This is hard to confess publicly but I must because it is a prime example of how GMAT takes advantage of our weaknesses – A couple of days back, I answered a 650 level question of weighted averages incorrectly. Those of you who have been following my blog would understand that it was an unpleasant surprise – to say the least. I know my weighted averages quite well, thank you! For this comedown, I blame the treachery of GMAT because it knows how to get you when you become too complacent. The takeaway here is – no matter how easy and conventional the question seems, you MUST read it carefully.
Let me share that particular question with you. I will also share two solutions which give you two different answers. It is an exercise for you to figure out which one is the correct solution (that is, if one of them is the correct solution). Needless to say, the error in the solution(s) is conceptual and very easy to see (not some sly calculation mistake). It’s just that in your haste, it’s very easy to miss this important point. I hope to see some comments with some good explanations.
Question: The price of each hair clip is ¢ 40 and the price of each hair band is ¢ 60. Rashi selects a total of 10 clips and bands from the store, and the average (arithmetic mean) price of the 10 items is ¢ 56. How many bands must Rashi put back so that the average price of the items that she keeps is ¢ 52?
Price of each clip (Pc) = 40
Price of each band (Pb) = 60
Average price of each item (Pavg) = 56
Wc/Wb = (Pb – Pavg)/(Pavg – Pc) = (60 – 56)/(56 – 40) = 1/4 (our weighted average formula)
Since the total number of items is 10, number of clips = 1*2 = 2 and number of bands = 4*2 = 8
If the average price is changed to 52,
Wc/Wb = (Pb – Pavg)/(Pavg – Pc) = (60 – 52)/(52 – 40) = 2/3
Now the ratio has changed to 2:3. This gives us number of clips as 4 and number of bands as 6.
Since previously she had 8 bands and now she has 6 bands, she must have put back 2 bands.
Say the number of hair clips is C and the number of hair bands is 10 – C.
(40C + 60(10 – C))/10 = 56 (Using the formula: Average = Sum/Number of items)
On solving, you get C = 2
Number of clips is 2 and number of bands is (C – 2) = 8.
Now, let’s consider the scenario when she puts back some bands, say x.
(2*40 + (8 – x)*60)/(10 – x) = 52
On solving, you get x = 5
So she puts back 5 bands so that the average price is 52.
Obviously, there is only one correct answer. It’s your job to figure out whether it is (B) or (E) or some third option. Also what’s wrong with one or both of these solutions?
Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!