CAT 2025 Lesson : Time & Speed - Product Constancy

If a train decreases its speed from $80$ km/hr to $60$ km/hr, then it takes $5$ more hours to travel from Chennai to Mumbai. What is the distance between Chennai and Mumbai?

Solution

Method 1

The distance between Chennai and Mumbai is constant. When the train decreases its speed from $80$ km/hr to $60$ km/hr, time taken increases by $5$ hours.

When the speed decreases by $20$ km/hr, i.e.$\dfrac{1}{4}$ times its usual speed, there will be an increase of$\dfrac{1}{4 - 1}$ =$\dfrac{1}{3}$ times the initial time taken, which is given to be $5$ hours.

Where the initial time taken is $t$, $\dfrac{1}{3} \times t = 5$ ⇒ $t = 15$ hours

As the train takes $15$ hours while travelling at $80$ km/hr,

Distance between cities $= 80 \times 15 = 1200$ km

Alternatively

Ratio of speeds $= 80 : 60 = 4 : 3$

Ratio of time taken is the reciprocal of ratio of speeds.

∴ Ratio of time taken $= 3 : 4$
Let the initial time and new time taken be $\bm{3x}$ and $\bm{4x}$ respectively.
$4x - 3x$ = $x = 5$ hours

∴ Distance $= 80 \times 3x = 80 \times 3 \times 5$ = $\bm{1,200}$ km

Answer: $1200$ km

When Musafar travels at three-fifth his usual speed, he reaches office $16$ minutes late. How many minutes does he usually take to cover this distance?

Solution

Let the usual time taken to cover the distance be t.

This decrease of $\dfrac{2}{5}$ of usual speed will result in increase of $\dfrac{2}{5 - 2} = \dfrac{2}{3}$ of the usual time.

∴ $\dfrac{2}{3} \times t = 16$

⇒ $t = \bm{24}$ minutes

Alternatively

Ratio of speeds $= 1 : \dfrac{3}{5} = 5 : 3$

∴ Ratio of time taken $= 3 : 5$

Let the time taken be $3x$ and $5x$ respectively.
$5x - 3x = 16$
⇒ $x = 8$

He usually takes $3x = 3 \times 8 = \bm{24}$ minutes to travel to office.

Answer: $24$ minutes

Ram always leaves home for his office at $8$ am. If he travels at $40$ km/hr, he is $18$ minutes late. If he travels at $60$ km/hr, then he is $12$ minutes early. What is the time at which Ram is required to report for work?

Solution

Increase in speed from $40$ km/hr to $60$ km/hr has reduced time by $12 + 18 = 30$ minutes.

As Speed increased by $\dfrac{1}{2}$ of itself, time would decrease by $\dfrac{1}{2 + 1} = \dfrac{1}{3}$ of itself.

∴ Time taken at 40 km/hr $= 3 \times 30 = 90$ minutes

As Ram is late by $18$ minutes at $40$ km/hr, to be on time, he should reach in $90 - 18 = 72$ minutes

∴ Ram's reporting time $= 9:12$ am

Alternatively

Ratio of speeds $= 40 : 60 = 2 : 3$
∴ Ratio of time taken $= 3 : 2$

Let the time taken in the two scenarios be $3x$ and $2x$ respectively.
$3x - 2x = 12 + 18$
⇒ $x = 30$

When he travels at $40$ km/hr, he takes $3x = 90$ minutes.
As he is $18$ minutes late in this case, he is supposed to reach office in $90 - 18 = 72$ minutes.

∴ Ram's reporting time $= 9:12$ am

Answer: $9:12$ am