 # MSC #30: By Inspection:

“Which of the following is the equation of the straight line that passes through the points (3, 7) and (1, 5)? A) x + 3y = 5;  B) x − y = − 4;  C) 3x − 2y = 4; D) x + y = 8; E) x − y = −3.

Most students will solve this problem, which was given in the intermediate level of the 1st International Vedic Mathematics Olympiad (IVMO 2021) in September 15, 2021 using the conventional two-point form formula.

Vedic Math practitioners, on the other hand, will quickly apply a special technique which will use the VM sub-sutra, “the product of the means less the product of the extremes.”

But neither solution is necessary since the answer should have been obvious from the start, By Inspection, which is the last of our 30 MATH-Inic Specials for Christmas series.

We reserved this Sutra for the last of our 30 MATH-Inic Specials for Christmas because it depends largely on how much one has mastered the techniques presented in the previous specials that he can solve many problems By Inspection or By Mere Observation.

In the case of the IVMO question, it can be observed that in each of the two points given, the x coordinate is 4 less than the y coordinate so that we can immediately say that x – y = – 4.

In our featured example,385 x 999, the answer can be readily obtained by using two sutras, By one less than the one before and All from 9 and the last from 10.

The first part of the answer comes from 385 – 1 = 384, while the second part is the 10’s complement of 385: 1000 – 385 = 615. So 385 x 999 = 384,615.

Or after getting the first part 384, just get the 9’s complement of 384.

In case there is a confusion in what rule to follow, it is best to use the last by the last. The last digits of the multiplicands are 5 and 9 so the last digit of the product must be 5.

By Mere Observation of the figures, we can announce the answer.

The reason we chose to use the multiplicand 385 is because we want to show you an interesting application of this multiplication by 999 – in getting the recurring decimal equivalent of a fraction with a denominator of 13, specifically, 5/13.

1. 5 x 7 = 35 (By Inspection)
2. 35 x 11 = 385 (By Inspection)
3. 385 x 999 = 384,615(By Inspection)
4. 5/13 = 0. 384 615 384 615…

Note that:

1. 1001 = 7 x 11 x 13
2. 1/1001 = 0.000 999 000 999 …
3. n/(7 x 11 x 13) = (n x 999)/1,000,000)
4.  n/13 = (n x 7 x 11 x 999)/1,000,000