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30MSC 2023 #30: By Mere Observation

30MSC 2023 #30: By Mere Observation

We reserved the easiest strategy for the last because its application depends on how many types of mental calculations you have mastered. The answers to many problems (especially multiple choice problems) can be determined without really solving – just by looking at the figures or By Mere Observation.

The example we used in 2023 MSC #23: Think of a Number: “What is the possible solution to the equation x + 1/x = 26/5”, can be solved By Mere Observation that 26/5 is equal to 5 1/5, which is the sum of 5 and its reciprocal 1/5.

Our example for this strategy was taken from the senior IVMO 2023: “What is the square root of [100 ( 2023  – 2000) + 4]?  a) 768; b) 144; c) 96; d) 48; e) 36”

If we know our MDAS, we can easily determine what the square number is by these mental steps:

  1. 2023 – 2000 = 23
  2. 23 x 100 = 2300
  3. 2300 + 4 = 2304  

The figure 2304 can be separated into two groups of two digits 23 and 04. So we can immediately conclude that the square root has two digits. Since 23 is between 16 and 25, the squares of 4 and 5 respectively, can safely say that the first or ten’s digit of the square root is 4.  Choice d has two digits and its 10’s digit is 4.

We can confirm, using one of our squaring techniques, that the square of 48 is 2304:

               48^2 = (50 – 2)^2 = 25-2|2^2 = 23|04

More examples of By Mere Observation are discussed in Chapter 10 of our forthcoming book, 30 Master Strategies in Computing, Vol III.

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