Important Quantitatude aptitude formulas- Time and Distance

Important Formulas - Time and Distance1. Basics

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

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

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

2. Convert kilometres per hour(km/hr) to metres per second(m/s)

$x\text{km/hr}=x\times \frac{5}{18}\text{m/s}$

3. Convert metres per second(m/s) to kilometres per hour(km/hr)

$x\text{m/s}=x\times \frac{18}{5}\text{km/hr}$

4. Average SpeedIf an object covers a certain distance at $x$ kmph and an equal distance at $y$ kmph, the average speed of the whole journey
$=\frac{2xy}{x+y}$ kmph

5. Relation Between Distance, Speed and Timea.) Speed and time are inversely proportional (when distance is constant)
$⟹\text{speed}\propto \frac{1}{\text{time}}$ (when distance is constant) 

b.) If the ratio of the speeds of A and B is $a:b,$ then, the ratio of the time taken by them to cover the same distance is
$\frac{1}{a}:\frac{1}{b}=b:a$

c.) Assume two objects A and B start at the same time in opposite directions from P and Q respectively. After passing each other, A reaches Q in $a$ seconds and B reaches P in $b$seconds. Then,
Speed of A : Speed of B $=\sqrt{b}:\sqrt{a}$

d.) An object covered a certain distance at a speed of $v$ kmph. If it had moved $v_1$ kmph faster, it would have taken $t_1$ hours less. If it had moved $v_2$ kmph slower, it would have taken $t_2$ hours more. Then,
$v=\frac{v_1{v}_2(t_1+t_2)}{v_1{t}_2-v_2{t}_1}\text{kmph}x=v{t}_1(1+\frac{v}{v_1})\text{km}$

Special Case:
If $t_1=t_2,v=\frac{2v_1{v}_2}{v_1-v_2}\text{kmph}$
6. Relative Speeda.) If two objects are moving in the same direction at $v_1$ m/s and $v_2$ m/s respectively where $v_1>v2,$ then their relative speed $=(v_1-v_2)$ m/s 

b.) Consider two objects A and B separated by a distance of $d$ metre. Suppose A and B start moving in the same direction at the same time such that A moves towards B at a speed of $a$ metre/second and B moves away from A at a speed of $b$ metre/second where $a>b.$ Then,
relative speed $=(a-b)$ metre/second
time needed for A to meet B $=\frac{d}{a-b}$ seconds

c.) If two objects are moving in opposite directions at $v_1$ m/s and $v_2$ m/s respectively, then their relative speed $=(v_1+v_2)$ m/s 

d.) Consider two objects A and B separated by a distance of $d$ metre. Suppose A and B start moving towards each other at the same time at $a$ metre/second and $b$ metre/second respectively. Then,
relative speed $=(a+b)$ metre/second
time needed for A and B to meet each other $=\frac{d}{a+b}$ seconds