How to solve algebraic word problems - SSAT Upper Level Quantitative
Card 1 of 112
Michael scores a 95, 87, 85, 93, and a 94 on his first 5 math tests. If he wants a 90 average, what must he score on the final math test?
Michael scores a 95, 87, 85, 93, and a 94 on his first 5 math tests. If he wants a 90 average, what must he score on the final math test?
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To solve for the final score:
Add the five past test scores and you get 454. Then set up an algebraic equation where you add 454 to 
, which is the final test score, and divide by six, because you want the average for 6 tests now. You make this equation equal to 90 because that is the average Michael wants and solve for 
:
To solve for the final score:
Add the five past test scores and you get 454. Then set up an algebraic equation where you add 454 to
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If David wants to drive to his friend's house, which is 450 miles away, in 6 hours, what is the average speed David has to drive at?
If David wants to drive to his friend's house, which is 450 miles away, in 6 hours, what is the average speed David has to drive at?
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Plug in the the values for distance and time, and solve for rate.
and
Plug in the the values for distance and time, and solve for rate.
and
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If the sum of the smallest and largest of three consecutive even numbers is 28, what is the value of the second largest number in the series?
If the sum of the smallest and largest of three consecutive even numbers is 28, what is the value of the second largest number in the series?
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The three numbers would be
, 
, and 
.
Add the first and third value and you get
and
.
The second largest value is 
.
The three numbers would be
Add the first and third value and you get
and
The second largest value is
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Beth and Sam are 500 miles apart. If Beth travels at 60mph and leaves her house at 1pm, what time will she arrive at Sam's house?
Beth and Sam are 500 miles apart. If Beth travels at 60mph and leaves her house at 1pm, what time will she arrive at Sam's house?
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Using 
, the time would be 
hours, which is 
hours and 
minutes. If you add that to 1pm, you get 9:20pm.
Using
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Greg is trying to fill a 16 oz. bottle with water. If Greg fills the bottle at 1 oz per second and the bottle leaks .2 oz per second, how long would it take for Greg to fill the bottle?
Greg is trying to fill a 16 oz. bottle with water. If Greg fills the bottle at 1 oz per second and the bottle leaks .2 oz per second, how long would it take for Greg to fill the bottle?
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You first find the rate at which the bottle is being filled at, which is
.
Then you divide the entire bottle, which is 
by the rate of 
, and you get 
.
You first find the rate at which the bottle is being filled at, which is
Then you divide the entire bottle, which is
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Barbara went to the store and bought a shirt for 
dollars. It had been discounted by 20%. What is the original price of the shirt?
Barbara went to the store and bought a shirt for
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To get the final price, you use 
as the variable for the original price. So,
, and divide by 0.8 on both sides of the equations, and you get
for the original price of the shirt.
To get the final price, you use
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One angle of a pentagon measures 
. The other four angles are congruent to one another. Give the measure of one of those angles in terms of 
.
One angle of a pentagon measures
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The total of the measures of the angles of a pentagon is
.
One angle measures 
; since the other four angles have the same measure as one another, we let 
be their common measure. Then we set up the equation and solve for 
:
The total of the measures of the angles of a pentagon is
One angle measures
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Two inlet pipes lead into a large water tank. One pipe can fill the tank in 45 minutes; the other can fill it in 40 minutes. To the nearest tenth of a minute, how long would it take the two pipes together to fill the tank if both were opened at the same time?
Two inlet pipes lead into a large water tank. One pipe can fill the tank in 45 minutes; the other can fill it in 40 minutes. To the nearest tenth of a minute, how long would it take the two pipes together to fill the tank if both were opened at the same time?
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Look at the work rates as "tanks per minute", not "minutes per tank".
The two pipes can fill the tank up at 
tanks per minute and 
tanks per minute.
Let 
be the time it took, in minutes, to fill the tank up. Then, since rate multiplied by time is equal to work, then two pipes filled up 
and 
tanks; together, they filled up 
tank - one tank. This sets up the equation to be solved:
minutes
Look at the work rates as "tanks per minute", not "minutes per tank".
The two pipes can fill the tank up at
Let
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The inlet pipe to a large water tank can fill the tank in 25 minutes; the drain of the tank can empty it in 55 minutes.
Once, the drain was left open by mistake when the tank was being filled. The mistake was not caught until the tank was full. To the nearest tenth of a minute, how long did it take to fill the tank?
The inlet pipe to a large water tank can fill the tank in 25 minutes; the drain of the tank can empty it in 55 minutes.
Once, the drain was left open by mistake when the tank was being filled. The mistake was not caught until the tank was full. To the nearest tenth of a minute, how long did it take to fill the tank?
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Look at the work rates as "tanks per minute", not "minutes per tank".
The inlet pipe can fill the tank at a rate of 
tank per minute. The drain is emptying the tank at a rate of 
tank per minute.
Let 
be the time it took, in minutes, to fill the tank up. Then, since rate multiplied by time is equal to work, then the inlet pipe let in 
tank of water, but the drain let out 
tank. Since they were working against each other, the work of the latter is subtracted from that of the former, and we can set up and solve the equation:
minutes
Look at the work rates as "tanks per minute", not "minutes per tank".
The inlet pipe can fill the tank at a rate of
Let
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John and Julie represented West High in a math contest. John outscored Julie by 16 points; as a team, they scored 80 points.
David and Dana represented East High in the same contest. Dana outscored David by 10 points; as a team, they scored 60 points.
Arrange the four students from highest score to lowest score.
John and Julie represented West High in a math contest. John outscored Julie by 16 points; as a team, they scored 80 points.
David and Dana represented East High in the same contest. Dana outscored David by 10 points; as a team, they scored 60 points.
Arrange the four students from highest score to lowest score.
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Let Julie's score be 
. Then John's score was 
. Since the sum of their scores was 80,
Julie scored 32, and John scored 16 higher, or 48.
Let David's score be 
. Then Dana's score was 
. Since the sum of their scores was 60,
David's score was 25, and Dana's score was 10 higher, or 35.
In descending order of score, the students were John, Dana, Julie, David.
Let Julie's score be
Julie scored 32, and John scored 16 higher, or 48.
Let David's score be
David's score was 25, and Dana's score was 10 higher, or 35.
In descending order of score, the students were John, Dana, Julie, David.
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Which of the following sentences is represented by the equation
?
Which of the following sentences is represented by the equation
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The expression 
can be written as "the product of five and a number".
The expression 
can be written as " the difference of the number and seven"; the expression 
can be written as "three times the difference of the number and seven".
Therefore, the equation
can be written as
"The product of five and a number is equal to three times the difference of the number and seven".
The expression
The expression
Therefore, the equation
can be written as
"The product of five and a number is equal to three times the difference of the number and seven".
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Four of the five numbers of a set are:
If the average of the five numbers is 
, give the fifth number in terms of 
.
Four of the five numbers of a set are:
If the average of the five numbers is
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Call the fifth number 
. If the average of the five numbers is 
, then 
is the sum of the numbers divided by five:
Solve for 
in the equation:
The fifth number is 
.
Call the fifth number
Solve for
The fifth number is
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Bobby is five years older than his sister Geri. In ten years, Geri's age will be thirty years less than twice Bobby's age. How old is Bobby now?
Bobby is five years older than his sister Geri. In ten years, Geri's age will be thirty years less than twice Bobby's age. How old is Bobby now?
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Let 
be Bobby's age. Since Geri is five years younger, her age is five years subtracted from Bobby's age, or 
.
In ten years, both will have had ten years added to their ages, so Bobby will be 
and Geri will be 
.
Twice Bobby's age will be 
, and Geri will be thirty years less than this, or 
. Set the two expressions for Geri's age in five years equal to each other and solve for 
.
Distribute:
Collect like terms:
Apply the subtraction property of equality and collect like terms:
Apply the addition property of equality:
Bobby is fifteen years old.
Let
In ten years, both will have had ten years added to their ages, so Bobby will be
Twice Bobby's age will be
Distribute:
Collect like terms:
Apply the subtraction property of equality and collect like terms:
Apply the addition property of equality:
Bobby is fifteen years old.
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The mean of 
, 
, 
, and 
is 125; the mean of 
, 
, 
, and 
is 150. Which of the following gives the sum of 
and 
if the mean of 
, 
, 
, 
, 
, and 
is 
?
The mean of
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Since the mean of the four numbers 
, 
, 
, and 
is 125,
Similarly,
Add the two sums:
The mean of 
, 
, 
, 
, 
, and 
is 
, so
So:
Since the mean of the four numbers
Similarly,
Add the two sums:
The mean of
So:
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Eddie, Freida, Grant, Helene, and Ira represented Washington High in a math contest. The team score was the sum of the three highest scores. Grant outscored Eddie and Freida; Helene outscored Grant; Freida outscored Ira. Which three students' scores were added to determine the team score?
Eddie, Freida, Grant, Helene, and Ira represented Washington High in a math contest. The team score was the sum of the three highest scores. Grant outscored Eddie and Freida; Helene outscored Grant; Freida outscored Ira. Which three students' scores were added to determine the team score?
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Let 
be Eddie's, Freida's, Grant's, Helene's and Ira's scores. Each of the following statements can be translated into inequalities as follows:
Grant outscored Eddie and Freida:
Helene outscored Grant:
Freida ourtscored Ira:
The first and third statements can be combined to form the three-part inequality:
The second, third, and fourth statements can be combined to form the four-part inequality:
Since Helene and Grant were the top two finishers, their scores were counted. However, it cannot be determined which student finished third from these statements. Therefore, insufficient information is given to answer the question.
Let
Grant outscored Eddie and Freida:
Helene outscored Grant:
Freida ourtscored Ira:
The first and third statements can be combined to form the three-part inequality:
The second, third, and fourth statements can be combined to form the four-part inequality:
Since Helene and Grant were the top two finishers, their scores were counted. However, it cannot be determined which student finished third from these statements. Therefore, insufficient information is given to answer the question.
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Which of the following sentences is represented by the equation
?
Which of the following sentences is represented by the equation
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is the square root of 
, which in turn can be written as "the sum of a number and five"; 
can subsequently be written as "the square root of the sum of a number and five". Since 
, 
is six greater than 
, the number, so the equation can be stated as "The square root of the sum of a number and five is six greater than the number."
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In a math contest, the team from Jefferson High comprised Velma, Wendell, Yancy, and Zack. Velma outscored Zack by 12 points, Zack outscored Yancy by 22 points, and Wendell scored 98 points. The team score was 391. Order the four students from greatest score to least score.
In a math contest, the team from Jefferson High comprised Velma, Wendell, Yancy, and Zack. Velma outscored Zack by 12 points, Zack outscored Yancy by 22 points, and Wendell scored 98 points. The team score was 391. Order the four students from greatest score to least score.
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If we let 
be Yancy's xcore, then, since Zack outscored Yancy by 22, Zack scored 
. Velma outscored Zack by 12, so Velma scored 
.
The sum of the four scores is the team score, or 391. Since Wendell scored 98, we can add the four expressions to get 391, then solve for 
:
Yancy scored 79; Zack scored 22 more than Yancy, or 101; Velma scored 12 more than Zack, or 113. Wendell scored 98, so the correct ordering, greatest to least, is: Velma, Zack, Wendell, Yancy.
If we let
The sum of the four scores is the team score, or 391. Since Wendell scored 98, we can add the four expressions to get 391, then solve for
Yancy scored 79; Zack scored 22 more than Yancy, or 101; Velma scored 12 more than Zack, or 113. Wendell scored 98, so the correct ordering, greatest to least, is: Velma, Zack, Wendell, Yancy.
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What is the mean of the set below?
What is the mean of the set below?
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The first step is to convert the set 
to fractions that have a common denominator of 12. This gives us:
The mean is then calculated by dividing the sum of the numbers in the set by the number of items in the set.
The sum of the items in the set is: 
There are 4 items in the set, so the sum must be divided by 4 (or multiplied by 
).
This results in:
The first step is to convert the set
The mean is then calculated by dividing the sum of the numbers in the set by the number of items in the set.
The sum of the items in the set is:
There are 4 items in the set, so the sum must be divided by 4 (or multiplied by
This results in:
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In his first seven games of the season LeBron scores 11, 31, 22, 32, 41, 32, and 22 points. How many points would he have to score in his next game in order to average 30 points per game for his first eight games of the season?
In his first seven games of the season LeBron scores 11, 31, 22, 32, 41, 32, and 22 points. How many points would he have to score in his next game in order to average 30 points per game for his first eight games of the season?
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For this question you can simply add up the points LeBron has scored thus far (191), determine how many points he would need through 8 games to be averaging 30 per game (240) and subtract to get 49.
If that's more adding than you want to do, though, you can also take the total number of points that LeBron is under 30 in each of the games when he failed to reach that mark 
and add these to get 
. Then combine this with the total number of points that he is over 30 in the rest of the games 
to get a net of 
. So in his 8th game he will need to score the 30 points, but 19 extra to make up for the deficit: 
.
For this question you can simply add up the points LeBron has scored thus far (191), determine how many points he would need through 8 games to be averaging 30 per game (240) and subtract to get 49.
If that's more adding than you want to do, though, you can also take the total number of points that LeBron is under 30 in each of the games when he failed to reach that mark
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The price of a nugget is 
units, which is two more than twice the value of a bronze object. What is the price of the bronze object?
The price of a nugget is
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Write the word problem in terms of a mathematical equation. Let 
be the value of the bronze object.
Solve for 
.
The value of the bronze object is 
units.
Write the word problem in terms of a mathematical equation. Let
Solve for
The value of the bronze object is
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