Most Important Formula

$$\text{Distance}=\text{Speed}\times\text{Time}$$

Derived forms:

$$\text{Speed}=\frac{\text{Distance}}{\text{Time}}$$ $$\text{Time}=\frac{\text{Distance}}{\text{Speed}}$$

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Unit Conversion

Must memorize.

1 m/s = 3.6 km/hr

1 km/hr = 5/18 m/s

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Conversion Shortcut

km/hr → m/s

× 5/18

m/s → km/hr

× 18/5

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Relative Speed

Same Direction

$$|S_1-S_2|$$

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Opposite Direction

$$S_1+S_2$$

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TRAINS

One of the most frequently asked topics.

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Train Crossing a Pole

$$\text{Time} =========== \frac{\text{Length of Train}} {\text{Speed}}$$

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Train Crossing a Platform

$$\text{Time} =========== \frac{\text{Train Length}+\text{Platform Length}} {\text{Speed}}$$

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Two Trains Crossing

Opposite Direction

$$\text{Time} =========== \frac{L_1+L_2}{S_1+S_2}$$

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Same Direction

$$\text{Time} =========== \frac{L_1+L_2}{|S_1-S_2|}$$

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Average Speed

Important trap.

Wrong approach:

(60 + 40)/2

This works only when distances are equal.

For equal distances:

$$\text{Average Speed} ==================== \frac{2ab}{a+b}$$

Example:

60 km/hr

40 km/hr

Average:

$$\frac{2\times60\times40}{60+40} =============================== 48\text{ km/hr}$$

Not 50 km/hr.

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STD Shortcuts

If speed doubles:

Time becomes half.

If time doubles:

Speed becomes half.

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2. BOATS & STREAMS

An extension of Speed, Time & Distance.

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Terms

Boat Speed = b

Stream Speed = s

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Downstream Speed

Boat moves with the current.

$$b+s$$

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Upstream Speed

Boat moves against the current.

$$b-s$$

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Finding Boat Speed

$$\frac{\text{Downstream}+\text{Upstream}}{2}$$

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Finding Stream Speed

$$\frac{\text{Downstream}-\text{Upstream}}{2}$$

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Example

Downstream = 12 km/hr

Upstream = 8 km/hr

Boat speed:

$$\frac{12+8}{2}=10\text{ km/hr}$$

Stream speed:

$$\frac{12-8}{2}=2\text{ km/hr}$$

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Boats Shortcut

Remember:

Downstream = Add

Upstream = Subtract

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3. WORK & TIME

One of the most important placement topics.

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Basic Rule

If A finishes work in:

10 days

One day’s work:

$$\frac{1}{10}$$

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General Formula

If a person completes work in (n) days:

$$\text{One Day's Work} ===================== \frac{1}{n}$$

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Together Work

If

A’s one-day work:

$$\frac{1}{10}$$

B’s one-day work:

$$\frac{1}{15}$$

Together:

$$\frac{1}{10}+\frac{1}{15}$$

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Shortcut Formula

If A completes work in (a) days and B in (b) days,

together they finish in:

$$\frac{ab}{a+b} \text{ days}$$

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Example

A = 10 days

B = 15 days

Together

$$\frac{10\times15}{10+15} = \frac{150}{25} = 6 \text{ days}$$

Efficiency Method

Very important.

Suppose:

A = 10 days

B = 20 days

Efficiency ratio:

$$2:1$$

Efficiency is inversely proportional to time.

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Shortcut Table

Days	Efficiency
10	6
12	5
15	4
20	3
30	2

Frequently useful.

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4. PIPES & CISTERNS

Same concept as Work & Time.

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Pipe Filling

Treat as positive work.

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Pipe Emptying

Treat as negative work.

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Example

Fill:

$$\frac{1}{10}$$

Empty:

$$\frac{1}{15}$$

Net work:

$$\frac{1}{10}-\frac{1}{15}$$

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Shortcut

Filling = +

Emptying = -

Always remember this.

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5. ALGEBRA

Identities

Must memorize.

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Identity 1

$$(a+b)^2 ======= a^2+2ab+b^2$$

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Identity 2

$$(a-b)^2 ======= a^2-2ab+b^2$$

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Identity 3

$$a^2-b^2 ======= (a-b)(a+b)$$

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Identity 4

$$(a+b)^3 ======= a^3+b^3+3ab(a+b)$$

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Identity 5

$$(a-b)^3 ======= a^3-b^3-3ab(a-b)$$

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Factorization

Difference of Squares

$$a^2-b^2 ======= (a-b)(a+b)$$

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Perfect Square

$$x^2+2xy+y^2 =========== (x+y)^2$$

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6. LINEAR EQUATIONS

One Variable

Example:

$$2x+5=15$$

Solution:

$$2x=10$$ $$x=5$$

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Shortcut

Move constants first.

Then divide by the coefficient.

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Two Variables

Example:

$$2x+y=7$$ $$x+y=4$$

Subtract the equations:

$$x=3$$

Substitute back:

$$y=1$$

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Elimination Method

Most useful in aptitude.

Steps:

Make coefficients equal

Subtract or add equations

Find one variable

Substitute back

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7. QUADRATIC EQUATIONS

Very important.

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Standard Form

$$ax^2+bx+c=0$$

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Factorization Method

Example:

$$x^2-5x+6=0$$

Factors:

$$(x-2)(x-3)=0$$

Roots:

$$x=2,;3$$

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Quadratic Formula

Most important.

$$x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

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Discriminant

$$D=b^2-4ac$$

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Nature of Roots

If (D>0)

Two distinct real roots

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If (D=0)

Equal real roots

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If (D<0)

No real roots

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MOST IMPORTANT FORMULAS

Speed, Time & Distance

$$D=ST$$

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Work & Time

$$\text{Work} =========== \text{Rate}\times\text{Time}$$

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Boats & Streams

Downstream:

$$b+s$$

Upstream:

$$b-s$$

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Quadratic Formula

$$x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$$