Problem Solving

Hello Madhav,

In this question, you observe that all the options have a variable associated with them. Additionally, there are these fractions which just make things more complex than simplifying them. This should tell you that trying to solve this question using Pure algebra would take more time and effort.

Dillesh has already done a great job on explaining the Algebraic approach to solving this question. As such, I’’ll try to shed some light on how to solve this question by plugging in values and eliminating options.

First things first, I’d look at \(\frac{99}{4}\) and \(\frac{101}{4}\). \(\frac{99}{4}\) = 24.75 and \(\frac{101}{4}\) = 25.25. The distance between these two numbers is exactly half. If we divide this into two equal halves, i.e. between 24.75 and 25 and between 25 and 25.25, each of these smaller intervals would span 0.25 (or a quarter). Clear till now?

A question in your mind at this stage could be – “why did he pick 25 of all numbers?”. My answer would be – “ Observe the options. Also, \(\frac{100}{4}\) (25) is exactly in between \(\frac{99}{4}\) and \(\frac{101}{4}\)”.

The question says that the sack should hold ‘m’ pounds of rice and this ‘m’ should like in the above interval. Now, time to ask ourselves some questions.

1) Is ‘m’ a really so small a number that, if I add it with 25, I will get ¼??? The answer clearly is a ‘NO’. We know that ‘m’ is a fairly sizeable number in the range of 24 to 26 and when you add such a number to 25, you will definitely NOT get a number which is equal to ¼ or less than ¼. That’s when you eliminate options B and D.
The possible answers at this stage are A, C or E.

2) Now, can I take ‘m’ = \(\frac{99}{4}\) or \(\frac{101}{4}\)?? Clearly not, because the question mentions that ‘m has to be IN BETWEEN these numbers. When the question says ‘IN BETWEEN’, it means that the extreme values (or boundary conditions) cannot be included.
But, if m=\(\frac{99}{4}\) or \(\frac{101}{4}\), |m-25| = ¼ (remember that whether you get 0.25 or -0.25 inside the modulus sign, the output will always be 0.25). Since m cannot be 99/4 or 101/4, |m-25| = ¼ i.e. option C can be eliminated.

3) Can I take m = 25.5 or m = 24.5? Clearly not, because either of these values are not between \(\frac{99}{4}\) and \(\frac{101}{4}\). But, in the case of 24.5 or 25.5, |m-25|> 0.25. Since we cannot take these values, |m-25|>0.25 is not a valid answer. Option A can be eliminated.

The only option left is E, which HAS to be the answer. If you come to think of it, the distance between \(\frac{99}{4}\) and \(\frac{101}{4}\) i.e. 0.5. This is distributed evenly on either side of 25 and therefore, regardless of where ‘m’ is, the distance of ‘m’ from 25 will always be less than half of 0.5 i.e. will always be less than 0.25.

When you have variables in the question and the options, a viable approach would be to plug in simple values and evaluate the options. As you evaluate options, you eliminate the ones that don’t satisfy the constraints. The option left has to be your answer.

When you are plugging in values, remember to plug in values not only from the range provided but, also from outside the range so that you can disprove options/statements.

With reference to your question on material for absolute values, here’s my 2 cents. If you have enrolled for the Online course, the video on Inequalities and Absolute values is more than sufficient to learn all the concepts and hacks related to Absolute values. Look no further.

If you have enrolled for the Classroom course, please get in touch with our support team and request them for an E-book on Inequalities. That should suffice!

Hope that helps!

Option E is the correct answer