Use of ratios - 1

A motorboat takes a total of 20 hours to travel 32 km upstream and 24 km downstream. Find the total time that it would take to travel 10 km upstream and 10 km downstream, given that its speed downstream is thrice its speed upstream?

1. 4 hrs 20 mins 2. 6 hrs 3. 6 hrs 40 mins 4. 8 hrs

The traditional way would again be solving the equation forming equations...

If the speed upstream is x, then the speed downstream is 3x and hence we have
. Solving the equation would involve the following steps in the least...
Now the question required us to find . Plugging the value of x, we find the answer to be 40/6 i.e. 6 hrs and 40 mins.

Since the equation was an easy one to solve, we did not have to resort to any plugging of values to reduce our work.

While the above is a simple question, one must make efforts to do away with equations for two reasons...

1. Equations, when solved, rarely give us a better understanding. They just throw up the answer

2. Mind thinks much much faster than any pencil work. So while doing mental work might appear ardous initially, once mastered they not only save a lot of time but also make our minds very agile.

Look at how we could have solved the above questions using only reasoning and doing all the work in our mind.... and then see if this method has given you more insight into the problem than solving the equation.

Distances covered upstream and downstream are in the ratio 4 : 3 and the respective speeds are in the ratio 1 : 3. Thus the ratio of the time taken for upstream and for downstream would be in the ratio 4 : 1 (time is distance divided by speed). Dividing the total time 20 hours in the ratio 4 : 1, we get the time for the upstream and downstream journey to be 16 hrs and 4 hrs respectively.

The boat takes 4 hrs to travel 24 km downstream (from this we can find the downstream speed as 6 kmph, though not needed). To travel 10 km downstream it will take i.e. 1 hr 40 mins. To travel 10 km upstream it will take thrice as much time i.e. 5 hours. Thus total time needed is 6 hrs 40 mins.

It is a very good idea to be very conversant with ratios. Try the following standard escalator problem using ratios only:

Shyama and Vyom walk up an escalator (moving stairway). The escalator moves at a constant speed. Shyama takes three steps for every two of Vyom’s steps. Shyama gets to the top of the escalator after having taken 25 steps, while Vyom (because his slower pace lets the escalator do a little more of the work) takes only 20 steps to reach the top. If the escalator were turned off, how many steps would they have to take to walk up?

1. 40 2. 50 3. 60 4. 80

Off to the beach to celebrate the new year. So next post only after 3rd January. Mean-time have a blasting new year.