Algebraic Fractions - PSAT Math
Card 1 of 644
Simplify the expression
.
Simplify the expression
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The expression must first be factored in order to be solvable. Both the numerator and the denominator can be factored, which would give
.
can be divided from both the numerator and the denominator to give
The expression must first be factored in order to be solvable. Both the numerator and the denominator can be factored, which would give
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Simplify the expression
.
Simplify the expression
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The expression can be rewritten as
The expression can be rewritten as
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If 
, what is the value of 
?
If
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To solve this question, substitute -5 in for x in the numerator and denominator. Remember that the square of a negative number is positive.
45 / -9 = -5
To solve this question, substitute -5 in for x in the numerator and denominator. Remember that the square of a negative number is positive.
45 / -9 = -5
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If (t-5)/2t =13/19, what is the value of t?
If (t-5)/2t =13/19, what is the value of t?
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We start by cross multiplying to get the equation 26t=19t-95. We then subtract 19t from 26t giving us 7t= -95. We then divide by 7, giving us t= (-95)/7.
We start by cross multiplying to get the equation 26t=19t-95. We then subtract 19t from 26t giving us 7t= -95. We then divide by 7, giving us t= (-95)/7.
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x = 1/2
What does 1/x + 1/(x + 4) equal?
x = 1/2
What does 1/x + 1/(x + 4) equal?
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1/x + 1/(x+4) =
1/(1/2) + 1/ (1/2 + 4) =
1/ (1/2) + 1 / (9/2) =
2 + 2/9
20/9
1/x + 1/(x+4) =
1/(1/2) + 1/ (1/2 + 4) =
1/ (1/2) + 1 / (9/2) =
2 + 2/9
20/9
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If 
, then what is the value of 
?
If 
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cross multiply:
(6)(19) = 9x
114=9x
x = 38/3
cross multiply:
(6)(19) = 9x
114=9x
x = 38/3
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If x/3 = 50, then what is x/10 equal to?
If x/3 = 50, then what is x/10 equal to?
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1. Solve for x in x/3 = 50
2. x = 150
3.Substitute 150 for x in x/10
4. x = 15
1. Solve for x in x/3 = 50
2. x = 150
3.Substitute 150 for x in x/10
4. x = 15
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The numerator of a fraction is the sum of 4 and 5 times the denominator. If you divide the fraction by 2, the numerator is 3 times the denominator. Find the simplified version of the fraction.
The numerator of a fraction is the sum of 4 and 5 times the denominator. If you divide the fraction by 2, the numerator is 3 times the denominator. Find the simplified version of the fraction.
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Let numerator = N and denominator = D.
According to the first statement,
N = (D x 5) + 4.
According to the second statement, N / 2 = 3 * D.
Let's multiply the second equation by –2 and add itthe first equation:
–N = –6D
+\[N = (D x 5) + 4\]
=
–6D + (D x 5) + 4 = 0
–1D + 4 = 0
D = 4
Thus, N = 24.
Therefore, N/D = 24/4 = 6.
Let numerator = N and denominator = D.
According to the first statement,
N = (D x 5) + 4.
According to the second statement, N / 2 = 3 * D.
Let's multiply the second equation by –2 and add itthe first equation:
–N = –6D
+\[N = (D x 5) + 4\]
=
–6D + (D x 5) + 4 = 0
–1D + 4 = 0
D = 4
Thus, N = 24.
Therefore, N/D = 24/4 = 6.
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The maximum number of sweaters that Lauren can sew every day is equal to s, and the amount, in cents, that she charges for each sweater is equal to c. Which of the following expressions is equivalent to the maximum amount of money that Lauren can make, in dollars, after three weeks?
The maximum number of sweaters that Lauren can sew every day is equal to s, and the amount, in cents, that she charges for each sweater is equal to c. Which of the following expressions is equivalent to the maximum amount of money that Lauren can make, in dollars, after three weeks?
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The amount of money that Lauren can make depends on the number of sweaters that she can make. If she makes at most s sweaters a day, then we can multiply the number of days that she works by s to determine the total number of sweaters she makes.
total number of sweaters = (s)(number of days)
We are told to consider a time interval of three weeks. Because there are seven days in one week, the number of days over this period of time would equal 3(7), or 21 days. In other words, there are 21 days in three weeks Thus, the number of sweaters is equal to the product of s and 21.
total number of sweaters = (s)(21)
Now that we have the number of sweaters Lauren can make, we can multiply this by the cost of each sweater, which is equal to c cents, in order to obtain the amount of money she eared.
amount of money earned = (number of sweaters)(cost of each sweater)
amount of money earned = s(21)(c)
However, because the price of each sweater is given in terms of cents, the amount of money s(21)(c) will be equal to the number of cents she makes. The question, though, asks us to find the amount of money in dollars. We must use a conversion factor to change the number of cents to dollars. Remember that there are 100 cents per dollar.
The amount of money that Lauren can make depends on the number of sweaters that she can make. If she makes at most s sweaters a day, then we can multiply the number of days that she works by s to determine the total number of sweaters she makes.
total number of sweaters = (s)(number of days)
We are told to consider a time interval of three weeks. Because there are seven days in one week, the number of days over this period of time would equal 3(7), or 21 days. In other words, there are 21 days in three weeks Thus, the number of sweaters is equal to the product of s and 21.
total number of sweaters = (s)(21)
Now that we have the number of sweaters Lauren can make, we can multiply this by the cost of each sweater, which is equal to c cents, in order to obtain the amount of money she eared.
amount of money earned = (number of sweaters)(cost of each sweater)
amount of money earned = s(21)(c)
However, because the price of each sweater is given in terms of cents, the amount of money s(21)(c) will be equal to the number of cents she makes. The question, though, asks us to find the amount of money in dollars. We must use a conversion factor to change the number of cents to dollars. Remember that there are 100 cents per dollar.
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Find x.
Find x.
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Cross multiply:
Cross multiply:
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If 
, what is 40 percent of 
?
If
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To find 40 percent of 
multiply 
The result is 
To find 40 percent of
The result is
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Solve for 
.
Solve for
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First distribute the fractions:
Combine like terms:
First distribute the fractions:
Combine like terms:
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Solve for 
.
Solve for
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First distribute to eliminate the parentheses:
Then combine like terms:
First distribute to eliminate the parentheses:
Then combine like terms:
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Simplify: 
Simplify:
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Recall that dividing is equivalent multiplying by the reciprocal. Therefore, ((x - 4) / (1 / 2)) / (1 / (x + 4)) = ((x - 4) * 2) * (x + 4) / 1.
Let's simplify this further:
(2x – 8) * (x + 4) = 2x2 – 8x + 8x – 32 = 2x2 – 32
Recall that dividing is equivalent multiplying by the reciprocal. Therefore, ((x - 4) / (1 / 2)) / (1 / (x + 4)) = ((x - 4) * 2) * (x + 4) / 1.
Let's simplify this further:
(2x – 8) * (x + 4) = 2x2 – 8x + 8x – 32 = 2x2 – 32
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If 3x = 12, y/4 = 10, and 4z = 9, what is the value of (10xyz)/xy?
If 3x = 12, y/4 = 10, and 4z = 9, what is the value of (10xyz)/xy?
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Solve for the variables, the plug into formula.
x = 12/3 = 4
y = 10 * 4 = 40
z= 9/4 = 2 1/4
10xyz = 3600
Xy = 160
3600/160 = 22 1/2
Solve for the variables, the plug into formula.
x = 12/3 = 4
y = 10 * 4 = 40
z= 9/4 = 2 1/4
10xyz = 3600
Xy = 160
3600/160 = 22 1/2
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Find the inverse equation of:
Find the inverse equation of:
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To solve for an inverse, we switch x and y and solve for y. Doing so yields:
To solve for an inverse, we switch x and y and solve for y. Doing so yields:
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Evaluate 
when x=11. Round to the nearest tenth.
Evaluate when x=11. Round to the nearest tenth.
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Wherever there is an x, plug in 11 and perform the given operations. The numerator will be equal to 83 and the denominator will be equal to 46. 83 divided by 46 is equal to 1.804… and since the second decimal place is not greater than or equal to 5, the first decimal place stays the same when rounding so the final answer is 1.8.
Wherever there is an x, plug in 11 and perform the given operations. The numerator will be equal to 83 and the denominator will be equal to 46. 83 divided by 46 is equal to 1.804… and since the second decimal place is not greater than or equal to 5, the first decimal place stays the same when rounding so the final answer is 1.8.
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Solve 
Solve
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The common denominator of the left side is x(x–1). Multiplying the top and bottom of 1/x by (x–1) yields
Since this statement is true, there are infinitely many solutions.
The common denominator of the left side is x(x–1). Multiplying the top and bottom of 1/x by (x–1) yields
Since this statement is true, there are infinitely many solutions.
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Mary walked to school at an average speed of 2 miles per hour and jogged back along the same route at an average speed of 6 miles per hour. If her total traveling time was 1 hour, what was the total number of miles in the round trip?
Mary walked to school at an average speed of 2 miles per hour and jogged back along the same route at an average speed of 6 miles per hour. If her total traveling time was 1 hour, what was the total number of miles in the round trip?
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Since Mary traveled 3 times as quickly coming from school as she did going to school (6 miles per hour compared to 2 miles per hour), we know that Mary spent only a third of the time coming from school as she did going. If x represents the number of hours it took to get to school, then x/3 represents the number of hours it took her to return.
Knowing that the total trip took 1 hour, we have:
x + x/3 = 1
3x/3 + 1x/3 = 1
4_x_/3 = 1
x = 3/4
So we know it took Mary 3/4 of an hour to travel to school (and the remaining 1/4 of an hour to get back).
Remembering that distance = rate * time, the distance Mary traveled on her way to school was (2 miles per hour) * (3/4 of an hour) = 3/2 miles. Furthermore, since she took the same route coming back, she must have traveled 3/2 of a mile to return as well.
Therefore, the the total number of miles in Mary's round trip is 3/2 miles + 3/2 miles = 6/2 miles = 3 miles.
Since Mary traveled 3 times as quickly coming from school as she did going to school (6 miles per hour compared to 2 miles per hour), we know that Mary spent only a third of the time coming from school as she did going. If x represents the number of hours it took to get to school, then x/3 represents the number of hours it took her to return.
Knowing that the total trip took 1 hour, we have:
x + x/3 = 1
3x/3 + 1x/3 = 1
4_x_/3 = 1
x = 3/4
So we know it took Mary 3/4 of an hour to travel to school (and the remaining 1/4 of an hour to get back).
Remembering that distance = rate * time, the distance Mary traveled on her way to school was (2 miles per hour) * (3/4 of an hour) = 3/2 miles. Furthermore, since she took the same route coming back, she must have traveled 3/2 of a mile to return as well.
Therefore, the the total number of miles in Mary's round trip is 3/2 miles + 3/2 miles = 6/2 miles = 3 miles.
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According the pie chart, the degree measure of the sector representing the number of workers spending 5 to 9 years in the same role is how much greater in the construction industry chart than in the financial industry chart?
According the pie chart, the degree measure of the sector representing the number of workers spending 5 to 9 years in the same role is how much greater in the construction industry chart than in the financial industry chart?
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Since the values in the pie charts are currently in terms of percentages (/100), we must convert them to degrees (/360, since within a circle) to solve the question. The "5 to 9 years" portion for the financial and construction industries are 18 and 25 percent, respectively. As such, we can cross-multiply both:
18/100 = x/360
x = 65 degrees
25/100 = y/360
y = 90 degrees
Subtract: 90 – 65 = 25 degrees
Alternatively, we could first subtract the percentages (25 – 18 = 7), then convert the 7% to degree form via the same method of cross-multiplication.
Since the values in the pie charts are currently in terms of percentages (/100), we must convert them to degrees (/360, since within a circle) to solve the question. The "5 to 9 years" portion for the financial and construction industries are 18 and 25 percent, respectively. As such, we can cross-multiply both:
18/100 = x/360
x = 65 degrees
25/100 = y/360
y = 90 degrees
Subtract: 90 – 65 = 25 degrees
Alternatively, we could first subtract the percentages (25 – 18 = 7), then convert the 7% to degree form via the same method of cross-multiplication.
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