# Computing the Average Speed

Only 75 out of 729 or only about 10% of the participants in the Intermediate bracket of the 3^{rd} MATH-Inic Vedic Mathematics National got the correct average speed of Abe.

488 or nearly 70% answered 37.5 kph which is obtained by merely adding the uphill speed of 30Kph and the downhill speed of 45kph and dividing the sum by 2. This solution is wrong since Abe obviously travelled longer in going to Baguio than in coming back home.

The common approach to this problem is to assume the distance from Abe’s place to Baguio City which is, of course, the same as the distance from Baguio to his place. The best assumption to make is that the distance should be the Least Common Multiple of the numerical values of the speeds 30 kph and 45 kph. Note that any assumption for the distance would serve the purpose, but using the LCM would lead to easier calculations.

The LCM of 30 and 45 is 90. So, it is assumed that it would take Abe 90 km/30 kph or 3 hours to travel to Baguio City and 90km/45 Kph or 2 hours to come home. Therefore, he travelled a total of (90 + 90) kilometers in (3 + 2) hours. His average speed for the entire trip is thus, 180km/5 hours = 36kph.

We can further simplify our computation by having one of the speeds as a base. Let us take 30kph as the base: 30Kph is treated as zero while 45 is regarded as 45 – 30 = 15Kph. The average speed is then 30Kph + (0 x 30 + 2 x 15)/(3+2) = 30kph + 30/5 kph = 30 + 6 or 36kph.

But we can also use the see-saw technique we used in finding the average age of call center agents in a past issue of this newsletter.

First, we note that the average speeds, 30 kph and 45 kph are in the ratio of 2:3, so, over the same distance, the ratio of the travel time is 3:2. The average for the entire trip must be nearer to the lower speed of 30kph. Now if we partition the difference of the two speeds, 45 – 30 = 15 into two parts with a ratio of 2:3, we will have 6:9.

By the see-saw principle, 3 x 6 = 2 x 9. We can simply add 6 to 30 to get an average of 36 kph for the entire trip.