Free sample of our Advanced College Math Placement Test Practice PDF.
Math Placement Test – College-Level Practice Problems:
Here are some CPT advanced college-level math sample problems and solutions from our College Math Placement Test Practice PDFs.
1) 7.23 + .004 + .513 = ?
2) 8.13 × 3.1 = ?
3) Two people are going to give money to a foundation for a project. Person A will provide one-half of the money. Person B will donate one-eighth of the money. What fraction represents the unfunded portion of the project?
4) If 5x − 2(x + 3) = 0, then x = ?
5) Simplify the following equation: (x + 3y)2
6) (x + 3y)(x − y) = ?
7) 58 x 52 = ?
8) Determine how many 3 letter permutations can be derived from the five letter set: F U N K Y
9) Find the coordinates (x, y) of the midpoint of the line segment on a graph that connects the points (−5, 3) and (3, −5).
When you download our College Math Placement Test practice PDF, you will get all 200 college math practice questions.
There are in-depth-solutions for each of the problems.
Math Placement Test – Solutions
Our College Math Placement Test practice PDF includes solutions and explanations for each problem.
Here is a sample of the solutions we provide with the College Math Placement Test practice PDF.
Math Practice – Solutions and Explanations:
Answer 1 – Decimals
7.23 + .004 + .513 = ?
ANSWER: 7.747
Do problems like this on the scratch paper provided to you during the test.
You have to line up all of the decimals in order to correctly solve the problem.
7.230
0.004
0.513
7.747
Study the solution above and notice how all of the decimals are lined up in a column.
Also notice that it will be easier to solve the problem if you add zeros at the beginning and end of the numbers stated in the question.
So, for example, 7.23 becomes 7.230 in the solution. Likewise .004 is restated as 0.004 and .513 as 0.513.
Answer 2 – Multiplication
8.13 × 3.1 = ?
ANSWER: 25.203
First you have got to do the long multiplication by hand.
When you get to the final solution, check that you have got the decimal point in the right place.
The position of the decimal point in the final product is three places from the right.
That is because 8.13 has 2 decimal places and 3.1 has 1 decimal place.
Adding these together, 1 plus 2 equals 3 places. The multiplication should look like this:
8.13
x 3.1
.813
24.390
25.203
Answer 3 – Fractions
Two people are going to give money to a foundation for a project. Person A will provide one-half of the money. Person B will donate one-eighth of the money. What fraction represents the unfunded portion of the project?
ANSWER: 3/8
In any problem like this, all of the constituent fractions must add up to 1, which represents 100% participation by all of the contributors.
Let’s put in the variable U for the unfunded portion of the project:
A + B + U = 1
Now put in the fractions that are stated in the problem:
1/2 + 1/8 + U = 1
Remember that the denominator is the number on the bottom of the fraction.
All of the denominators need to be the same. This is called finding the lowest common denominator.
Now find the LCD (lowest common denominator) to solve the problem:
4/8 + 1/8 + U = 1
5/8 + U = 1
U = 1 − 5/8
U = 3/8
Answer 4 – Algebraic Equations with One Variable
If 5x − 2(x + 3) = 0, then x = ?
ANSWER: 2
For CPT math problems like this one, first you need to perform the multiplication on the items in parentheses:
5x − 2(x + 3) = 0
5x − 2x − 6 = 0
The next step is to isolate the integers to one side of the equation:
5x − 2x − 6 + 6 = 0 + 6
3x = 6
Then solve for x:
3x = 6
x = 6/3
x = 2
Answer 5 – Expanding Polynomials
Simplify the following equation: (x + 3y)2
ANSWER: x2 + 6xy + 9y2
(x + 3y)2 = (x + 3y)(x + 3y)
For algebra problems like this you need to use a method called “FOIL” (First – Outside – Inside – Last).
FIRST: Find the product of the terms on the left side of each set of parentheses.
x × x = x2
OUTSIDE: Find the product of the terms on the outer sides of each set of parentheses.
x × 3y = 3xy
INSIDE: Find the product of the terms on the inner sides of each set of parentheses.
3y × x = 3xy
LAST: Find the product of the terms on the right-hand sides of each set of parentheses.
3y × 3y = 9y2
Add these 4 products to get the solution:
x2 + 3xy + 3xy + 9y2 =
x2 + 6xy + 9y2
Answer 6 – Polynomials and the FOIL Method
(x + 3y)(x − y) = ?
ANSWER: x2 + 2xy − 3y2
You can practice the FOIL method one more time with this solution.
FIRST: x × x = x2
OUTSIDE: x × − y = − xy
INSIDE: 3y × x = 3xy
LAST: 3y × − y = −3y2
Then add the 4 products together for the solution:
x2 − xy + 3xy − 3y2 =
x2 + 2xy − 3y2
Answer 7 – Laws of Exponents
58 × 52 = ?
ANSWER: 510
You will see quite a few questions on the examination on exponent laws.
First of all, you need to know the difference between base numbers and exponents.
5 is the base number for this equation. 8 and 2 are the exponents.
Since the base number is the same and you need to multiply, you simply add the exponents.
On the other hand, in order to divide, you subtract the exponents:
58 ÷ 52 =
58 − 2 =
56
Answer 8 – Permutations
Determine how many 3 letter permutations can be derived from the five letter set: F U N K Y
ANSWER: 60
The formulas for permutations and combinations look similar, but these are two distinct mathematical concepts.
Permutations take into account the order of the items in each group, but combinations do not.
For instance, the letter set F U N is one permutation and U N F is another permutation.
However, F U N and U N F are the same combination.
You should use this formula for permutations:
N! / (N − S)! =
N represents the amount of items in the set.
Here we have 5 items in the group because there are five letters in the set F U N K Y.
S represents the size of the permutations.
We have been asked to find 3 letter permutations, so S is 3.
The exclamation point means that you have to multiply the stated number by every positive integer less than it.
So here is how to do the solution:
N = 5 and S = 3
N! / (N − S)! =
(5 × 4 × 3 × 2 × 1) / (5 − 3)! =
(5 × 4 × 3 × 2) / (2 × 1) =
120 / 2 =
60
Answer 9 – Coordinate Geometry and Midpoints
Find the coordinates x, y of the midpoint of the line segment on a graph that connects the points (−5, 3) and (3, −5).
ANSWER: (−1, −1)
You will also see several questions on coordinate geometry on the test.
For the points ( x1, y1) and (x2, y2), the midpoint is:
(x1 + x2) / 2 , (y1 +y2) / 2
Now calculate the midpoints like this:
(−5 + 3) / 2 = midpoint x
−2 / 2 = midpoint x
−1 = midpoint x
(3 + − 5) / 2 = midpoint y
−2 / 2 = midpoint y
−1 = midpoint y
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