Solutions: Topics in PreAlgebra
If you would like to better understand the curriculum, here are the learning objectives for this unit.
Order of Operations
Exercise  Problem  Solution  
1.  9 + 6 x (8 – 5)  9 + 6 x (8 – 5) = 9 + 6 x 3 = 9 + 18 = 27 

2.  (14 – 5) ÷ (9 – 6)  (14 – 5) ÷ (9 – 6) = 9 ÷ (9 – 6) = 9 ÷ 3 = 3 

3.  5 x 8 + 6 ÷ 6 – 12 x 2  5 x 8 + 6 ÷ 6 – 12 x 2 = 40 + 6 ÷ 6 – 12 x 2 = 40 + 1 – 12 x 2 = 40 + 1 – 24 = 41 – 24 = 17 

4. 


5.  A caterer charges a setup fee of $50, plus $20 per person. How much will the caterer charge if 35 people attend the party, and the customer has a coupon for $100 off the total?  50 + 20 x 35 – 100 = 50 + 700 – 100 = 750 – 100 = 650 Answer: $650 
Order of Operations With Exponents
Exercise  Problem  Solution 
1.  3^{2} x 4^{3}  3^{2} x 4^{3} = 9 x 4^{3} = 9 x 64 = 576 
2.  27 – 256 ÷ 4^{3}  27 – 256 ÷ 4^{3} = 27 – 256 ÷ 64 = 27 – 4 = 23 
3.  9 x (5 + 3)^{2} – 144  9 x (5 + 3)^{2} – 144 = 9 x 8^{2} – 144 = 9 x 64 – 144 = 576 – 144 = 432 
4.  7 + 3 x 2^{4} ÷ 6  7 + 3 x 2^{4} ÷ 6 = 7 + 3 x 16 ÷ 6 = 7 + 48 ÷ 6 = 7 + 8 = 15 
5.  A carpenter charges $10 per square foot to lay a floor. If a squareshaped hallway is 6 feet along one side, and the customer has a coupon for $25 off the total, then how much will the floor cost?  10 x 6^{2} – 25 = 10 x 36 – 25 = 360 – 25 = 335 Answer: $335 
Order of Operations With Integers
Exercise  Problem  Solution  
1.  18 – (^{–}6)^{2} ÷ 9  18 – (^{–}6)^{2} ÷ 9 = 18 – 36 ÷ 9 = 18 – 4 = 14 or ^{+}14 

2.  [49 + 3(49)] ÷ ^{–}17  [49 + 3(49)] ÷ ^{–}17 = [49 + 3(^{–}5)] ÷ ^{–}17 = [49 + ^{–}15] ÷ ^{–}17 = 34 ÷ ^{–}17 = ^{–}2 

3.  5^{3} + (^{–}9)(^{+}2) ÷ 6  5^{3} + (^{–}9)(^{+}2) ÷ 6 = 125 + (^{–}9)(^{+}2) ÷ 6 = 125 + ^{–}18 ÷ 6 = 125 + ^{–}3 = 122 or ^{+}122 

4. 


5.  (^{–}4)(^{–}3) ÷ 6 – 2[5 – ^{–}8 + (6 ÷ 2)]  (^{–}4)(^{–}3) ÷ 6 – 2[5 – ^{–}8 + (6 ÷ 2)] = (^{–}4)(^{–}3) ÷ 6 – 2[5 – ^{–}8 + 3] = (^{–}4)(^{–}3) ÷ 6 – 2[13 + 3] = (^{–}4)(^{–}3) ÷ 6 – 2[16] = (^{–}4)(^{–}3) ÷ 6 – 32 = 12 ÷ 6 – 32 = 2 – 32 = ^{–}30 
Writing Algebraic Expressions
Exercise  Problem  Solution 
1.  Fifteen less than twice a number  2x – 15 
2.  Three times a number increased by seventeen  3a + 17 
3.  The product of nine and a number, decreased by six  9m – 6 
4.  Thirty divided by seven times a number  None of the above. 
5.  Jenny earns $30 a day working part time at a supermarket. Write an algebraic expression to represent the amount of money she will earn in d days.  30d 
Writing Algebraic Equations
Exercise  Problem  Solution 
1.  Twice a number is twentyeight.  2x = 28 
2.  The quotient of thirtynine and a number is three.  
3.  Five less than three times a number is fortysix.  3t – 5 = 46 
4.  Three is twentyone divided by the sum of a number and five.  None of the above. 
5.  Eric had $197 in his savings account before he was paid his weekly salary. His current savings balance is $429. How much money does Eric earn each week?  197 + n = 429 
Practice Exercises
Exercise  Problem  Solution  
1.  Evaluate this arithmetic expression: 7 x 8 – 6 ÷ 2 + 18 x 3 
7 x 8 – 6 ÷ 2 + 18 x 3 = 56 – 6 ÷ 2 + 18 x 3 = 56 – 3 + 18 x 3 = 56 – 3 + 54 = 53 + 54 = 107 Answer: 107 or ^{+}107 

2.  Evaluate this arithmetic expression: 
Answer: 5 or ^{+}5 

3.  Evaluate this arithmetic expression: 15 – 2 x 3^{2} ÷ 6 
15 – 2 x 3^{2} ÷ 6 = 15 – 2 x 9 ÷ 6 = 15 – 18 ÷ 6 = 15 – 3 = 12 Answer: 12 or ^{+}12 

4.  Evaluate this arithmetic expression: 36 ÷ (74)^{2} + 21 
36 ÷ (74)^{2} + 21 = 36 ÷ 3^{2} + 21 = 36 ÷ 9 + 21 = 4 + 21 = 25 Answer: 25 or ^{+}25 

5.  Evaluate this arithmetic expression: [54 – 3(2 + 4)] ÷ ^{–}9 
[54 – 3(2 + 4)] ÷ ^{–}9 = [54 – 3(6)] ÷ ^{–}9 = [54 – 18] ÷ ^{–}9 = 36 ÷ ^{–}9 = ^{–}4 Answer: ^{–}4 

6.  Evaluate this arithmetic expression: 
Answer: ^{–}3 

7.  Which algebraic expression correctly represents this phrase? Twentyfive less than three times a number. 
3x – 25  
8.  Which algebraic expression correctly represents this phrase? The quotient of a number and seven, decreased by ten. 

9.  Which sentence correctly represents this algebraic equation? 2w – 9 = 32 
Nine less than twice a number is thirtytwo.  
10.  Which sentence correctly represents this algebraic equation? 
The quotient of fortytwo and a number is seven. 
Challenge Exercises
Exercise  Problem  Solution  
1.  An agent charges $150 per gig to book a rock band, plus $75 per month for travel expenses. What was his monthly fee if he booked 6 gigs for the band last month?  150 x 6 + 75 = 900 + 75 = 975 Answer: $975 

2.  A gardener charges $8 per square foot to lay sod. If a square garden is 7 feet along each side, how much will he charge to lay sod on it?  8 x 7^{2} = 8 x 49 = 392 Answer: $392 

3.  Six people in a club will share the expenses of a party that costs $240. How much will Katie pay for her share of the party if the club also credits her $8?  240 ÷ 6 – 8 = 40 – 8 = 32 Answer: $32 

4.  Evaluate the following expression: 


5.  Jesse spends $5 a day on lunch. Which algebraic expression correctly represents the amount of money he will spend on lunch in x days?  5x  
6.  Which algebraic expression correctly represents this phrase? The quotient of twelve and seven times a number, decreased by five 

7.  Which algebraic equation correctly represents this sentence? A number increased by eight is nineteen. 
y + 8 = 19  
8.  Which algebraic equation correctly represents this sentence? Twentyfive is three times a number, decreased by eight. 
25 = 3w – 8  
9.  Which sentence correctly represents this algebraic equation? 15 = 16y – 3 
None of the above.  
10.  Which sentence correctly represents this algebraic equation? 
The quotient of three times a number and five is thirty. 