Linear / Rational / Variable Equations - ACT Math
Card 0 of 1287
Two cars start 25 mile apart and drive away from each other in opposite directions at speeds of 50 and 70 miles per hour. In approximately how many minutes will they be 400 miles apart?
Two cars start 25 mile apart and drive away from each other in opposite directions at speeds of 50 and 70 miles per hour. In approximately how many minutes will they be 400 miles apart?
The cars have a distance from each other of 25 + 120t miles, where t is the number of hours, 25 is their initial distance and 120 is 50 + 70, or their combined speeds. Solve this equation for 400:
25 + 120t = 400; 120t = 375; t = 3.125
However, the question asked for minutes, so we must multiply this by 60:
3.125 * 60 = 187.5 minutes.
The cars have a distance from each other of 25 + 120t miles, where t is the number of hours, 25 is their initial distance and 120 is 50 + 70, or their combined speeds. Solve this equation for 400:
25 + 120t = 400; 120t = 375; t = 3.125
However, the question asked for minutes, so we must multiply this by 60:
3.125 * 60 = 187.5 minutes.
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√(3x) = 9
What is x?
√(3x) = 9
What is x?
To solve, remove the radical by squaring both sides
(√3x) 2 = 92
3x = 81
x = 81/3 = 27
To solve, remove the radical by squaring both sides
(√3x) 2 = 92
3x = 81
x = 81/3 = 27
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√(8y) + 18 = 4
What is y?
√(8y) + 18 = 4
What is y?
First, simplify the equation:
√(8y) + 18 = 4
√(8y) = -14
Then square both sides
(√8y) 2 = -142
8y = 196
y = 196/8 = 24.5
First, simplify the equation:
√(8y) + 18 = 4
√(8y) = -14
Then square both sides
(√8y) 2 = -142
8y = 196
y = 196/8 = 24.5
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If you drove at an average speed of 78 miles per hour, what distance, in miles, did you drive in 140 minutes?
If you drove at an average speed of 78 miles per hour, what distance, in miles, did you drive in 140 minutes?
140 minutes is 7/3 of an hour. Multiplied by the speed of 78mph, we obtain 182 miles traveled.
140 minutes is 7/3 of an hour. Multiplied by the speed of 78mph, we obtain 182 miles traveled.
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What is the slope of the line 7y – 4x = 27
What is the slope of the line 7y – 4x = 27
Adding 4x to both sides of the equation and dividing by 7 yields a slope of 4/7.
Adding 4x to both sides of the equation and dividing by 7 yields a slope of 4/7.
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If the expression x4 + 3cx – 2 is equal to 5 when x = **–**1, what is the value of c?
If the expression x4 + 3cx – 2 is equal to 5 when x = **–**1, what is the value of c?
Plugging in **–**1 into the equation for x and solving for c yields **–**2.
Plugging in **–**1 into the equation for x and solving for c yields **–**2.
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Which of the lines below is not parallel to 4x – 3y = 17 ?
Which of the lines below is not parallel to 4x – 3y = 17 ?
For a line to be parallel with another, the slopes must be equal. All of the equations have a slope of 4/3 except 8x +6y=34.
For a line to be parallel with another, the slopes must be equal. All of the equations have a slope of 4/3 except 8x +6y=34.
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A given university has an average professor pay of $40,000 a year and an average administrator pay of $45,000 per year. If the ratio of professors to administrators is 4 to 3, and the total pay for professors and administrators in a year is $40,415,000, how many professors does the college have?
A given university has an average professor pay of $40,000 a year and an average administrator pay of $45,000 per year. If the ratio of professors to administrators is 4 to 3, and the total pay for professors and administrators in a year is $40,415,000, how many professors does the college have?
Set up a system of linear equations based on our data:
40,000P + 45,000A = 40,415,000
P/A = 4/3
To make things easiest, solve the second equation for A in terms of P:
A = (3/4) P
Replace this value into the first equation:
40,000P + 45,000 * (3/4)P = 40,415,000
Simplify:
40,000P + 33,750P = 40,415,000
73,750P = 40,415,000
P = 548 (The number of professors)
Set up a system of linear equations based on our data:
40,000P + 45,000A = 40,415,000
P/A = 4/3
To make things easiest, solve the second equation for A in terms of P:
A = (3/4) P
Replace this value into the first equation:
40,000P + 45,000 * (3/4)P = 40,415,000
Simplify:
40,000P + 33,750P = 40,415,000
73,750P = 40,415,000
P = 548 (The number of professors)
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If y = 4 and 6y = 10z + y, then z = ?
If y = 4 and 6y = 10z + y, then z = ?
- Substitute y in the equation for 4.
- You now have 6 * 4 = 10z + 4
- Simplify the equation: 24 = 10z + 4
- Subtract 4 from both sides: 24 – 4 = 10z + 4 – 4
- You now have 20 = 10z
- Divde both sides by 10 to solve for z.
- z = 2.
- Substitute y in the equation for 4.
- You now have 6 * 4 = 10z + 4
- Simplify the equation: 24 = 10z + 4
- Subtract 4 from both sides: 24 – 4 = 10z + 4 – 4
- You now have 20 = 10z
- Divde both sides by 10 to solve for z.
- z = 2.
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A sequence of numbers is: 2, 5, 8, 11. Assuming it follows the same pattern, what would be the value of the 20th number?
A sequence of numbers is: 2, 5, 8, 11. Assuming it follows the same pattern, what would be the value of the 20th number?
This goes up at a constant number between values, making it an arthmetic sequence. The first number is 2, with a difference of 3. Plugging this into the arithmetic equation you get An = 2 + 3 (n – 1). Plugging in 20 for n, you get a value of 59.
This goes up at a constant number between values, making it an arthmetic sequence. The first number is 2, with a difference of 3. Plugging this into the arithmetic equation you get An = 2 + 3 (n – 1). Plugging in 20 for n, you get a value of 59.
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The first four numbers of a sequence are 5, 10, 20, 40. Assuming the pattern continues, what is the 6th term of the sequence?
The first four numbers of a sequence are 5, 10, 20, 40. Assuming the pattern continues, what is the 6th term of the sequence?
Looking at the sequence you can see that it doubles each term, making it a geometric sequence. Since it doubles r = 2 and the first term is 5. Plugging this into the geometric equation you get An = 5(2)n–1. Setting n = 6, you get 160 as the 6th term.
Looking at the sequence you can see that it doubles each term, making it a geometric sequence. Since it doubles r = 2 and the first term is 5. Plugging this into the geometric equation you get An = 5(2)n–1. Setting n = 6, you get 160 as the 6th term.
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Given f(x) = x2 – 9. What are the zeroes of the function?
Given f(x) = x2 – 9. What are the zeroes of the function?
The zeroes of the equation are where f(x) = 0 (aka x-intercepts). Setting the equation equal to zero you get x2 = 9. Since a square makes a negative number positive, x can be equal to 3 or –3.
The zeroes of the equation are where f(x) = 0 (aka x-intercepts). Setting the equation equal to zero you get x2 = 9. Since a square makes a negative number positive, x can be equal to 3 or –3.
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Give the lines y = 0.5x+3 and y=3x-2. What is the y value of the point of intersection?
Give the lines y = 0.5x+3 and y=3x-2. What is the y value of the point of intersection?
In order to solve for the x value you set both equations equal to each other (0.5x+3=3x-2). This gives you the x value for the point of intersection at x=2. Plugging x=2 into either equation gives you y=4.
In order to solve for the x value you set both equations equal to each other (0.5x+3=3x-2). This gives you the x value for the point of intersection at x=2. Plugging x=2 into either equation gives you y=4.
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If 12x + 3 = 2(5x + 5) + 1, what is the value of x?
If 12x + 3 = 2(5x + 5) + 1, what is the value of x?
Starting with 12x + 3 = 2(5x + 5) + 1, we start by solving the parenthesis, giving us 12x + 3 = 10x + 11. We then subtract 10x from the right side and subtract three from the left, giving us 2x = 8; divide by 2 → x = 4.
Starting with 12x + 3 = 2(5x + 5) + 1, we start by solving the parenthesis, giving us 12x + 3 = 10x + 11. We then subtract 10x from the right side and subtract three from the left, giving us 2x = 8; divide by 2 → x = 4.
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If you multiply two integers together and then add 5, the result is 69. Which of the following could not be the sum of the two integers?
If you multiply two integers together and then add 5, the result is 69. Which of the following could not be the sum of the two integers?
The equation is xy + 5 = 69, making xy equal to 64. If we factor 64, we see that 1x64, 2x32, 4x16 and 8x8 all equal 16 when the two numbers are added together, so 24 is the only possibility that does not work.
The equation is xy + 5 = 69, making xy equal to 64. If we factor 64, we see that 1x64, 2x32, 4x16 and 8x8 all equal 16 when the two numbers are added together, so 24 is the only possibility that does not work.
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If 5 + x is 5 more than 5,what is the value of 2_x_?
If 5 + x is 5 more than 5,what is the value of 2_x_?
5 more than 5 = 10
5 + x = 10
Subtract 5 from each side of the equation: x = 5 → 2_x_ = 10
5 more than 5 = 10
5 + x = 10
Subtract 5 from each side of the equation: x = 5 → 2_x_ = 10
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Three consecutive positive numbers have the sum of 15. What is the product of these numbers?
Three consecutive positive numbers have the sum of 15. What is the product of these numbers?
Define the variables as x = the first number, x + 1, the second number, and x + 2 the thrid number.
The sum becomes x + x + 1 + x + 2 = 15 so 3x + 3 = 15. Subtract 3 from both sides of the equation to get 3x = 12 → 3x/3 = 12/3 → x = 4
The three numbers are 4, 5, and 6 and their product is 120.
Define the variables as x = the first number, x + 1, the second number, and x + 2 the thrid number.
The sum becomes x + x + 1 + x + 2 = 15 so 3x + 3 = 15. Subtract 3 from both sides of the equation to get 3x = 12 → 3x/3 = 12/3 → x = 4
The three numbers are 4, 5, and 6 and their product is 120.
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A company makes toy boats. Their monthly fixed costs are $1500. The variable costs are $50 per boat. They sell boats for $75 a piece. How many boats must be sold each month to break even?
A company makes toy boats. Their monthly fixed costs are $1500. The variable costs are $50 per boat. They sell boats for $75 a piece. How many boats must be sold each month to break even?
The break-even point is where the costs equal the revenues
Fixed Costs + Variable Costs = Revenues
1500 + 50_x_ = 75_x_
Solving for x results in x = 60 boats sold each month to break even.
The break-even point is where the costs equal the revenues
Fixed Costs + Variable Costs = Revenues
1500 + 50_x_ = 75_x_
Solving for x results in x = 60 boats sold each month to break even.
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Solve for 
.
Cross multiplying leaves 
, which is not possible.
Cross multiplying leaves
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A store sells potatoes for $0.24 and tomatoes for $0.76. Fred bought 12 individual vegetables. If he paid $6.52 total, how many potatoes did Fred buy?
A store sells potatoes for $0.24 and tomatoes for $0.76. Fred bought 12 individual vegetables. If he paid $6.52 total, how many potatoes did Fred buy?
Set up an equation to represent the total cost in cents: 24P + 76T = 652. In order to reduce the number of variables from 2 to 1, let the # tomatoes = 12 – # of potatoes. This makes the equation 24P + 76(12 – P) = 652.
Solving for P will give the answer.
Set up an equation to represent the total cost in cents: 24P + 76T = 652. In order to reduce the number of variables from 2 to 1, let the # tomatoes = 12 – # of potatoes. This makes the equation 24P + 76(12 – P) = 652.
Solving for P will give the answer.
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