How to evaluate algebraic expressions - GRE Quantitative Reasoning
Card 1 of 648
Quantitative Comparison
A
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B
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Quantitative Comparison
A
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B
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Since both values are negative, in both situations the final result of the fraction will be positive regardless of the absolute value sign. Additionally, the definition of the exponent "–1" means the value is becoming its reciprocal. Thus, "x" → "1/x" and "1/y" → "y", equating to y/x.
Since both values are negative, in both situations the final result of the fraction will be positive regardless of the absolute value sign. Additionally, the definition of the exponent "–1" means the value is becoming its reciprocal. Thus, "x" → "1/x" and "1/y" → "y", equating to y/x.
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Quantitative Comparison
0 < x < 1
A
---
(2x + 5)/(x2)
B
---
5x
Quantitative Comparison
0 < x < 1
A
---
(2x + 5)/(x2)
B
---
5x
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Since A is a fraction with an exponential term in the denominator, its maximum value is when x is at a minimum. In B, the maximum value is when x approaches its maximum. Therefore, we can check whether there is overlap between the two quantities: No matter how close to either 0 or 1 x reaches, A will always be greater than B. (In fact, the minimum value for A is ~7, while the maximum value of B is ~5)
Be sure to keep your value of x consistent when plugging between the two fractions! The question asks for when they have the same x-value, not for when they are solved independently.
Since A is a fraction with an exponential term in the denominator, its maximum value is when x is at a minimum. In B, the maximum value is when x approaches its maximum. Therefore, we can check whether there is overlap between the two quantities: No matter how close to either 0 or 1 x reaches, A will always be greater than B. (In fact, the minimum value for A is ~7, while the maximum value of B is ~5)
Be sure to keep your value of x consistent when plugging between the two fractions! The question asks for when they have the same x-value, not for when they are solved independently.
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If the average (arithmetic mean) of the above terms is 
, what is the median?
If the average (arithmetic mean) of the above terms is
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Since we know the mean is 7, and the number of terms is 5, we can first find the value of x:
((x – 2)+ (x) + (x + 1) + (x + 4) + (x + 7))/5 = (5x + 10)/5 = 7
x = 5
Since the terms are already listed in order, we can simpy plug in x = 5 to the middle value to get 5 + 1 = 6 as the median term.
Since we know the mean is 7, and the number of terms is 5, we can first find the value of x:
((x – 2)+ (x) + (x + 1) + (x + 4) + (x + 7))/5 = (5x + 10)/5 = 7
x = 5
Since the terms are already listed in order, we can simpy plug in x = 5 to the middle value to get 5 + 1 = 6 as the median term.
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Quantitative Comparison
Quantity A: x
Quantity B: 2_x_
Quantitative Comparison
Quantity A: x
Quantity B: 2_x_
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For a quantitative comparison question such as this one, it is best to first plug in the numbers 0, 1, and –1. Plugging in 0 gets the same answer for both columns. Plugging in 1 makes Quantity B bigger. Plugging in –1 makes Quantity A bigger. Therefore the answer cannot be determined.
For a quantitative comparison question such as this one, it is best to first plug in the numbers 0, 1, and –1. Plugging in 0 gets the same answer for both columns. Plugging in 1 makes Quantity B bigger. Plugging in –1 makes Quantity A bigger. Therefore the answer cannot be determined.
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A store sells 17 coffee mugs for $169. Some of the mugs are $12 each and some are $7 each. How many $7 coffee mugs were sold?
A store sells 17 coffee mugs for $169. Some of the mugs are $12 each and some are $7 each. How many $7 coffee mugs were sold?
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The answer is 7.
Write two independent equations that represent the problem.
x + y = 17 and 12_x_ + 7_y_ = 169
If we solve the first equation for x, we get x = 17 – y and we can plug this into the second equation.
12(17 – y) + 7_y_ = 169
204 – 12_y_ + 7_y_ =169
–5_y_ = –35
y = 7
The answer is 7.
Write two independent equations that represent the problem.
x + y = 17 and 12_x_ + 7_y_ = 169
If we solve the first equation for x, we get x = 17 – y and we can plug this into the second equation.
12(17 – y) + 7_y_ = 169
204 – 12_y_ + 7_y_ =169
–5_y_ = –35
y = 7
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How many square tiles, each with an area of 49 square inches, must be used to completely cover a floor with a width of 98 inches and a length of 49 inches.
How many square tiles, each with an area of 49 square inches, must be used to completely cover a floor with a width of 98 inches and a length of 49 inches.
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The answer is 98.
If the area of each square tile is 49 then each side of the tile is 7. The width of the floor is 98 so 14 tiles would fit across. The length of the floor is 49 so 7 tiles would fit down.
14 tiles * 7 tiles = 98 tiles total
The answer is 98.
If the area of each square tile is 49 then each side of the tile is 7. The width of the floor is 98 so 14 tiles would fit across. The length of the floor is 49 so 7 tiles would fit down.
14 tiles * 7 tiles = 98 tiles total
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Kim has 22 coins made up of quarters, nickles, and dimes that total $2.45. Kim has twice as many nickles as quarters. How many dimes does she have?
Kim has 22 coins made up of quarters, nickles, and dimes that total $2.45. Kim has twice as many nickles as quarters. How many dimes does she have?
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The answer is 7.
Let us first write down three equations that represent the problem:
n + d + q = 22
2_q_ = n
5_n_ + 10_d_ + 25_q_ = 245
Lets plug the second equation into the first and third equations:
(2_q)_ + d + q = 22
5(2_q_) + 10_d_ + 25_q_ = 245
Solve the first equation for d and plug into the last equation:
d = 22 – 3_q_
10q + 10(22 – 3_q_) + 25_q_ = 245
Solve for q.
220 – 30_q_ + 25_q_ + 10_q_ = 245
5_q_ = 25
q = 5
Therefore, n = 10 and d = 7
The answer is 7.
Let us first write down three equations that represent the problem:
n + d + q = 22
2_q_ = n
5_n_ + 10_d_ + 25_q_ = 245
Lets plug the second equation into the first and third equations:
(2_q)_ + d + q = 22
5(2_q_) + 10_d_ + 25_q_ = 245
Solve the first equation for d and plug into the last equation:
d = 22 – 3_q_
10q + 10(22 – 3_q_) + 25_q_ = 245
Solve for q.
220 – 30_q_ + 25_q_ + 10_q_ = 245
5_q_ = 25
q = 5
Therefore, n = 10 and d = 7
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A specialty socket wrench, typically priced at $29.99, is on sale for 30% off. An additional 45% is discounted at the register. What is the final sale price of the wrench?
A specialty socket wrench, typically priced at $29.99, is on sale for 30% off. An additional 45% is discounted at the register. What is the final sale price of the wrench?
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The answer is $11.55
The original cost is $29.99 but we are going to discount 30%, meaning we will only pay 70%. The new prices is 29.99 x 0.70 = $20.99.
The new price is then dicounted an additional 45%, meaning we will only pay 55% of the new price. The final price is 20.99 x 0.55 = $11.55.
The answer is $11.55
The original cost is $29.99 but we are going to discount 30%, meaning we will only pay 70%. The new prices is 29.99 x 0.70 = $20.99.
The new price is then dicounted an additional 45%, meaning we will only pay 55% of the new price. The final price is 20.99 x 0.55 = $11.55.
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Quantitative Comparison
x is an integer.
Quantity A: (x + 1)2
Quantity B: (x – 1)2
Quantitative Comparison
x is an integer.
Quantity A: (x + 1)2
Quantity B: (x – 1)2
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When picking numbers, we should always try to plug in the numbers 0, 1, and –1 first.
First try 0:
(0 + 1)2 = 1
(0 – 1)2 = 1
Here the two quantities are equal.
Now try 1:
(1 + 1)2 = 4
(1 – 1)2 = 0.
Here Quantity A is greater.
Therefore the relationship cannot be determined.
When picking numbers, we should always try to plug in the numbers 0, 1, and –1 first.
First try 0:
(0 + 1)2 = 1
(0 – 1)2 = 1
Here the two quantities are equal.
Now try 1:
(1 + 1)2 = 4
(1 – 1)2 = 0.
Here Quantity A is greater.
Therefore the relationship cannot be determined.
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Barry's workout today consists of 10 squats every minute on the minute and 6 situps every other minute for 1 hour. How many squats and situps does Barry do in total?
Barry's workout today consists of 10 squats every minute on the minute and 6 situps every other minute for 1 hour. How many squats and situps does Barry do in total?
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squats: 10 squats * 60 minutes = 600 squats
situps: 6 situps * 30 minutes = 180 situps
total = 600 + 180 = 780
squats: 10 squats * 60 minutes = 600 squats
situps: 6 situps * 30 minutes = 180 situps
total = 600 + 180 = 780
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Quantitative Comparison
Quantity A: 
Quantity B: 
Quantitative Comparison
Quantity A:
Quantity B:
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If x = 0, (x – 5)2 = 25 and (x + 5)2 = 25, so the quantities are equal.
If x = 1, (x – 5)2 = (–4)2 = 16 and (x + 5)2 = 62 = 36, so B is greater.
This is a contradiction, so the answer cannot be determined.
If x = 0, (x – 5)2 = 25 and (x + 5)2 = 25, so the quantities are equal.
If x = 1, (x – 5)2 = (–4)2 = 16 and (x + 5)2 = 62 = 36, so B is greater.
This is a contradiction, so the answer cannot be determined.
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Fran has eight more chickens that Jan. If Fran gives two of her chickens to Jan, Fran will have twice as many chickens as Jan. How many chickens does Fran currently have (before giving two away to Jan)?
Fran has eight more chickens that Jan. If Fran gives two of her chickens to Jan, Fran will have twice as many chickens as Jan. How many chickens does Fran currently have (before giving two away to Jan)?
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Set up two different equations:
and 
Note the importance in the second equation of accounting for the exchange of 2 chickens for both quantities: Jan loses the two that she gives to Fran, and Fran gains the two that she receives.
Solve for 
by substituting 
for 
in the second equation:
Solving for 
results in 
.
Set up two different equations:
Note the importance in the second equation of accounting for the exchange of 2 chickens for both quantities: Jan loses the two that she gives to Fran, and Fran gains the two that she receives.
Solve for
Solving for
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Let 
and 
be integers such that 
and 
.
Quantity A Quantity B
0
Let
Quantity A Quantity B
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The quantity 
produces a minimum of 
and a maximum of 4, which are less than and greater than 0, respectively. Therefore, the answer cannot be determined from the information given.
The quantity
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Let 
be a positive integer such that one less than three-eighths of 
is the third prime integer. What is the value of 
?
Let
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The third prime integer is 5, since 1 is not considered a prime number.
Working backwards,
The third prime integer is 5, since 1 is not considered a prime number.
Working backwards,
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Quantity A Quantity B
35 
Quantity A Quantity B
35
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You don't actually need to solve for the values of x and y. You just want to know the sum of x and y. Adding the two equations together you get
You don't actually need to solve for the values of x and y. You just want to know the sum of x and y. Adding the two equations together you get
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In a four-digit positive interger 
, the thousand's digit is three times the units digit.
Quantitiy A Quantitiy B
Unit's digit of 
4
In a four-digit positive interger
Quantitiy A Quantitiy B
Unit's digit of
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If we set the quantities equal, and the units digit of 
is 4, then the thousands digit which is three times the units digit would be 12, which doesnt make sense at all because a digit must be one of the integers 0 through 9. So 4 is too big to be the units digit of 
. Therefore Quantity B is greater.
If we set the quantities equal, and the units digit of
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The arithmetic mean of 
is 6. What is the value of 
?
The arithmetic mean of
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The arithmetic mean is the sum of all numbers in a series divided by the number of items in the series. There are 5 items in the series, so they must sum to
.
If we subtract 
from 
, you are left with 
so therefore 
must sum to 
.
The arithmetic mean is the sum of all numbers in a series divided by the number of items in the series. There are 5 items in the series, so they must sum to
If we subtract
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If 
, 
and 
, what is the value of 
?
If
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If you add all three equations together you get 
If you add all three equations together you get
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Ten years ago, when Molly was born, Ginny was twice as old as Lily. Now Ginny is twice as old as Molly. How old is Lily now?
Ten years ago, when Molly was born, Ginny was twice as old as Lily. Now Ginny is twice as old as Molly. How old is Lily now?
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If Molly was born 10 years ago, she is now 10 years old. If Ginny is twice Molly's age now, she is 20 years old, which made her 10 years old when Molly was born. If Ginny was 10 years old when Molly was born and was twice as old as Lily, Lily must have been 5 years old. If Lily was 5 years old 10 years ago, she must be 15 years old currently.
If Molly was born 10 years ago, she is now 10 years old. If Ginny is twice Molly's age now, she is 20 years old, which made her 10 years old when Molly was born. If Ginny was 10 years old when Molly was born and was twice as old as Lily, Lily must have been 5 years old. If Lily was 5 years old 10 years ago, she must be 15 years old currently.
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Apoorwa has $117 in $1, $5, and $20 bills. She has twice as many $20's as $1's and the number of $20's times the number of $1's is one more than then number of $5's. How many $5's does Apoorwa have?
Apoorwa has $117 in $1, $5, and $20 bills. She has twice as many $20's as $1's and the number of $20's times the number of $1's is one more than then number of $5's. How many $5's does Apoorwa have?
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The answer is 7.
First we know we have three variables so we need three different equations. Let x represent $1 bills, y represent $5 bills, and z represent $20 bills. Looking at the word problem we know that our equations are:
-
1_x_ + 5_y_ + 20_z_ = 117
-
2_x_ = z
-
xz – 1 = y
Plug equation 2 into equation 3 to get y = x(2_x_) – 1. Now plug this new equation 3 and equation 2 into equation 1 giving us
so 
but we know this isn't the answer becuase we can't have negative dollar bills and we also can't have 0.1 parts of a dollar bill.
and 
thus we know we have two $1 bills
this is the number of $5 bills
The answer is 7.
First we know we have three variables so we need three different equations. Let x represent $1 bills, y represent $5 bills, and z represent $20 bills. Looking at the word problem we know that our equations are:
-
1_x_ + 5_y_ + 20_z_ = 117
-
2_x_ = z
-
xz – 1 = y
Plug equation 2 into equation 3 to get y = x(2_x_) – 1. Now plug this new equation 3 and equation 2 into equation 1 giving us
so
and
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