CCAT Mathematical Reasoning: Weighted Averages

Maths: Weighted Averages

This is a somewhat time consuming question type, which is asked in some (not all) CCAT tests. It will be among the last two or three questions in the test. The questions themselves are not particularly difficult. If you get the question in your test, you can decide, based on time remaining, whether to attempt it or skip.

It is easiest to explain this problem type with an example:

Q1: Suppose that in an election, voter preference is sharply divided along religious lines. 40% of people belonging to religion R1 will vote for Candidate A, the rest will vote for Candidate B. 70% of people belonging to religion R2 will vote for Candidate A, the rest for Candidate B. Which of the following represents the lowest percentage of voters that must belong to religion R1 on election day, in order that Candidate B wins the election?

56%, 61%,65%, 67%, 70%

Answer:

Let the number of people belonging to Religion R1 = r1.

Let the number of people belonging Religion R2 = r2.

40% of r1, and 70% of r2 vote for A.

Hence, number of votes for A = 0.4r1 + 0.7r2

Similarly, votes for B = 0.6r1 + 0.3r2

For B to win, 0.6r1 + 0.3r2 > 0.4r1 + 0.7r2

Or, 0.2r1 > 0.4r2

Or, r1 > 2r2

This is the condition needed for B to win.

Now, we have to find the percentage of r1, and this is equal to

[(r1) / (r1 + r2)] * 100

Now r1 must be greater than 2r2, so at a minimum, r1 = 2r2

Hence, the bolded part becomes

[(2r2) / (2r2 + r2)] * 100

= (2r2/3r2) * 100

= 66.67%

But, r1 must actually be greater than this. So the smallest value of r1, greater than 66.67%, in the answer choices, is 67%, and this is the correct answer.

Now, try this (solution follows):

Q-1: A company has white collar and blue collar employees. There is a plan to build either a library or a gym. Employees have voted to show their preference. 20% of white collar employees vote for the gym, and others for the library. 40% of blue collar employees vote for the library and others for the gym. What should be the minimum percentage of white collar employees in order that the library gets built?

Solution:

Q-1: A company has white collar and blue collar employees. There is a plan to build either a library or a gym. Employees have voted to show their preference. 20% of white collar employees vote for the gym, and others for the library. 40% of blue collar employees vote for the library and others for the gym. What should be the minimum percentage of white collar employees in order that the library gets built?

22%, 23%, 24%, 25%, 26%

Answer-1:

Let the number of white collar employees = w

Let the number of blue collar employees = b

Then, votes for library = 0.8w + 0.4b

Votes for gym = 0.2w + 0.6b

For the library to be built, it should get more votes:

Or, 0.8w + 0.4b > 0.2w + 0.6b

Or, 0.6w > 0.2b

Or, 3w > b

So, at a minimum, w = b/3 for the library to be built.

Now, percentage of white collar employees = (w / w+b) * 100

Let us substitute the value of w from the bolded step:

= ( (b/3) / (b/3 + b) ) * 100

= ( (⅓) / (4/3) ) * 100

= 25%

Hence, the percentage of white collar employees must be more than 25% for the library to get built. (If it is 25%, then both the gym and library would get equal votes).

The only choice which fits is 26%.

Coming up next: CCAT Logical Reasoning — Assertions

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