Approximation Techniques for speed calculations
While dealing with calculation intensive sections like Quant and DI, it is very important to pick the right questions use the best methods. As we grow, we tend to get habituated in our day to day calculations and employ conventional methods without thinking whether it is the best for a given scenario. Many a time there will be an approach that is much easier than the conventional methods to solve a given problem. Most of the so called calculation intensive questions are not that scary if we think a bit before solving them. As a rule of thumb, always spend few seconds to identify the best approach before start solving.
Some useful methods are given below which can help in our calculations.
To approximate Actual values
If (and only if) we need to find the actual value of a given fraction, represent the numerator as sum or difference of terms related to denominator.
1449 = 1320 + 132 – 3
1449/132 = 10 + 1 – a small value ≈ little less than 11 (actual value is 10.977)
36587 / 123 =
36587 = 36900 – 246 – 61.5 - …
36587 / 123 = 300 – 2 - 0.5 – a small value ≈ little less than 297.5 (actual is 297.455)
1569 / 12 =
1569 = 1200 + 360 + 8.4 + 0.6
1569 / 12 = 100 + 30 + 0.7 + 0.05 = 130.75
This method should suffice for the level of accuracy expected in our exams.
Another method is to reduce the complexity of fraction and then solve. Complexity of a fraction can be directly related to the complexity of its denominator. If we simplify denominator, we simplify the fraction. Add to or subtract from the denominator to make it an easier value (like add 2 to 1998 to get 2000 or subtract 16 from 116 to get 100).
While adjusting the denominator always remember to BALANCE the fraction. Balancing fraction is not just adding/subtracting the same number to/from the numerator that we used to change the denominator.
Consider a fraction p/q = n; then p = qn.
If we add a number x to q, we need to add nx to p to balance the fraction. Also if q is reduced by a number x, p needs to be reduced by nx.
Here the approximation comes while fixing n. If the given options are separated well enough from each other and simplification of denominator is pretty obvious, then this method can be employed. If we have closer options it is better to stick with the method we discussed first.
1569 / 12 = ?
Here if we make the denominator as 10 we can tell the value in no time. To do so, we need to subtract 2 from denominator. Numerator is more than 130 times the denominator (n ≈ 130). Hence to balance the fraction we need to subtract 2 * 130 from numerator.
1569 / 12 ≈ 1309 / 10 ≈ 130.9 (actual value is 130.75)
To Approximate relative values
Most of the DI questions revolves around sorting the given numbers/fractions or finding its relative position (lesser/greater than) based on a reference value. If we don’t need the actual value, DON’T find the actual value.
Find the largest and smallest value among the below fractions
56/298, 46/374, 138/493, 37/540, 670/2498
We will do the first level approximation by guesstimating the given fractions. Try to represent the given numbers in 1/x format. While arranging fractions we usually try to represent the given fractions with the same denominator after finding the LCM of all denominators. But we are here to solve faster using approximation. We will take an easier route, Make the numerator same, i.e. one.
56/298, we know 56 * 6 > 298 = > 56/298 > 1/6. Note that we didn’t find the actual value of 56 * 6; we just want to get the closest multiple of 56 to the number 298.
56/298 = Greater than 1/6
46/374= Less than 1/8
138/493 = Greater than 1/4
37/540 = Greater than 1/15
670/2498 = Greater than 1/4
We don’t have any confusion in finding the smallest which is 37/540 (1/15 is less than other values). But we have 2 candidates fighting for the largest fraction title, 138/493 and 670/2498. We will consider only those two and try to get an approximate value. We will try both methods discussed before for finding the actual value.
138 = 98.6 + 24.65 + 12.325 + …
138/493 ≈ 0.2 + 0.05 + 0.025 + small value ≈ greater than 0.275
670 = 499.6 + 124.9 + 49.96 – 4.46
670/2498 ≈ 0.2 + 0.5 + 0.02 – small value ≈ less than 0.27
Hence 138/493 is the largest.
We can see denominator is close to 3.5 times numerator. Hence if we increase denominator by x, we need to balance the fraction by increasing numerator by x/3.5. We will get an easier fraction if we can write denominator as 500 by adding 7. We also need to add 7/3.5 = 2 to the numerator.
138/493 ≈ 140/500 ≈ 0.28
Similarly for 670/2498, here we can get a neat fraction by adding 2 to the denominator. And here as 2 is negligible compared to the denominator we can very well skip the balancing part and write fraction as 670/2500 = 0.268
Hence, 138/493 is the largest.
Here we wrote 670/2500 = 0.268. How?
670/2500 = 67/250, we can get denominator as 1000 by multiplying both sides by 4. Hence 67/250 = 268/1000 = 0.268
We used the same logic while ‘cleaning up’ 140/500. Multiply both sides with 2 to get denominator as 1000. Fraction becomes 280/1000 = 0.028
Here, instead of finding actual values of all five fractions and comparing them we just played with the relative values of the fractions and found actual values only for two cases which were required to get the answer.
Another usual DI question type is to find the relative position of a given value based on a reference value. This question comes like ‘How many students scored marks more than class average (Reference value)’ , ‘How many players has strike rate higher than Sachin (Reference value)’ etc…
How many of the given values are greater than 0.7
11/13, 25/34, 33/46, 44/65, 56/81
As we are asked to find only the relative values (with respect to 0.7) don’t jump into finding actual values. Take few seconds to write the below statement which will help us in solving faster.
If x/y > 0.7, x > 0.7 y, 10x > 7y
So we need to find all fractions where 10 times numerator is greater than 7 times y. multiplying both sides with 10 is to ease the calculation and simplify the comparison
Take fractions one by one
Three fractions (11/13, 25/34 and 33/46) are greater than 0.7
Most of us have higher comfortable level with multiplication than division. To find relative values based on a reference point, convert division into multiplication. This way we can get our answers faster without messing with our accuracy.
In our example 56/81 = 0.69, still we were able to find it is lesser than 0.7 without doing any complicated or time consuming stuff. Sweet, right!