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CDS Mathematics · Formula sheet

Time, Speed and Distance formulas

11 formulas and 11 common traps for CDS Mathematics Time, Speed and Distance, grouped by subtopic.

Full notes with worked examples

Average Speed and Speed–Time Ratios

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Average speed

Equal distances

vˉ=2aba+b\bar v = \dfrac{2ab}{a + b}

Speed and time over a fixed distance

Fixed distance

v1t1=v2t2v_1 t_1 = v_2 t_2

Common traps

Not the average of the speeds

Going at 6060 and returning at 4040 gives 4848, not 5050. The plain average is right only when equal TIMES, not equal distances, are spent at each speed.

Early plus late

'4040 minutes early at one speed and 4040 minutes late at another' means the two times differ by 8080 minutes, not 4040 and not 00.

Speed Changes and Equations

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Faster speed, less time

Speed change

dv−dv+x=y  ⇒  v(v+x)=dxy\dfrac dv - \dfrac{d}{v + x} = y \;\Rightarrow\; v(v + x) = \dfrac{dx}{y}

Two conditions, two unknowns

Linear in reciprocals

d1a+d2b=T1,e1a+e2b=T2\dfrac{d_1}{a} + \dfrac{d_2}{b} = T_1, \quad \dfrac{e_1}{a} + \dfrac{e_2}{b} = T_2

Common traps

Slower means more time

If the speed is REDUCED, the new time is longer: write dv−x−dv=y\dfrac{d}{v - x} - \dfrac dv = y. Reversing the order gives a negative time difference and nonsense roots.

Solve for the reciprocals

The equations are linear in 1a\dfrac1a and 1b\dfrac1b, not in aa and bb. Clearing denominators first produces a messy quadratic system that is easy to get wrong.

Relative Speed: Chasing and Meeting

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Chasing: the same direction

Catching up

t=gapu−vt = \dfrac{\text{gap}}{u - v}

Meeting: opposite directions

After meeting

uv=t2t1\dfrac uv = \sqrt{\dfrac{t_2}{t_1}}

Common traps

After the theft or after the start?

The owner catches up one hour after setting off, which is one and a half hours after the theft. Options pair the right distance with the wrong reference time.

Distance from which end?

'How far from Q' asks for the distance the train starting at Q has covered, not the one from P. Compute both; they must add to the total.

Trains: Length and Crossing

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Passing poles, platforms and bridges

Crossing time

t=L+Pvt = \dfrac{L + P}{v}

Trains passing moving objects

Relative crossing

t=L1+L2u±vt = \dfrac{L_1 + L_2}{u \pm v}

Common traps

Clearing the last stone

Passing 9191 km-stones spans 9090 km between the first and last, plus the train's length to clear the last one completely.

A man in a train is a point

When a train crosses a PERSON in another train, the other train's length is a distractor. Adding it gives a speed that matches no option.

Boats and Streams

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Downstream and upstream

Boat and stream

b=d+u2,s=d−u2b = \dfrac{d + u}{2}, \qquad s = \dfrac{d - u}{2}

Common traps

Half the difference, not the difference

Downstream 1010 and upstream 22 km/hr give a stream of 44 km/hr, not 88. The difference d−ud - u is TWICE the stream's speed.

Races

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Starts, beats and laps

Beat by a distance

vA:vB=D:(D−x)v_A : v_B = D : (D - x)

Common traps

Count only complete laps

Gaining 3.23.2 laps means passing the other runner 33 times, not 3.23.2 and not 44. A pass happens each time a WHOLE extra lap is gained.

Clocks and Angles

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Angles between the hands

Angle between the hands

θ=∣30h−5.5m∣\theta = |30h - 5.5m|

Common traps

The hour hand moves too

At 8:208{:}20 the hour hand is not on the 88; it has moved 10∘10^\circ toward the 99. Ignoring that gives 120∘120^\circ instead of 130∘130^\circ.

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