Equations / Inequalities - ACT Math
Card 0 of 2799
Abby works at a car dealership and receives a commission "c" which is a percent of the profit the dealership makes from the sale, which is the difference between the price "p" of the car and the value "v" of the car. How much, in dollars, does the dealership earn per transaction?
Abby works at a car dealership and receives a commission "c" which is a percent of the profit the dealership makes from the sale, which is the difference between the price "p" of the car and the value "v" of the car. How much, in dollars, does the dealership earn per transaction?
To show that c is of the profit of the transaction, we must represent the profit as the difference between the price and the value of the car, or "(p – v)"
To show that Abby's commission in dollars is a percentage of the profit, we use 0.01 * c to convert the commission she earns to a percent.
To shift the earnings from Abby to the dealership (which is what the question requires of us), we must take 1 – 0.01c since this will accommodate for the remaining percentage. For example, it shifts 75% (0.75) to 25% (1 – 0.75 or 0.25).
Putting this all together, we get a final expression of:
(p – v)(1 – 0.01c) = dealership earnings
Check answer with arbitrary values: letting p = 300, v = 200, and c = 20, we get a value of 80 which makes sense as the $100 profit must be distributed evenly between Abby ($20) and the dealership ($80).
To show that c is of the profit of the transaction, we must represent the profit as the difference between the price and the value of the car, or "(p – v)"
To show that Abby's commission in dollars is a percentage of the profit, we use 0.01 * c to convert the commission she earns to a percent.
To shift the earnings from Abby to the dealership (which is what the question requires of us), we must take 1 – 0.01c since this will accommodate for the remaining percentage. For example, it shifts 75% (0.75) to 25% (1 – 0.75 or 0.25).
Putting this all together, we get a final expression of:
(p – v)(1 – 0.01c) = dealership earnings
Check answer with arbitrary values: letting p = 300, v = 200, and c = 20, we get a value of 80 which makes sense as the $100 profit must be distributed evenly between Abby ($20) and the dealership ($80).
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3x + 9i2 – 5x = 17
What is x?
3x + 9i2 – 5x = 17
What is x?
i = 
i2 = -1
3x + 9i2 – 5x = 17
3x + 9(–1) – 5x = 17
–2x – 9 = 17
–2x = 26
x = –13
i =
i2 = -1
3x + 9i2 – 5x = 17
3x + 9(–1) – 5x = 17
–2x – 9 = 17
–2x = 26
x = –13
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Solve for x:
3x + 4y = 26
–5x + 12y = 14
Solve for x:
3x + 4y = 26
–5x + 12y = 14
Eliminate y and solve for x.
3x + 4y = 26 (multiply by –3)
–5x + 12y = 14
(–3)3x +(–3) 4y = (–3)26
–5x + 12y = 14
–9x +-12y = –78
–5x + 12y = 14
–14x + 0y = –64
x = –64/–14 = 32/7
Eliminate y and solve for x.
3x + 4y = 26 (multiply by –3)
–5x + 12y = 14
(–3)3x +(–3) 4y = (–3)26
–5x + 12y = 14
–9x +-12y = –78
–5x + 12y = 14
–14x + 0y = –64
x = –64/–14 = 32/7
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Tommy's and Sara's current ages are represented by t and s, respectively. If in five years, Tommy will be twice as old as Sara, which of the following represents t in terms of s?
Tommy's and Sara's current ages are represented by t and s, respectively. If in five years, Tommy will be twice as old as Sara, which of the following represents t in terms of s?
Tommy's current age is represented by t, and Sara's is represented by s. In five years, both Tommy's and Sara's ages will be increased by five. Thus, in five years, we can represent Tommy's age as 
and Sara's as 
.
The problem tells us that Tommy's age in five years will be twice as great as Sara's in five years. Thus, we can write an algebraic expression to represent the problem as follows:
In order to solve for t, first simplify the right side by distributing the 2.
Then subtract 5 from both sides.
The answer is 
.
Tommy's current age is represented by t, and Sara's is represented by s. In five years, both Tommy's and Sara's ages will be increased by five. Thus, in five years, we can represent Tommy's age as
The problem tells us that Tommy's age in five years will be twice as great as Sara's in five years. Thus, we can write an algebraic expression to represent the problem as follows:
In order to solve for t, first simplify the right side by distributing the 2.
Then subtract 5 from both sides.
The answer is
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10 gallons of paint will cover 75 ft2. How many gallons of paint will be required to paint the area of a rectangular wall that has a height of 8 ft and a length of 24 ft?
10 gallons of paint will cover 75 ft2. How many gallons of paint will be required to paint the area of a rectangular wall that has a height of 8 ft and a length of 24 ft?
First we need the area or the rectangle. 24 * 8 = 192. So now we know that 10 gallons will cover 75 ft2 and x gallons will cover 192 ft2. We set up a simple ratio and cross multiply to find that 75_x_ = 1920.
x = 25.6
First we need the area or the rectangle. 24 * 8 = 192. So now we know that 10 gallons will cover 75 ft2 and x gallons will cover 192 ft2. We set up a simple ratio and cross multiply to find that 75_x_ = 1920.
x = 25.6
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What number decreased by 7 is equal to 10 increased by 7?
What number decreased by 7 is equal to 10 increased by 7?
The best way to solve this problem is to translate it into an equation, "decreased" meaning subtract and "increased" meaning add:
x – 7 = 10 + 7
x = 24
The best way to solve this problem is to translate it into an equation, "decreased" meaning subtract and "increased" meaning add:
x – 7 = 10 + 7
x = 24
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n/17 = 54/9
What is the value of n?
n/17 = 54/9
What is the value of n?
Cross multiply: 9n = 54 * 17 → n = (54 * 17)/9 = 102.
This also can be solved by reducing the right hand side of the equation, so n/17 = 6.
Cross multiply: 9n = 54 * 17 → n = (54 * 17)/9 = 102.
This also can be solved by reducing the right hand side of the equation, so n/17 = 6.
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If a%b = (2b + 3a)/(6ab), what would have a greater value, 2%3 or 3%2?
If a%b = (2b + 3a)/(6ab), what would have a greater value, 2%3 or 3%2?
First find 2%3 = (2 * 3 + 3 * 2)/(6 * 2 * 3) = 12/36 = 1/3, then 3%2 = (2 * 2 + 3 * 3)/(6 * 3 * 2) = 13/36 which is greater.
First find 2%3 = (2 * 3 + 3 * 2)/(6 * 2 * 3) = 12/36 = 1/3, then 3%2 = (2 * 2 + 3 * 3)/(6 * 3 * 2) = 13/36 which is greater.
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Which of the following is a factor of the polynomial x_2 – 6_x + 5?
Which of the following is a factor of the polynomial x_2 – 6_x + 5?
Factor the polynomial by choosing values that when FOIL'ed will add to equal the middle coefficient, 3, and multiply to equal the constant, 1.
x_2 – 6_x + 5 = (x – 1)(x – 5)
Because only (x – 5) is one of the choices listed, we choose it.
Factor the polynomial by choosing values that when FOIL'ed will add to equal the middle coefficient, 3, and multiply to equal the constant, 1.
x_2 – 6_x + 5 = (x – 1)(x – 5)
Because only (x – 5) is one of the choices listed, we choose it.
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Solve for x: (x2 – x) / (x – 1) = 1
Solve for x: (x2 – x) / (x – 1) = 1
Begin by multiplying both sides by (x – 1):
x2 – x = x – 1
Solve as a quadratic equation: x2 – 2x + 1 = 0
Factor the left: (x – 1)(x – 1) = 0
Therefore, x = 1.
However, notice that in the original equation, a value of 1 for x would place a 0 in the denominator. Therefore, there is no solution.
Begin by multiplying both sides by (x – 1):
x2 – x = x – 1
Solve as a quadratic equation: x2 – 2x + 1 = 0
Factor the left: (x – 1)(x – 1) = 0
Therefore, x = 1.
However, notice that in the original equation, a value of 1 for x would place a 0 in the denominator. Therefore, there is no solution.
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Solve 3x2 + 10x = –3
Solve 3x2 + 10x = –3
Generally, quadratic equations have two answers.
First, the equations must be put in standard form: 3x2 + 10x + 3 = 0
Second, try to factor the quadratic; however, if that is not possible use the quadratic formula.
Third, check the answer by plugging the answers back into the original equation.
Generally, quadratic equations have two answers.
First, the equations must be put in standard form: 3x2 + 10x + 3 = 0
Second, try to factor the quadratic; however, if that is not possible use the quadratic formula.
Third, check the answer by plugging the answers back into the original equation.
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Let f(x) = 2_x_3 + 7_x_2 - 4_x_ and g(x) = 4_x_ - _x_3. If k is a positive number such that f(k) = 0, then what is g(k)?
Let f(x) = 2_x_3 + 7_x_2 - 4_x_ and g(x) = 4_x_ - _x_3. If k is a positive number such that f(k) = 0, then what is g(k)?
First, we need to find the possible values of k such that f(k) = 0.
We can use the definition of f(x) to write an expression for k.
f(x) = 2_x_3 + 7_x_2 – 4_x_
f(k) = 2_k_3 + 7_k_2 – 4_k_ = 0
In order to solve this equation, we will want to factor as much as we can. We can immediately see that we could take out a k from all three terms.
2_k_3 + 7_k_2 – 4_k_ = 0
k(2_k_2 + 7_k –_ 4) = 0
Now, we need to factor 2_k_2 + 7_k –_ 4 by grouping. When we multiply the outer two coefficients we get (2)(–4) = –8. The middle coefficient is 7. This means we need to think of two numbers that multiply to give us –8, but add to give us 7. These two numbers are 8 and –1. We can now rewrite 2_k_2 + 7_k –_ 4 as follows:
2_k_2 + 7_k –_ 4 = 2_k_2 + 8_k_ – k – 4
Group the first two terms and the second two terms.
2_k_2 + 8_k_ – k – 4 = (2_k_2 + 8_k_ ) + (–k – 4)
Next, factor out a 2_k_ from the first two terms and a –1 from the last two terms.
(2_k_2 + 8_k_ ) + (–k – 4) = 2_k_(k + 4) + –1(k + 4)
Then, factor out k + 4 from the 2_k_ and the –1.
2_k_(k + 4) + –1(k + 4) = (k + 4)(2_k –_ 1)
Thus, 2_k_2 + 7_k –_ 4, when fully factored, equals (k + 4)(2_k –_ 1).
Now, let's go back to the equation k(2_k_2 + 7_k –_ 4) = 0 and substitute (k + 4)(2_k –_ 1) for 2_k_2 + 7_k –_ 4.
k(2_k_2 + 7_k –_ 4) = k(k + 4)(2_k –_ 1) = 0
We now have three factors, and we can set each equal to zero to find the possible values of k.
The first factor is k. This means k = 0 is one value for k.
The next factor is k + 4.
k + 4 = 0
k = –4
The last factor is 2_k_ – 1.
2_k_ – 1 = 0
2_k_ = 1
k = 1/2
The values of k for which f(k) = 0 are 0, –4, and 1/2. However, we are told that k is positive, so this means that k can only be 1/2.
Ultimately, the problem asks us to find g(k). This means we must use the equation for g(x) to find g(1/2).
g(x) = 4_x_ – _x_3
g(1/2) = 4(1/2) – (1/2)3
= 2 – (1/8)
= 15/8
The answer is 15/8.
First, we need to find the possible values of k such that f(k) = 0.
We can use the definition of f(x) to write an expression for k.
f(x) = 2_x_3 + 7_x_2 – 4_x_
f(k) = 2_k_3 + 7_k_2 – 4_k_ = 0
In order to solve this equation, we will want to factor as much as we can. We can immediately see that we could take out a k from all three terms.
2_k_3 + 7_k_2 – 4_k_ = 0
k(2_k_2 + 7_k –_ 4) = 0
Now, we need to factor 2_k_2 + 7_k –_ 4 by grouping. When we multiply the outer two coefficients we get (2)(–4) = –8. The middle coefficient is 7. This means we need to think of two numbers that multiply to give us –8, but add to give us 7. These two numbers are 8 and –1. We can now rewrite 2_k_2 + 7_k –_ 4 as follows:
2_k_2 + 7_k –_ 4 = 2_k_2 + 8_k_ – k – 4
Group the first two terms and the second two terms.
2_k_2 + 8_k_ – k – 4 = (2_k_2 + 8_k_ ) + (–k – 4)
Next, factor out a 2_k_ from the first two terms and a –1 from the last two terms.
(2_k_2 + 8_k_ ) + (–k – 4) = 2_k_(k + 4) + –1(k + 4)
Then, factor out k + 4 from the 2_k_ and the –1.
2_k_(k + 4) + –1(k + 4) = (k + 4)(2_k –_ 1)
Thus, 2_k_2 + 7_k –_ 4, when fully factored, equals (k + 4)(2_k –_ 1).
Now, let's go back to the equation k(2_k_2 + 7_k –_ 4) = 0 and substitute (k + 4)(2_k –_ 1) for 2_k_2 + 7_k –_ 4.
k(2_k_2 + 7_k –_ 4) = k(k + 4)(2_k –_ 1) = 0
We now have three factors, and we can set each equal to zero to find the possible values of k.
The first factor is k. This means k = 0 is one value for k.
The next factor is k + 4.
k + 4 = 0
k = –4
The last factor is 2_k_ – 1.
2_k_ – 1 = 0
2_k_ = 1
k = 1/2
The values of k for which f(k) = 0 are 0, –4, and 1/2. However, we are told that k is positive, so this means that k can only be 1/2.
Ultimately, the problem asks us to find g(k). This means we must use the equation for g(x) to find g(1/2).
g(x) = 4_x_ – _x_3
g(1/2) = 4(1/2) – (1/2)3
= 2 – (1/8)
= 15/8
The answer is 15/8.
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In a group of 24 children, there are twice as many girls as there are boys. How many boys are there?
In a group of 24 children, there are twice as many girls as there are boys. How many boys are there?
Since there are twice as many girls as boys, we know that 2_b_ = g.
Since there are 24 total, we know that b + g = 24.
Substituting the first equation into the second equation yields
b + 2_b_ = 24
3_b_ = 24
b = 8
Since there are twice as many girls as boys, we know that 2_b_ = g.
Since there are 24 total, we know that b + g = 24.
Substituting the first equation into the second equation yields
b + 2_b_ = 24
3_b_ = 24
b = 8
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For each megawatt of power that it produces, a power plant spends $30,000 on coal, $50,000 on salaries, and $20,000 on general expenses. If the plant produces 250 megawatts, how much does it spend on salaries and general expenses combined?
For each megawatt of power that it produces, a power plant spends $30,000 on coal, $50,000 on salaries, and $20,000 on general expenses. If the plant produces 250 megawatts, how much does it spend on salaries and general expenses combined?
The problem is simple if you do not mix up your values and their meanings. What we know is that for 250 megawatts, we should have 250 times the expenses in each of our categories. Therefore, for our data, we know the following.
For salaries, we have $50,000 * 250 or $12,500,000.
For general expenses, we have $20,000 * 250 or $5,000,000
Therefore, our total is: $12,500,000 + $5,000,000 = $17,500,000
We also could have calculated this by adding $50,000 and $20,000 together to get $70,000 per megawatt. This would be $70,000 * 250 = $17,500,000 as well.
The problem is simple if you do not mix up your values and their meanings. What we know is that for 250 megawatts, we should have 250 times the expenses in each of our categories. Therefore, for our data, we know the following.
For salaries, we have $50,000 * 250 or $12,500,000.
For general expenses, we have $20,000 * 250 or $5,000,000
Therefore, our total is: $12,500,000 + $5,000,000 = $17,500,000
We also could have calculated this by adding $50,000 and $20,000 together to get $70,000 per megawatt. This would be $70,000 * 250 = $17,500,000 as well.
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Given that 
, what is the value of 
?
Given that
First, we must solve the equation for 
by subtracting 
from both sides:
Then we must add 
to both sides:
First, we must solve the equation for
Then we must add
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The length, in meters, of a rectangular fence is 4 more than twice its width. Which of the following gives the length (l) in terms of width (w) of the rectangular fence?
The length, in meters, of a rectangular fence is 4 more than twice its width. Which of the following gives the length (l) in terms of width (w) of the rectangular fence?
To find the length, we must take twice the width, meaning to multiply the width by 2. Then we must take 4 more than that number, meaning we must add 4 to the number. Combining these, we get:
l = 2_w_ + 4
To find the length, we must take twice the width, meaning to multiply the width by 2. Then we must take 4 more than that number, meaning we must add 4 to the number. Combining these, we get:
l = 2_w_ + 4
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Given the following equation:
y = –3_x_ – 5
What is y when x = –2?
Given the following equation:
y = –3_x_ – 5
What is y when x = –2?
Plug in x and evaluate the equation.
Plug in x and evaluate the equation.
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A theme park charges $10 for adults and $5 for kids. How many kids tickets were sold if a total of 548 tickets were sold for a total of $3750?
A theme park charges $10 for adults and $5 for kids. How many kids tickets were sold if a total of 548 tickets were sold for a total of $3750?
Let c = number of kids tickets sold. Then (548 – c) adult tickets were sold. The revenue from kids tickets is $5_c_, and the total revenue from adult tickets is $10(548 – c). Then,
5_c_ + 10(548 – c) = 3750
5_c_ + 5480 – 10_c_ = 3750
5_c_ = 1730
c = 346.
We can check to make sure that this number is correct:
$5 * 346 tickets + $10 * (548 – 346) tickets = $3750 total revenue
Let c = number of kids tickets sold. Then (548 – c) adult tickets were sold. The revenue from kids tickets is $5_c_, and the total revenue from adult tickets is $10(548 – c). Then,
5_c_ + 10(548 – c) = 3750
5_c_ + 5480 – 10_c_ = 3750
5_c_ = 1730
c = 346.
We can check to make sure that this number is correct:
$5 * 346 tickets + $10 * (548 – 346) tickets = $3750 total revenue
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Two palm trees grow next to each other in Luke's backyard. One of the trees gets sick, so Luke cuts off the top 3 feet. The other tree, however, is healthy and grows 2 feet. How tall are the two trees if the healthy tree is now 4 feet taller than the sick tree, and together they are 28 feet tall?
Two palm trees grow next to each other in Luke's backyard. One of the trees gets sick, so Luke cuts off the top 3 feet. The other tree, however, is healthy and grows 2 feet. How tall are the two trees if the healthy tree is now 4 feet taller than the sick tree, and together they are 28 feet tall?
Let s stand for the sick tree and h for the healthy tree. The beginning information about cutting the sick tree and the healthy tree growing is actually not needed to solve this problem! We know that the healthy tree is 4 feet taller than the sick tree, so h = s + 4.
We also know that the two trees are 28 feet tall together, so s + h = 28. Now we can solve for the two tree heights.
Plug h = s + 4 into the second equation: (s + 4) + s = 28. Simplify and solve for h: 2_s_ = 24 so s = 12. Then solve for h: h = s + 4 = 12 + 4 = 16.
Let s stand for the sick tree and h for the healthy tree. The beginning information about cutting the sick tree and the healthy tree growing is actually not needed to solve this problem! We know that the healthy tree is 4 feet taller than the sick tree, so h = s + 4.
We also know that the two trees are 28 feet tall together, so s + h = 28. Now we can solve for the two tree heights.
Plug h = s + 4 into the second equation: (s + 4) + s = 28. Simplify and solve for h: 2_s_ = 24 so s = 12. Then solve for h: h = s + 4 = 12 + 4 = 16.
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Jim got a job trimming trees. If he trims trees from noon to 6pm, with the exception of a 30 minute lunch break, Jim can trim 55 trees. How long does it take him to trim one tree?
Jim got a job trimming trees. If he trims trees from noon to 6pm, with the exception of a 30 minute lunch break, Jim can trim 55 trees. How long does it take him to trim one tree?
Once you subtract the 30 minute break, you are left with 5 and a half hours. You multiply that by 60 to get 330 minutes. You then divide that by 55 trees, to get 6 minutes per tree.
Once you subtract the 30 minute break, you are left with 5 and a half hours. You multiply that by 60 to get 330 minutes. You then divide that by 55 trees, to get 6 minutes per tree.
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