In a survey of potential presidential candidates A and B, 30 percent liked A and 48 percent liked B. If the percentage of people who liked only one candidate is twice the percentage who liked both candidates, then what is the percentage of people that liked neither candidate?

I dont understand 78-x=3x, if the percentage of people who liked at least one candidate is twice the percentage of people who liked both candidates, isn't it 78-x=2x?

Perhaps it will help to visualize the problem with a Venn diagram. We see that x + y = 30, y + z = 48, and that x + y + z + U = 100.

percentage that liked only one candidate is x + z percentage that liked both candidates is y percentage that liked neither candidate is U

We have 3 equations to work with (listed above). So lets translate the word problem into symbols:

% liked one candidate = 2 (% liked both) means x + z = 2 ( y )

We want to solve for U, so we have U = 100 - y - (x + z). So we make the substitution U = 100 - 3y, so 3y = 100 - U = 78- y. So 3y = 78 - y. We can now solve for y and substitute to determine the value for U.

Please, be very concentrated when reading question statement. It says “If the percentage of public who liked only one candidate is twice the percentage who liked both candidates”. In the terms of denotation we used it is: (30 – x) + (48 – x) = 2x

For better comprehension, take a look at this Venn diagram:

Let's denote by x the percentage of people that liked both candidates. Then the percentage of people that liked candidate A only is (30 – x). The percentage of people that liked candidate B only is (48 – x). The total percentage of people that liked only one candidate is (30 – x) + (48 – x). We know that it is twice the percentage of people that like, both candidates. Therefore we get equation: (30 – x) + (48 – x) = 2x 78 = 4x x = 19,5

We can find percentage of people that like neither candidate if we deduct from 100% those that like only one candidate as also those who like both candidates.

100 – x – (30 – x) – (48 – x) = (100 – 78) + x = 22 + 19,5 = 41,5 The percent of people that liked neither candidate is 41,5%

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