ACT Math : Linear / Rational / Variable Equations
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Example Question #1 : Linear / Rational / Variable Equations
n/17 = 54/9
What is the value of n?
None of the above.
102
54
93.5
85
102
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.
Example Question #1 : Linear / Rational / Variable Equations
If 12x + 3 = 2(5x + 5) + 1, what is the value of x?
6
5
3
7
4
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.
Example Question #2 : Linear / Rational / Variable Equations
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?
34
24
20
16
65
24
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.
Example Question #1 : Linear / Rational / Variable Equations
Three consecutive positive numbers have the sum of 15. What is the product of these numbers?
20
75
30
45
120
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.
Example Question #2 : Linear / Rational / Variable Equations
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?
50
80
60
100
75
60
The break-even point is where the costs equal the revenues
Fixed Costs + Variable Costs = Revenues
1500 + 50x = 75x
Solving for x results in x = 60 boats sold each month to break even.
Example Question #1 : Equations / Inequalities
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?
l = 2w * 4
l = 2w + 4
l = 2w – 4
l = (1/2)w – 4
l = (1/2)w + 4
l = 2w + 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 = 2w + 4
Example Question #1 : Linear / Rational / Variable Equations
Given the following equation:
y = –3x – 5
What is y when x = –2?
5
–1
1
2
3
1
Plug in x and evaluate the equation.
Example Question #5 : Linear / Rational / Variable Equations
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?
5 minutes
6 minutes
7 minutes
5.5 minutes
6.5 minutes
6 minutes
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.
Example Question #3 : How To Find The Solution To An Equation
If 16x + 4 = 5(3x + 1) + 2, what is the value of x?
4
3
2
1
5
3
By the order of operations, we multiply 5 by 3x + 1
Add like terms to get 16x + 4 = 15x + 7
Subtract 15x from both sides, then subtract 4 from both sides.
Then you have x = 3.
Example Question #6 : Linear / Rational / Variable Equations
Solve for x.
4(x + 3) = 16
x = 2
x = 12
x = 1
Undefined
x = 4
x = 1
4(x + 3) = 16
First you must multiply out the expression to 4x + 12 = 16. Next subtract 12 from both sides to isolate the x leaving 4x = 4. Divide both sides by 4 to get x = 1.
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