# CAT Quantitative Aptitude Questions | CAT Ratios, Mixtures, Alligations and Averages Questions

###### CAT Questions | Averages | Averages - Integers

The question is from CAT Ratio, Mixtures and Averages. Given the product of n digits, our task is to minimize the average of those numbers which is related to the sum of those numbers. CAT exam is known to test on basics rather than high funda ideas. A range of CAT questions can be asked from Ratios and Proportions, Mixtures, Alligations and Averages. Make sure you master the topics. 2IIMs CAT questions bank provides you with CAT questions that can help you gear for CAT Exam CAT 2020.

Question 47: If the product of n distinct positive integers is nn. What is the minimum value of their average if n = 6?

1. 6
2. 10
3. $$frac{46}{6}\\$ 4. 8 ## Learn from 4 time 100%iler.5,000 off on Live Online CAT Course Signup Now! #### 2IIM : Best Online CAT Coaching. ## Best CAT Coaching in Chennai #### CAT Coaching in Chennai - CAT 2020Online Batches Available Now! ### Explanatory Answer ##### Method of solving this CAT Question from Ratio, Mixtures and Averages: How you would relate the product with its average? For the average to be minimum, the numbers should be as close to each other as possible. Why? Let us take a simpler example to check this out. Let us say product of two natural numbers is 64. We could take the two extremes 1 * 64 and 8 * 8. In the first instance the average is 32.5, in the second one the average is 8. The closer the numbers are, the smaller their sum and average are going to be. In our case, we are looking at 66. If the numbers need not have been distinct, then 6 * 6 * 6 * 6 * 6 * 6 would work best. Now, let us anchor around 6 and try to figure out a mechanism of keeping numbers as close to 6 as possible. We need to look for 1. 6 numbers multiplying to give 26 * 36 2. With the numbers as close to 6 as possible Let us get in a 6 obviously, then include 8 and 9 as well. 6 * 8 * 9 = 24 * 33 So, the other 3 numbers should multiply to give 22 * 33. There are more 3’s than 2’s, so perhaps we could include 18. So, we could have 6, 8, 9, 18 – these 4 multiply to give 25 * 35. The last 2 could be 2 and 3. So, we could have 2, 3, 6, 8, 9, 18. These numbers add up to 46 giving us an average of 46/6. Between 2 and 18 the only other numbers that are factors of 66 that we have missed out on are 4 and 12. We will need to see if we can have some combination that includes either one or both of these numbers that can bring down the average. Scenario I – Including 12. Let us start by including 6 and 12. We have accounted for 23 * 32. We have more 2s than 3’s in the system so we can perhaps offset this by including a 9. If we had 6, 9 and 12, the product would be 23 * 34. The remaining 3 numbers should give us 23 * 32. We have already included 6, 9 and 12. So, if we need more multiples of 3, we need either 3 or 18. Merely having 3 does not help as we still need to accommodate 2 more 3’s. If we include 18, our average will go higher anyway. We will have 1, 4, 6, 9, 12, 18. Let us rejig our approach. We need six 3’s. We can include 3, 6, 9 and 12 but that gets us to only 5 3’s. So, we need to have 18 or 24. 24 is too large and would mangle the average so we NEED to have an 18 in the mix. From 3, 6, 9, 12 we need to find 4 more 3’s. We cannot drop 9 from the mix as the other 3 would give us only three more 3’s. So, we will DEFINITELY need to have 9 also. So, we have 9 and 18 in the mix. And 2 of 3, 6 and 12 in the mix. We could have 3, 6, 9, 18 , or 3, 9, 12, 18 or 6, 9, 12, 18 – 3, 6, 9, 18 – Product of remaining two numbers should be 24. Best case scenario would be 3, 6, 9, 18, 2, 8 yielding a sum of 46 3, 9, 12, 18 - – Product of remaining two numbers should be 23. Best case scenario would be 3, 9, 12, 18, 2, 4 yielding a sum of 48. 6, 9, 12, 18 – – Product of remaining two numbers should be 22. Best case scenario would be 6, 9, 12, 18, 1, 4 yielding a sum of 50. Overall best-case scenario is 3, 6, 9, 18, 2, 8 yielding a sum of 46 and an average of 46/6 The question is "What is the minimum value of their average if n = 6?" ##### Hence, the answer is "$\frac{46}{6}\\$". Choice C is the correct answer. ###### Nail CAT 2020 with 2IIM. Live Online Course with 100+ Live Sessions 5,000 Off! ###### Best Indore IPM & Rohtak IPM CoachingSignup and sample 9 full classes for free. Register now! ###### Already have an Account? ###### CAT Coaching in ChennaiCAT 2020Enroll at 30,000/- Online Classroom Batches Starting Now! ###### Best CAT Coaching in ChennaiPrices slashed by Rs 5000/- Attend a Demo Class ##### Where is 2IIM located? 2IIM Online CAT Coaching A Fermat Education Initiative, 58/16, Indira Gandhi Street, Kaveri Rangan Nagar, Saligramam, Chennai 600 093 ##### How to reach 2IIM? Phone:$91) 44 4505 8484
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