# 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 1^{st} 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

**or**

*By Inspection***.**

*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**.

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

**Note that:**

**1001 = 7 x 11 x 13****1/1001 = 0.000 999 000 999 …****n/(7 x 11 x 13) = (n x 999)/1,000,000)****n/13 = (n x 7 x 11 x 999)/1,000,000**