A = P(1+\frac{r}{100})^n \ (\text{CI}) \qquad A = P(1+\frac{rn}{100}) \ (\text{SI}) \]

Test your recall

Can you recall the percentage change formula?

Reveal formula

\[ \%\Delta = \frac{\text{New}-\text{Old}}{\text{Old}} \times 100 \]

GATE traps 4 traps

Successive % change isn't additive

Successive percentage changes don't simply add -- a% then b% gives a net change of a + b + ab/100, and forgetting the cross term ab/100 is the single most common percentage mistake on GATE-style aptitude questions.

Discount is on Marked Price

Discount is always applied to Marked Price, never Cost Price -- computing it on the wrong base is a frequent, easy-to-miss error in profit/loss problems.

Know which quantity is unknown

'X% of Y' and 'Y is X% of what' are inverse setups -- misreading which quantity is the unknown flips the entire equation; read the problem twice before setting it up.

Scale ratios to a common term

When combining two ratios sharing a common term, the common term must be scaled to a common value (LCM) before combining -- skipping this step gives a wrong combined ratio.

Sheet 2

Time-Speed-Distance & Time-Work

3 formulas

Core concepts

• Relative speed: same direction |S1-S2|; opposite directions S1+S2
• Combined work rate = sum of individual rates; time together = 1 / (sum of rates)

Key formulas

Distance-Speed-Time ☆

\[ \text{Distance} = \text{Speed} \times \text{Time} \]

Average Speed (Equal Distances) ☆

\[ \frac{2 S_1 S_2}{S_1+S_2} \]

harmonic mean

Crossing Time ☆

\[ \frac{\text{train length}+L}{\text{speed}} \]

Test your recall

Can you recall the average speed (equal distances) formula?

Reveal formula

\[ \frac{2 S_1 S_2}{S_1+S_2} \]

GATE traps 4 traps

Equal distances need the harmonic mean

Average speed for EQUAL distances (not equal times) uses the harmonic mean formula 2.S1.S2/(S1+S2), NOT the simple average (S1+S2)/2 -- using arithmetic mean here is the most common time-speed-distance mistake.

Add both lengths when crossing a platform

Train-crossing-a-platform problems must add BOTH the train's own length AND the platform's length to the distance travelled -- forgetting the train's length is a very common setup error.

A leak subtracts from the fill rate

When a leak/outlet pipe works against inlet pipes, its rate is SUBTRACTED -- if the net rate comes out negative, the tank never fills (or empties instead); always sanity-check the sign of the final answer.

Downstream adds, upstream subtracts current

Boat problems: downstream speed = boat speed + current; upstream speed = boat speed - current -- mixing these up flips the whole answer, an easily avoidable slip.

Sheet 3

Permutations, Combinations & Probability

5 formulas

Key formulas

Permutations & Combinations ☆

\[ {}^{n}P_r = \frac{n!}{(n-r)!} \qquad {}^{n}C_r = \frac{n!}{r!(n-r)!} \]

Circular Permutations ☆

\[ (n-1)! \]

n distinct objects

Permutations with Repetition ☆

\[ \frac{n!}{p_1!\,p_2!\cdots p_k!} \]

Union of Events ☆

\[ P(A \cup B) = P(A)+P(B)-P(A \cap B) \]

Independent Events ☆

\[ P(A \cap B) = P(A)\,P(B) \]

only if independent

Test your recall

Can you recall the permutations & combinations formula?

Reveal formula

\[ {}^{n}P_r = \frac{n!}{(n-r)!} \qquad {}^{n}C_r = \frac{n!}{r!(n-r)!} \]

GATE traps 4 traps

Arrangement vs. selection wording

'Arrangement' problems need permutations (order matters); 'selection/grouping' problems need combinations (order doesn't) -- picking the wrong one based on surface wording, not actual meaning, is the most common setup error in aptitude word problems.

Reflections may need an extra ÷2

Circular permutation (n-1)! applies when only rotations are considered identical -- if reflections are ALSO considered identical, divide by an extra factor of 2. GATE sometimes specifies this explicitly and it's easy to miss.

Repeated letters need dividing out

When objects repeat (e.g. letters of a word), naive n! overcounts -- divide by the factorial of each repeated group's count; forgetting this division is a very common combinatorics mistake.

'At least one' via the complement

'At least one' probability questions are almost always faster via the complement: P(at least one) = 1 - P(none) -- directly summing 'exactly 1 + exactly 2 + ...' is slower and more error-prone.

Sheet 4

Ages, Mixtures & Alligation

2 formulas

Core concepts

• Mean price in a mixture always lies BETWEEN the two component prices -- sanity-check your alligation ratio against this
• Ages problems: set up two linear equations from 'present age' and 'age N years ago/hence', then solve simultaneously

Key formulas

Alligation Rule ☆

\[ \frac{\text{Qty cheaper}}{\text{Qty dearer}} = \frac{CP_{dearer} - \text{Mean}}{\text{Mean} - CP_{cheaper}} \]

Repeated Replacement ☆

\[ \text{Final} = \text{Initial} \times \left(1 - \frac{x}{V}\right)^n \]

removing & replacing x out of V, n times

Test your recall

Can you recall the alligation rule formula?

Reveal formula

\[ \frac{\text{Qty cheaper}}{\text{Qty dearer}} = \frac{CP_{dearer} - \text{Mean}}{\text{Mean} - CP_{cheaper}} \]

GATE traps 3 traps

Alligation ratio order matters

The alligation ratio gives QUANTITIES in the order (dearer's excess) : (cheaper's shortfall) from the mean price -- inverting this ratio is the most common alligation mistake.

Replacement formula applies to concentration

Repeated-replacement problems apply (1 - x/V)^n to the CONCENTRATION, not the absolute quantity removed each round -- the amount of the original substance removed in later rounds is smaller in absolute terms even though the same volume x is drawn out each time.

Age shifts apply to everyone equally

'K years ago' and 'K years hence' both add or subtract K from EVERY person's current age equally -- a common setup error is applying it to only one person's age in the equation.