train problem

[defeated_soldier]

Rajdhani Express leaves Chennai towards Delhi at 3:10p.m and travels uniformally at 120 km/hr. August Kranti Rajdhani Express leaves Delhi towards Mumbai at 12:20 p.m and travels uniformally at 80 km/hr. Both the trains cross at Baroda at 4:30 p.m..On a particular day, Rajdhani leaves at 3:20 p.m ...When will the two trains cross ?

My solution:
---------------

let the distance be between Delhi and chennai = x km.

relative speed =120+80=200km/hr

so, they would meet after x/200 hours .

they meet Baroda at 4:30 p.m

so, the eqn is....
x/200=4:30-3:10 = 4/3 hours.

=> x=4*200/3

so , when the trian leaves by 3:20 p.m ....then time should take 3:20+x/200 =>3:20+1:20 =4:40 pm

but answer in my boo is 4:32 pm .....whtas wrong in my calculation ?

 

[soroban]

Hello, defeated_soldier!

The problem's wording can be misleading . . . and I got a different answer.
I've checked and double-checked . . . I believe their answer is wrong.

AKR Express leaves Delhi towards Mumbai at 12:20 pm and travels uniformly at 80 km/hr.
R Express leaves Chennai towards Delhi at 3:10 pm and travels uniformly at 120 km/hr.
Both trains cross at Baroda at 4:30 pm.

On a particular day, Rajdhani leaves at 3:20 pm.
When will the two trains cross ?

Both trains have uniform speeds. . . but do not travel the same length of time.

The AKR Express leaves at 12:20 pm and crosses Baroda at 4:30 pm.
\(\displaystyle \;\;\)It travels for \(\displaystyle 4\frac{1}{6}\) hours at 80 km/hr.
\(\displaystyle \;\;\)Its distance is: \(\displaystyle \,\left(\frac{25}{6}\right)(80)\:=\:\frac{1000}{3}\) km.

The R Express leaves at 3:10 and crosses Baroda at 4:30 pm.
\(\displaystyle \;\;\)It travels for \(\displaystyle 1\frac{1}{3}\) hours at 120 km/hr.
\(\displaystyle \;\;\)Its distance is: \(\displaystyle \,\left(\frac{4}{3}\right)(120)\:=\;160\) km.

The total distance is: \(\displaystyle \,\frac{1000}{3}\,+\,160\:=\:\frac{1480}{3}\:=\:493\frac{1}{3}\) km.


On that particular day, AKR Express has a three-hour headstart.
From 12:20 pm to 3:20 pm, it has travelled: \(\displaystyle 3\,\times\,80\:=\:240\) km already.

At 3:20 pm, the trains have a combined speed of 200 km/hr
\(\displaystyle \;\;\)to cover the remaining \(\displaystyle \,\frac{1480}{3}\,-\,240\:=\:\frac{760}{3}\) km.

The time required is: \(\displaystyle \,\frac{\frac{760}{3}}{200}\:=\:\frac{760}{600}\;=\;\frac{76}{60}\text{ hours}\:=\:1\text{ hour, }16\text{ minutes}\)

Therefore, the trains meet 1 hour, 16 minutes after 3:20 pm . . . at 4:36 pm.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The book answer of \(\displaystyle 4:32\) is definitely wrong.

AKR Express leaves at 12:20 and travels until 4:32.
\(\displaystyle \;\;\)It travels for \(\displaystyle 4\frac{1}{5}\) hours at 80 km/hr.
\(\displaystyle \;\;\)Its distance is: \(\displaystyle \,\left(\frac{21}{5}\right)(80)\:=\:336\) km.

R Express leaves at 3:20 and travels until 4:32.
\(\displaystyle \;\;\)It travels for \(\displaystyle 1\frac{1}{5}\) hours at 120 kn/hr.
\(\displaystyle \;\;\)Its distance is: \(\displaystyle \,\left(\frac{6}{5}\right)(120)\:=\:144\) km.

Their combined distances is: \(\displaystyle \,336\,+\,144\:=\:480\) km.

They are still \(\displaystyle 13\frac{1}{3}\) km apart!

 

[steve_b]

I also got 4:36 as an answer, but I did the problem quickly and didn't check it, so I thought sure it was wrong. I kept getting numbers with 1/3 and 2/3 attached to them. Glad soroban verified it!

Steve