Cpt Review

Review for CPT Placement Examination Math 099 and 100A Elementary Algebra The following exercises are a sample of the concepts and skills covered in the CPT for new undergraduate students enrolling at St. Thomas University. The answer key for these problems is located at the end of this document. 1. Simplify −1 − ( −7) − ( −4)

2. Simplify

1 2  5 − +−  2 3  6

3. Simplify −7 + 2 + (−2) + 4

4. Simplify:

7 3 • 9 7

3 4 5. Simplify 9 − 16 −

6. Evaluate:

yz when y = 7 and z = -11 y+z

7. Evaluate: d = rt if

8.

r=

6x y+2 and t = y 2x

Evaluate − c − 5b; when c = −1 and b = −2

9. 8x + 5x + 2x + 4x = 114, then 5x + 3 =

10. Solve

1 3 a − = 2 4 5

Review for CPT Placement Examination Math 099 and 100A 1 11. Solve for h: V = h ( a + b ) 3

12. If r = 5z then 15z = 3 y, then r = ?

13. Solve the following equation for A :

2A

/3 = 8 + 4A

14. If one-third of a number is added to three times the number, the result is 30. Find the number. 15. If Leah is 6 years older than her sister, Sue, and John is 5 years older than Leah, and the total of their ages is 41. Then how old is Sue? 16. If 300 jellybeans cost you x dollars. How many jellybeans can you purchase for 50 cents at the same rate?

17. Solve

x x ≥ 2+ 3 6

18. Subtract (8m + 6m − 4) − (5m − 2m + 2) 3

19. Expand (2 x − 7)

2

2

3

2

6 x 4 − 8 x3 + 9 20. Simplify 3x3

 3m−2 n0  21. Simplify  −7 4   m n 

−2

22. Solve the equation 5 x −20 x + 20 x = 0 3

23. Simplify

3k − 21 k 2 − 5k − 14

2

Review for CPT Placement Examination Math 099 and 100A

24. Simplify

x 2 + x+7 x−7

10m2 n 4 15mn6 ÷ 25. Simplify 9m3n 21m2 n

26. Solve

4 3 1 − = m 2m 2

27. If the average of three numbers is V. If one of the numbers is Z and another is Y, what is the remaining number?

28. Simplify

29. Simplify:

49 + 82 + 62 + 1 49 x10 y 5 7 x2 y

30. The formula to calculate the circumference of a circle is C = 2π r , where C is the circumference and r is the radius of the circle. What is the radius of the circle that has a circumference of 6π ? 31. (x – 1) is the factor of which polynomial?

x 2 − x − 2 or 2x2 − 5 x + 3

32. Simplify:

x x x −  −  23 4

33. Find the area of a circle if the radius is (x + 3).

Review for CPT Placement Examination Math 099 and 100A 34. Find the perimeter of a rectangle when the width is x and the length is x + 3. 35. The length of the hypotenuse in a right triangle is 20 centimeters. If the length of one leg is 16 centimeters, find the length of the other leg.

Review for CPT Placement Examination Math 099 and 100A Answer key 1. Simplify −1 − ( −7) − ( −4) Remember: Two consecutive negatives becomes a positive. Follow order of operations from left to right. -1 + 7 + 4 6+4 10

2. Simplify

1 2  5 − +−  2 3  6

First, find least common denominator of 2, 3 and 6 which is 6. Second, find equivalent fractions with denominator 6. Third, follow order of operations from left to right.

1 1× 3 3 = = 2 2×3 6 2 2× 2 4 = = 3 3× 2 6 5 5 ×1 5 − =− =− 6 6 ×1 6 3 4  5  3 − 4 + ( −5 ) −1 + ( −5 ) −6 − +−  = = = = −1 6 6  6 6 6 6

3. Simplify

−7 + 2 + (−2) + 4

First, simplify inside the absolute value (i.e. within bars). Second, perform the absolute value. The absolute value of a number is always positive. Third, follow order of operations from left to right.

Review for CPT Placement Examination Math 099 and 100A

−7 + 2 + (−2) + 4 −5 + (−2) + 4 5 + (−2) + 4 3+ 4 7 4. Simplify:

7 3 • 9 7

Simplify the fractions before you multiply (this is called cross cancellation). 1

1

3

1

7 3 1 •1 1 • = = 9 7 3 •1 3

3 4 5. Simplify 9 − 16 −

3 3 9 4 − ÷ − is another way to write the division 9 4 16 − 16 −

To divide by a fraction, you must multiply by the reciprocal of the fraction. −1

−4

3 9 3 16 −/• 3 −16 // 4 − ÷− = − •− = = 4 16 4 9 4/• 9/ 3 1

6. Evaluate:

3

yz when y = 7 and z = -11 y+z

Substitute the variables by its values and follow order of operations. Fractions are division problems, so dividing two negatives yield a positive.

( 7 )( −11) = −77 = 77 yz = y + z 7 + ( −11) −4 4 7. Evaluate: d = rt if

r=

Review for CPT Placement Examination Math 099 and 100A

6x y+2 and t = y 2x

Substitute the variables by its values and follow order of operations. Caution: You cannot cancel out terms! 3

 6 x   y + 2  6/ x/ ( y + 2 ) 3 ( y + 2 ) 3 ( y + 2 ) d = rt =   = = = 2/ xy 1y y /  y   2x  1 8. Evaluate − c − 5b; when c = −1 and b = −2 Substitute the variables by its values and follow order of operations.

−c − 5b = − ( −1) − 5 ( −2 ) = 1 + 10 = 11 9. 8x + 5x + 2x + 4x = 114, then 5x + 3 = First, solve the equation 8x + 5x + 2x + 4x = 114. Combine like terms. 8x + 5x + 2x + 4x = 114 19x=114 Divide by 19 both sides of the equation. 19 x 114 = 19 19

x=6 Second, substitute x = 6 into the expression 5x + 3 and follow order of operations. 5(6) + 3 = 30 + 3 = 33

Review for CPT Placement Examination Math 099 and 100A 10. Solve

1 3 a − = 2 4 5

Eliminate the fractions in the equation by multiplying each term of the equation by the least common denominator.

1 3 a 20 • − 20 • = 20 • 2 4 5 Simplify. 10 – 15 = 4a -5 = 4a Divide both sides of the equation by 4. −5 4 a = 4 4 5 − =a 4 1 11. Solve for h: V = h ( a + b ) 3

Eliminate the fraction in the equation by multiplying both sides of the equation by the denominator. 1 V • 3 = h (a + b) • 3 3 3V = h ( a + b )

Next, undo the multiplication by (a + b). Divide both sides of the equation by (a + b).

h (a + b) 3V = ( a + b) ( a + b) 3V =h ( a + b) or h=

3V (a + b)

Review for CPT Placement Examination Math 099 and 100A 12. If r = 5z then 15z = 3 y, then r = ? First, solve for z in the equation 15z = 3y.

15 z 3 y = 15 15 3 z= y 15 1 z= y 5 Second, replace the value of z that was found previously in the equation r = 5z. 1   1 r = 5z = 5  y  =  5 •  y = 1y = y 5   5 So, r = y.

13. Solve the following equation for A :

2A

/3 = 8 + 4A

Eliminate the fraction in the equation by multiplying each term of the equation by the denominator. 2A • 3 = 8•3 + 4A•3 3 2 A = 24 + 12 A

Solve for A.

2 A − 12 A = 24 + 12 A − 12 A −10 A 24 = −10 −10 24 12 A=− =− 10 5 14. If one-third of a number is added to three times the number, the result is 30. Find the number. Translate the problem into an equation. Watch the keywords! 1 n + 3n = 30 3

Review for CPT Placement Examination Math 099 and 100A Eliminate the fraction, combine like terms and solve for n.

1 n • 3 + 3n • 3 = 30 • 3 3 n + 9n = 90 10n 90 = 10 10 n=9 15. If Leah is 6 years older than her sister, Sue, and John is 5 years older than Leah, and the total of their ages is 41. Then how old is Sue? Translate the problem into an equation. Watch the keywords! L represents Leah’s age; S represents Sue’s age and J represents John’s age. L=6+S J=5+L L + S + J = 41 Substitute the value of L into the equation J = 5 + L J = 5 + (6 + S) J = 11 + S Substitute the values of L and J into the equation L + S + J = 41 and solve for S. (6 + S) + S + (11 + S) = 41 17 + 3S = 41 3S = 41 – 17 3S = 24 S=8 Find the numerical values for L and J by substituting S = 8. L = 6 + 8 = 14 J = 11 + 8 = 19. Sue is 8 years old. Leah is 14 years old. John is 19 years old. 16. If 300 jellybeans cost you x dollars. How many jellybeans can you purchase for 50 cents at the same rate? Set up a proportion. 300 jelly beans ? jelly beans = x dollars 50 cents

Review for CPT Placement Examination Math 099 and 100A Convert the 50 cents into dollars.

300 jelly beans ? jelly beans = x dollars 0.50 dollars Multiply each side of the equation by 0.50 dollars to solve for ? jelly beans.

300 jelly beans ? jelly beans • 0.50 dollars = • 0.50 dollars 0.50 dollars x dollars 300 jelly beans • 0.50 = ? jelly beans x 150 jelly beans = ? jelly beans x 17. Solve

x x ≥ 2+ 3 6

Eliminate the fractions in the equation by multiplying each term of the equation by the least common denominator.

x x •6 ≥ 2•6 + •6 3 6 2 x ≥ 12 + x Solve for x.

2 x − x ≥ 12 + x − x x ≥ 12 The graph of this inequality is

18. Subtract (8m + 6m − 4) − (5m − 2m + 2) 3

2

2

3

Convert the subtraction into an addition by changing the sign of each term of the polynomial after the subtraction symbol.

(8m

3

+ 6m2 − 4 ) + ( −5m2 + 2m3 − 2 )

Review for CPT Placement Examination Math 099 and 100A Combine like terms.

( 8m

3

+ 2m3 ) + ( 6m 2 − 5m 2 ) + ( −4 − 2 )

10m3 + m2 − 6

19. Expand (2 x − 7)

2

Apply the definition of exponents and FOIL.

(2 x − 7)2 = ( 2 x − 7 )( 2 x − 7 )

= 2 x ( 2 x ) + 2 x ( −7 ) − 7 ( 2 x ) − 7 ( −7 ) = 4 x 2 − 14 x − 14 x + 49 = 4 x 2 − 28 x + 49

6 x 4 − 8 x3 + 9 20. Simplify 3x 3 Divide each term of the numerator by the denominator and apply exponent rules.

6 x 4 − 8 x 3 + 9 6 x 4 8 x3 9 8 1 = − + = 2 x − + 3x3 3 x 3 3 x3 3 x 3 3 x3

 3m−2 n0  21. Simplify  −7 4   m n 

−2

First, simplify inside the parenthesis by applying exponent rules. −2

 3m−2 n0  −2 −( −7 ) 0 − 4 n  −7 4  = 3m m n  

(

) = (3m n ) −2

5 −4 −2

Second, eliminate the parenthesis by applying exponent rules.

( 3m n )

5 −4 −2

= 31( −2) m5( −2) n−4( −2) = 3−2 m−10 n8

Review for CPT Placement Examination Math 099 and 100A Last, simplify expression with positive exponents. Apply negative exponent rule.

1 1 n8 8 3 m n = 2 • 10 • n = 10 3 m 9m −2

−10 8

22. Solve the equation

5 x 3 −20 x 2 + 20 x = 0

First, factor completely the trinomial.

5 x 3 −20 x 2 + 20 x = 0 5x ( x2 − 4 x + 4) = 0

5 x ( x − 2 )( x − 2 ) = 0 Second, equal each factor to zero. Solve for x.

5 x ( x − 2 )( x − 2 ) = 0 5 x = 0 or x − 2 = 0 x = 0 or x = 2 23. Simplify

3k − 21 k − 5k − 14 2

First, factor completely the polynomials.

3( k − 7) 3k − 21 = k 2 − 5k − 14 ( k − 7 )( k − 2 ) Second, simplify rational expression by canceling common factors (i.e. k – 7).

3( k − 7) 3 = ( k − 7 )( k − 2 ) k − 2 24. Simplify

x 2 + x+7 x−7

First, find least common denominator (LCD). The LCD for this problem is (x +7)(x – 7). Second, find equivalent rational expressions with LCD of (x +7)(x – 7).

Review for CPT Placement Examination Math 099 and 100A

x  x−7 2  x+7  +   x+7 x−7 x−7 x+7 x ( x − 7)

+

2 ( x + 7)

( x + 7 )( x − 7 ) ( x − 7 )( x + 7 ) Third, apply distributive property and combine like terms. x2 − 7 x 2 x + 14 + ( x + 7 )( x − 7 ) ( x − 7 )( x + 7 ) x 2 − 7 x + 2 x + 14 ( x + 7 )( x − 7 ) x 2 − 5 x + 14 ( x + 7 )( x − 7 )

Last, factor the polynomial in the numerator and simplify rational expression by canceling common factors.

( x − 7 )( x + 2 ) = x + 2 x 2 − 5 x + 14 = ( x + 7 )( x − 7 ) ( x + 7 )( x − 7 ) x + 7 10m2 n 4 15mn6 ÷ 25. Simplify 9m3n 21m2 n To divide by a fraction, you must multiply by the reciprocal.

10m2 n4 21m2 n × 9m3n 15mn6 Cross cancel numerical factors and apply exponent rules. 2

7

10m n 21m n 10 m n • 21 m2 n 14m4 n5 14 × = = = 9m3n 15mn6 9 m3n • 15 mn6 9m4 n 7 9n 2 2 4

2

3

2 4

3

Review for CPT Placement Examination Math 099 and 100A 26. Solve

4 3 1 − = m 2m 2

Eliminate the fractions in the equation by multiplying each term of the equation by the least common denominator. Simplify and solve for m.

4 3 1 • 2m − • 2m = • 2m m 2m 2 8−3 = m 5=m or m=5 27. If the average of three numbers is V. If one of the numbers is Z and another is Y, what is the remaining number?

Translate the problem into an equation using the formula to calculate the average. Assume X is the third number.

Average = V=

Sum of DataValues Number of DataValues

X +Y + Z 3

Solve for X.

X +Y + Z •3 3 3V = X + Y + Z

V •3 =

3V − Y − Z = X + Y + Z − Y − Z 3V − Y − Z = X or X = 3V − Y − Z

Review for CPT Placement Examination Math 099 and 100A 28. Simplify

49 + 82 + 62 + 1

Follow order of operations. Simplify first the expression inside the second square root.

49 + 82 + 62 + 1 49 + 64 + 36 + 1 49 + 100 + 1 Perform the square roots from left to right.

49 + 100 + 1 7 + 10 + 1 Add from left to right. 17 + 1 18

29. Simplify:

49 x10 y 5 7 x2 y

Simplify first the expression inside the square root.

49 x10 y5 = 7 x8 y 4 2 7x y Apply the properties of radicals.

7 x8 y 4 = 7 • x8 • y 4 = 7 • x 4 • y 2 = x 4 y 2 7

Review for CPT Placement Examination Math 099 and 100A 30. The formula to calculate the circumference of a circle is C = 2π r , where C is the circumference and r is the radius of the circle. What is the radius of the circle that has a circumference of 6π ? Substitute C in the formula C = 2π r with 6π . Solve for r. C = 2π r 6π = 2π r 6π 2π r = 2π 2π 3= r or r =3

31. (x – 1) is the factor of which polynomial?

x 2 − x − 2 or 2x2 − 5 x + 3 Factor each trinomial.

x2 − x − 2

or

2x 2 − 5 x + 3

( x − 2 )( x + 1)

or

( 2 x − 3)( x − 1)

Therefore, (x – 1) is a factor of 2x 2 − 5 x + 3 .

32. Simplify:

x x x −  −  23 4

First, apply the distributive property. Careful with the signs!

x x x x x x x x2 x2 −  −  = −  − −  = − + 23 4 23 2 4 6 8 Second, add the rational expressions. Remember, to add rational expressions, it is necessary to have common denominators.

x 2  4  x 2  3  −4 x 2 3x 2 −4 x 2 + 3x 2 − x 2 −  +   = + = = 6  4 8 3 24 24 24 24

Review for CPT Placement Examination Math 099 and 100A 33. Find the area of a circle if the radius is (x + 3). Use the formula A = π r 2 and replace r with (x + 3). A = π r 2 = π ( x + 3)

2

Expand the binomial square.

( x + 3)

2

= ( x + 3)( x + 3) = x 2 + 3 x + 3x + 9 = x 2 + 6 x + 9

Replace ( x + 3) with x 2 + 6 x + 9 . 2

A = π r 2 = π ( x + 3) = π ( x 2 + 6 x + 9 ) 2

34. Find the perimeter of a rectangle when the width is x and the length is x + 3. Use the formula of perimeter and replace W with x and L with x + 3 P = 2L + 2W P = 2(x + 3) + 2x Simplify. P = 2x + 6 + 2x P = 4x + 6 35. The length of the hypotenuse in a right triangle is 20 centimeters. If the length of one leg is 16 centimeters, find the length of the other leg. Use Pythagoras theorem: a 2 + b 2 = c 2 where b = 16 and c = 20. Solve for a. a 2 + b2 = c2 a 2 + 162 = 202 a 2 + 256 = 400 a 2 = 400 − 256 a 2 = 144 a = 144 a = 12