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## Arithmetic - Mixtures Piglet Question

Today we continue to look at ways to achieve that much desired score of 51 in Quant. Obviously, we don’t need Sheldon Cooper’s smarts to realize that for that revered high score, we must do well on the high level questions but the actual question is – how to do well on the high level questions?

We will illustrate that with the help of a supremely beguiling official question today. We are sure you wouldn’t call an academician’s work exactly thrilling but questions like these do add a decent bit of joie de vivre to our lives. It’s hard to explain the gratification we get when it all falls into place in your mind and you light up with – “shoot, so simple, and yet, it seemed like a monster a few minutes back!” – we basically live for those moments!

Let us first give you some stats which indicate the difficulty level of this question:

95% of people find this question hard. Only 1/3rd of respondents answer it correctly (which includes the ton of people who had tried it before and hence knew the correct answer).

Let us give you the question now:

Question: Each piglet in a litter is fed exactly one-half pound of a mixture of oats and barley. The ratio of the amount of barley to that of oats varies from piglet to piglet, but each piglet is fed some of both grains. How many piglets are there in the litter?

Statement 1: Piglet A was fed exactly 1/4 of the oats today.
Statement 2: Piglet A was fed exactly 1/6 of the barley today.

First think, what concept does it test? Fractions? Ratios? Or is it just a word problem requiring algebraic manipulation?

Actually, none of these. We can look at the question and say straight away that the answer  is (C). It needs no manipulation and no calculation. Of course, what it does need is a solid understanding of the weighted averages principle!

For now, forget the data given in the question.

Consider this:

Say, 10% of total Oats and 20% of total Barley was fed to a piglet.

The question now is – Of the total food (Oats + Barley) what percentage was fed to this piglet?

We hope you agree that it will depend on the ratio of Oats and Barley. If the mixture was only oats, the piglet was fed 10% of the total food. If the mixture was only Barley, the piglet was fed 20% of the total mixture. If the mixture was half oats and half barley, the piglet was fed 15% of the total mixture. If the mixture was 1 part Oats for every 4 parts of Barley, the piglet was fed 18% of the mixture (it is just weighted average with weights being the amount of initial quantity of Oats and Barley). Whatever the case, the piglet was fed more than 10% of total food and less than 20% of total food if the mixture consisted of both Oats and Barley.

If this is not clear, look at this example:

Say a meal consists of a sandwich and a milkshake. You eat 1/2 of the sandwich and drink 1/2 of the milkshake. Can we say that you have had 1/2 of the meal? Sure.
If you eat only 1/4 of the sandwich and drink 1/4 of the milkshake, then you would have had only 1/4 of the meal.
What happens in case you eat 1/2 of the sandwich but drink only 1/4 of the milkshake? In that case, you have had less than 1/2 of the meal but certainly more than 1/4 of the meal, right?

Go through this again till you are satisfied with this logic.

If this sounds good, consider data given in the question – piglet A was fed 25% Oats (1/4 Oats) and 16.66% Barley (1/6 Barley). So definitely, the piglet was fed more than 16.66% (which is 1/6) of the total mixture and less than 25% (which is 1/4) of the total mixture (as reasoned above).  Stay with this idea.

Another piece of information from the question stem: the total food mixture was split equally among all the piglets. Since all piglets got the same quantity of food, we can say that all piglets were fed more than 1/6 of the total mixture but less than 1/4 of the total mixture. Number of piglets has to be an integer, say n. Then, each piglet gets the same amount of food i.e. 1/n of the total mixture. This 1/n must lie between 1/4 and 1/6. Note that the number of pigs i.e. n, must be a positive integer. What integer value can n take? Can it be 7? Will 1/7 lie between 1/6 and 1/4? No. 1/7 will be less than 1/6. Can n be 3? Will 1/3 lie between 1/4 and 1/6? No, because 1/3 will be greater than 1/4. n cannot be greater than 6 or less than 4 because it goes out of range. Only 1/5 lies between 1/4 and 1/6 (such that n is a positive integer). Hence n must be 5.

Notice that we did not need to do any calculations – just looking at the two statements, we can say that 1/n must lie between 1/4 and 1/6 and hence n must be 5.

Questions such as this one set GMAT apart from other tests. It tests you on basic concepts but how!!!