How to find percentage - GRE Quantitative Reasoning
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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?
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?
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
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
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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.
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.
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.
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.
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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.
. 
What percentage of the student's loan goes toward housing and tutiton?
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.
.
What percentage of the student's loan goes toward housing and tutiton?
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 = 
.
We then round the percentage to 77.5%.
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 =
We then round the percentage to 77.5%.
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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%?
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%?
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.
which equals 90%.
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
which equals 90%.
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of a 
alcohol solution is mixed with 
of a 
alcohol solution. What percentage of the new mixture is alcohol?
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 
of pure alcohol. To find the % alcohol content, we simply divide the amount of pure alcohol by the total volume, 
or 33% alcohol.
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.
Next, we will break down how much alcohol content is in the 6 gallons of 45% alcohol.
Therefore, the 10 gallon mixture of both mixtures now contains
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A man has three sons, aged 
. In 
the combined age of his sons will be equal to 
of this age. How old is the man currently?
A man has three sons, aged
To solve this problem, we must first add up the ages of all his sons.
. The combined age of his three sons is currently 
.
The problem states that in 
, the combined age of the man's three sons will be 
of his age.
Therefore we add 
to all his sons ages and sum them up once again.
. In 
, the combined age of his three sons will be 
.
Now to find what number 
is 
of, we simply divide 
by 
, 
. This means that in 
, the man will be 
old. To find how old the man is currently, we simply subtract 
from 
, this means that the man is currently 
old.
To solve this problem, we must first add up the ages of all his sons.
The problem states that in
Therefore we add
Now to find what number
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is 25 percent of 80
13 is 
percent of 65
Quantity A: 
Quantity B: 
13 is
Quantity A:
Quantity B:
To attempt this problem, translate the statements into mathematical equations. When you see the word 'is', it very often means 'equals:
is 25 percent of 80:
13 is 
percent of 65:
The two quantities are equal.
To attempt this problem, translate the statements into mathematical equations. When you see the word 'is', it very often means 'equals:
13 is
The two quantities are equal.
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30 is 
percent of 210
20 is 
percent of 160
Quantity A: 
Quantity B: 
30 is
20 is
Quantity A:
Quantity B:
To attempt this problem, translate the statements into mathematical equations. When you see the word 'is', it very often means 'equals:
30 is 
percent of 210:
20 is 
percent of 160:
Since Quantity A has the smaller denominator with an equivalent numerator, it is the larger value. Quantity A is greater.
To attempt this problem, translate the statements into mathematical equations. When you see the word 'is', it very often means 'equals:
30 is
20 is
Since Quantity A has the smaller denominator with an equivalent numerator, it is the larger value. Quantity A is greater.
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is 30 percent of 410
is 25 precent of 480
Quantity A: 
Quantity B: 
Quantity A:
Quantity B:
To attempt this problem, translate the statements into mathematical equations. When you see the word 'is', it very often means 'equals:
is 30 percent of 410
is 25 precent of 480
Quantity A is greater.
To attempt this problem, translate the statements into mathematical equations. When you see the word 'is', it very often means 'equals:
Quantity A is greater.
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is 
percent of 
is 
percent of 
Quantity A: 
Quantity B: 
Quantity A:
Quantity B:
To attempt this problem, translate the statements into mathematical equations. When you see the word 'is', it very often means 'equals:
is 
percent of 
:
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 
Again, this validates our result. We may not know what r and s are, but we know how they relate to each other, and that is enough.
Quantity B is greater.
To attempt this problem, translate the statements into mathematical equations. When you see the word 'is', it very often means 'equals:
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.
Again, this validates our result. We may not know what r and s are, but we know how they relate to each other, and that is enough.
Quantity B is greater.
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is what percent of 
?
To solve this problem, translate 
is what percent of 
into mathematical terms:
To solve this problem, translate
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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?
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?
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
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
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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?
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?
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:
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:
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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?
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?
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
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
Compare your answer with the correct one above
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.
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.
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.
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.
Compare your answer with the correct one above
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.
.
What percentage of the student's loan goes toward housing and tutiton?
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.
.
What percentage of the student's loan goes toward housing and tutiton?
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 = .
We then round the percentage to 77.5%.
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 =
We then round the percentage to 77.5%.
Compare your answer with the correct one above
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%?
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%?
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.
which equals 90%.
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
which equals 90%.
Compare your answer with the correct one above
of a alcohol solution is mixed with of a alcohol solution. What percentage of the new mixture is alcohol?
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 of pure alcohol. To find the % alcohol content, we simply divide the amount of pure alcohol by the total volume, or 33% alcohol.
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.
Next, we will break down how much alcohol content is in the 6 gallons of 45% alcohol.
Therefore, the 10 gallon mixture of both mixtures now contains
Compare your answer with the correct one above
A man has three sons, aged . In the combined age of his sons will be equal to of this age. How old is the man currently?
A man has three sons, aged
To solve this problem, we must first add up the ages of all his sons.
. The combined age of his three sons is currently .
The problem states that in , the combined age of the man's three sons will be of his age.
Therefore we add to all his sons ages and sum them up once again.
. In , the combined age of his three sons will be .
Now to find what number is of, we simply divide by , . This means that in , the man will be old. To find how old the man is currently, we simply subtract from , this means that the man is currently old.
To solve this problem, we must first add up the ages of all his sons.
The problem states that in
Therefore we add
Now to find what number
Compare your answer with the correct one above
is 25 percent of 80
13 is percent of 65
Quantity A:
Quantity B:
13 is
Quantity A:
Quantity B:
To attempt this problem, translate the statements into mathematical equations. When you see the word 'is', it very often means 'equals:
is 25 percent of 80:
13 is percent of 65:
The two quantities are equal.
To attempt this problem, translate the statements into mathematical equations. When you see the word 'is', it very often means 'equals:
13 is
The two quantities are equal.
Compare your answer with the correct one above