GRE Quantitative Reasoning - How to find percentage

In Dopolopolis, the first $20,000 dollars of a person's income is taxed at 6%, and any additional income they make is taxed at 10%. If Donny of Dopolopolis paid $2,000 in income tax this year, how much did Donny make?

Explanation

All of the answer choices are above $20,000, so assume that Donny paid the 6% on that:

That means of the $2,000 he paid in taxes

$800 was taken from any income he made above the $20,000 line, and was thus taxed at 10%:

Donny's income must therefore have been

On a 30 question test, Mike answered 60% of the first 20 questions correctly. What percent of the remaining 10 questions does he need to answer correctly for his grade on the entire test to be 70%?

Explanation

To earn a 70%, Mike needs to correctly answer 21 questions (70% of 30). So far, he has answered 12 questions correctly (60% of 20).

Therefore, on the last 10 questions he needs correct answers.

$\frac{9}{10}=\frac{90}{100}=0.9$

which equals 90%.

is what percent of ?

Explanation

To solve this problem, translate is what percent of into mathematical terms:

$72=\frac{x}{100}120$

$x\frac{120}{100}=72$

$x=\frac{100}{120}(72)=\frac{100}{12*10}(12*6)=\frac{100}{10}(6)=10(6)=60$

Forty percent of the 20 professors in a university's physics department are male. If two male professors and two female professors retire, what percentage of the department will be male after the change?

Explanation

Begin by finding out how many of the 20 professors are male:

That means that there are eight males and twelve females currently to sum up to twenty.

Now, two males and two females leave, meaning the faculty is reduced by four:

The amount of males has been reduced by two:

Therefore, the percentage of the faculty that is male can be found as the number of males divided by the total faculty size times 100:

$\frac{6}{16}100=37.5$

A college student takes out a loan of $8,000 per Quarter. A college student's average living cost is represented by the chart below.

. Living_cost_per_quarter

What percentage of the student's loan goes toward housing and tutiton?

Explanation

The sum cost of housing and tutition is $6199.24. To find the percentage of the loan that equals $6199.24, we can set up the equation:

Percentage = $\frac{$6199.24}{$8000}\times 100%=77.49%$ .

We then round the percentage to 77.5%.

A baker is able to sell doughnut holes made from his doughnuts in groups of 6 for $0.50. If he wastes one hole per dozen doughnuts, how much money does he fail to make when selling the holes that can be packaged from 22 dozen doughnuts?

None of the other answers

$3.33

$11

$1.83

Explanation

Given his waste rate, the baker would waste 22 holes. That is the equivalent 3 2/3 bags; however, since that would leave him with a 1/3 of a bag, the waste really is 4 bags of holes. He therefore fails to make 4 * 0.5 or $2 on this batch of 22 dozen doughnuts and holes.

A group of five students averaged 85 points on an exam taken out of 100 total points. If the addition of two additional students raises the group average to 88 points, what is the minimum score that one of those two students can receive? Assume that 100 is the highest score for the exam.

100

95.5

None of the other answers

Explanation

If the average score was 85, we know that the total points earned by the 5 students must be 85 * 5 or 425 points. Upon adding two new scores, we know that the formula for the average will be:

(425 + x) / 7 = 88, where x stands for the combined points earened by the two new students.

Solve for x: 425 + x = 616; x = 191

This means that both students must score 191 points total. If this is the case, we can assume that one student earns the maximum points, i.e. 100. This will provide the minimum amount that could be earned by the other: 191 – 100 = 91.

$\textup{4 gallons}$ of a alcohol solution is mixed with $\textup{6 gallons}$ of a alcohol solution. What percentage of the new mixture is alcohol?

Explanation

For this problem, we must figure out what % alcohol the 10 gallon combined mixture contains. We know that 4 gallons of 15% alcohol is mixed together with 6 gallons of 45% alcohol.

First we will break down how much alcohol content is within the 4 gallons of 15% alcohol. . This means that of the 4 gallons, 0.6 gallons can be considered pure alcohol.

Next, we will break down how much alcohol content is in the 6 gallons of 45% alcohol. . This means that of the 6 gallons, 2.7 gallons can be considered pure alcohol.

Therefore, the 10 gallon mixture of both mixtures now contains $2.7+0.6=3.3\textup{ gallons}$ of pure alcohol. To find the % alcohol content, we simply divide the amount of pure alcohol by the total volume, $\frac{3.3}{10}=0.33$ or 33% alcohol.

A pair of shoes originally cost $40. It is on sale and now only costs $32. The sale was for what percentage off the original cost?

20%

35%

15%

25%

Explanation

The difference in cost is $8 (40 – 32). The sale percentage can be found by figuring out this question: 8 is what percentage of 40? 8/40 = 0.20, or a 20% discount. You can check this. 40 * 0.20 = 8.

40 – 8 = 32

is percent of

Quantity A:

Quantity B:

Quantity B is greater.

Quantity A is greater.

The two quantities are equal.

The relationship cannot be determined.

Explanation

To attempt this problem, translate the statements into mathematical equations. When you see the word 'is', it very often means 'equals:

is percent of :

$s=\frac{r}{100}80=\frac{4}{5}r$

Now before we continue, look at this statement. We see that s is fourth fifths of r. That alone should tell us that r is greater! However, we can continue with the calculations to see what these values are, if only for the sake of robustness. However, on the GRE, once you find the answer like this, fill it in and move on.

is percent of