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Math-I (grade 8)

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  • Аннотация:
    Math lessons for students, grade 8

Vladimir Luzgin

 Math Lessons for Gifted Students

Grade 8

Center Impulse

Week-end and evening classes for gifted students grades 5-9
Canada, ON, L4K 1T7, Vaughan (Toronto),
80 Glen Shields Ave., Unit #10,
Phone (416)826-7270
vluzgin@hotmail.com

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Content

Click on the lesson!


Lesson 01.
Lesson 02.
Lesson 03.
Lesson 04.
Lesson 05.
Lesson 06.
Lesson 07.
Lesson 08.
Lesson 09.
Lesson 10.
Lesson 11.
Lesson 12.
Lesson 13.
Lesson 14.
Lesson 15.



Lesson 01


1. Solve the following motion problems.
a) If a car goes half the distance at 40 miles per hour, and then goes another half at 60 miles per hour, what is its average speed for the whole trip?
b) If a car goes half the time at 40 miles per hour, and then goes another half at 60 miles per hour, what is its average speed for the whole trip?
c) Pete can run around a circular track in 40 s. Frank, running in the opposite direction, meets Pete every 15 s. How many seconds does it take Frank to run around the track?

2. Simplify by combining like terms.
1) 3x + 4y - 5xy + 3y + 5yx - 4x aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa 2) 2ab + 0.7b2 - 5ab + 1.2b2 + 8ab
3) 2a2b - 8b2 + 5a2b + 5c2 - 3b2 + 4c2 aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa 4) 8x2y - 6xy2 - 7yx2 + 6y2x - x2y2
5) 3xy2 + 4x3 - 5x2y - 3x3 + 4x2y - 9xy2 aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa 6) 9xy2 + 6xy - 8xy2 - 8x2y + 6x2y2 - 6xy

3. Expand and simplify by combining like terms.
1) (0.1c - 0.4c2) - (0.1c - 0.5c2) aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa 2) (13x - 11y + 10z) - (- 15x + 10y - 15z)
3) (17a + 12b - 14c) - (11a - 10b - 14c) aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa 4) (3a2 + 8a - 4) - (3 + 8a - 5a2)
5) (b3 + 3b2 + 4b - 5) - (- 3b - 2b2 + b3) aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa 6) (7m2 - 4mn - n2) - (2m2 - mn + n2)
7) (3x2 + 5xy + 7x2y) - (5xy + 3x2) - (7x2y - 3x2) aaaaaaaaaaaaaaaaaaaaaaaa 8) (1/2 a + 1/3 b) - (5/2 a - 2/3 b) + (a + b)

4. Expand and simplify.
1) 6(2x - 3y) - 3(3x - 2y) aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa 2) 5(a - b) - 4(2a - 3b)
3) - 2(3x - 2y) - 5(2y - 3x) aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa 4) 7(4p + 3) - 6(5 + 7p)
5) (x2 - 1)(3x) - (x2 - 2)(2x) aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa 6) 2b(4a2 - 3b) - 3b(3a2 - 4b)

5. Find:
1) 11.25% of 51.2 aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa 2) 37.125% of 428/33
3) 562/3% of 55/119 aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa 4) 61/3% of 8124/247

6. Solve the following problems:
a) In a recent survey, 40% of the cars contained two or more people. Of those cars containing only one person, 25% contained a male. Find the percentage of all cars, which contained exactly one female and no male.
b) If x is 6% of y and y is 7% of z, find x in terms of z.
c) In a grade 8 class of 380 students, 70% can swim, 68% of the boys can swim, and 113 girls can swim. How many girls are in the class?
d) In a recent year, Statistics Canada reported that 8.9 million persons were employed and 810 000 were unemployed. If the size of the work force does not change, how many of the unemployed need to find jobs to bring the unemployment rate down to 4%?

7. Solve the following word problems:
a) James has two-fifths the amount that Lorna has, and Muriel has seven-ninths the amount that James has. Together, they have $770. Find how much each person has.
b) John, Bill, and Joe share a bill of $183. Bill pays 4/5 as much as John, and Joe pays 4/5 as much as Bill. How much does each pay?
c) Girish had $2 more than three times the amount that Joseph had. He gave Joseph $5 who then had one-half as much as Girish. How much did each person have at first?

8. Apply Pythegorean Theorem to solve the following problems.
a) A ladder 3.25 m long is leaning against a wall 3.75 m high. The foot of the ladder is 1.25 m from the base of the wall. Determine the distance, in meters, from the top of the ladder to the top of the wall.
b) A 6.5 m ladder is placed against a wall with the foot of the ladder 2.5 m from the wall. If the top of the ladder slips 0.8 m, how far will the bottom of the ladder slip?
c) Show that two isosceles triangles with sides 5 cm, 5 cm, 6 cm and 5 cm, 5 cm, 8 cm have the same area.
d) Five times the square on the hypotenuse of a right-angled triangle is equal to four times the sum of the squares on the medians drawn to the other two sides. Prove.

9. Solve the following Number Theory problems.
a) A field trip is being planned for some Grade 8 students. The number of people going on the trip, including students, teachers and parents, is 1161. Each bus can hold at most 49 passengers. If all busses have the same number of passengers, what is the least number of buses needed for the field trip?
b) A six-digit number is formed by repeating a three-digit number; for example, 265265 or 345345. What is the largest integer, which will divide all such numbers?
c) What is the smallest natural number by which 2 520 can be multiplied so that the result is a perfect square?

10. Evaluate (do without a calculator and show your work).
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Lesson 02


1. Solve the following motion problems.
a) A boat moved by an engine, which would propel it at 10 miles per hour in still water, makes a 1.5-hour trip downstream, and the return trip in 3.5 hours. Find the rate of the stream.
b) Two cyclists, 25 km apart, set out at the same time and meet in 50 min. Had they been cycling in the same direction, the faster would have overtaken the slower in 5 h. Find their cycling speeds.
c) A motorboat can travel upstream on a river at 18 km/h and downstream at 30 km/h. How far upstream can the boat travel if it leaves at 8:00 a.m. and must return by noon?

2. Simplify by combining like terms.
1) 5a2b + 10ab2 + 12a2b2 + 6ab2 - 12a2b - 8a2b2 + 7a2b aaaaaaaaaaaaaaaaaaaaaaaaaaaa 2) 4abc - 5a2bc + 6abc2 - 8abc + 7a2bc + 3abc2 - 2abc
3) 3xyz - 4x2yz + 5xy2z - 7xyz - 8xy2z + 6x2yz + 12xy aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa 4) 1.8x2y - 7.2xy + 0.8xy2 - 0.7x2y + 4.2xy2 - 4.8xy + xy2

3. Expand and simplify by combining like terms.
1) (0.3a - 1.2b) + (a - b) - (1.3a - 0.2b) aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa 2) 5x2 + 5x3 + (x3 - x2) - (- 2x3 + 4x2)
3) (1 - x + 4x2 - 8x3) + (2x3 + x2 - 6x - 3) - (5x3 + 8x2) aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa 4) (0.5a - 0.6b + 5.5) - (- 0.5a + 0.4b) + (1.3b - 4.5)
5) 0.6ab2 + (2a3 + b3 - (3ab2 - (a3 + 2.4ab2 - b3))) aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa 6) 12.5x2 + y2 - (8x2 - 5y2 - (- 10x2 + (5.5x2 - 6y2)))

4. Simplify and write it in descending powers of the terms. State the degree of each polynomial.
1) (12a2)(3ba) - (2ab)(3ab2) + (11ab)a
2) (2ab2)(4ab) - (3a2)(8a)(ba) - (2ab)(ab2)
3) (1.5xy2)( - 4xyz) - (4xyz)(5x2yz)
4) (2m)(4n) - (3a)(2b) - (0.2n)(5m) + b(5a) - 5mn + 8ab
5) 12ab - 0.2xy - (2a)(5b) + (6x)(0.2y) + a(-3b)
6) (2abc)(5a) + (15/7 a2)(7/12 bc) - (22/3 ab)( - 3/8 a)
7) (3nmk)(4n) - (3/8 nm)(22/3 nk) + (2/9 n2m)(- 41/2 k)

5. aaa 1) 3/5 is what percent of 1.5? aaaaaaaaaaaaaaaaa 2) 3.9 is what percent of 19.5?
3) 91/6 is what percent of 71/3? aaaaaaaaaaaaaaa 4) 21/4 is what percent of 33/7?

6. Solve the following problems:
a) The Royals basketball team took 48 shots in a basketball game and made 20 baskets. The Pirates basketball team took 64 shots and made 25 baskets. Express each team's ratio of number of baskets to number of shots taken as a percent. Which team had the better percentage?
b) A bowl contains 40 g of white rice and 60 g of brown rice. If 100 g of white rice is added to the mixture, what percentage of the new mixture is white rice?
c) Bobby purchased orange juice for $2.50, a pound of bacon for $3.00, a dozen eggs for $1.25, and a loaf of bread for $1.25. What percent of his total cost is for bacon?

7. Solve the following word problems:
a) Five men do a job and 6 men do another job costing $300 more than the first. If each of the 11 men receives the same amount, find the cost of each job.
b) The five members of a club decided to buy a used microcomputer, splitting the cost equally. Later, three new members joined the club and agreed to pay their share of the purchase price. This resulted in saving of $15.00 for each of the original five members. What was the original price of the used microcomputer?
c) A parking meter contains $36.85 in dimes and quarters. If there is a total of 223 coins, how many quarters does the meter contain?

8. Apply Pythegorean Theorem to solve the following problems.
a) Two men A and B set out from the same point at the same time. A goes north at 12 miles per hour and B goes east at 5 miles per hour. How far are they apart at the end of two hours?
b) Two ships leave from the same port at 11: 30 a.m. If one sails due east at 20 miles per hour and the other due south at 15 miles per hour, how many miles apart are the ships at 2: 30 p.m.?
c) A rectangular park is 400 m long and 300 m wide. If it takes 14 min to walk around the perimeter once, how long would it take to walk across a diagonal of the park?
d) Three masts 40', 100', 25' high are erected in a straight line, the middle one being 80' from each of the others. Calculate the length of cable required to connect the tops of the masts.

9. Solve the following Number Theory problems.
a) Find the smallest natural number by which 546 must be multiplied so that the number created is divisible by 180.
b) How many numbers less than 100 whose factors are half even and half odd?
c) How many numbers less than 2000 whose factors, other than 1, are all even? List all these numbers.

10. Evaluate. Do without a calculator and show your work [x and : are maltiplication and divison signs].
a) (35/48)3 x (6/7)3 x (13/5)2 aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa b) (53 : 62)4 x (2/5)5 x (3/5)7
c) (74 : 152)3 x (5/7)6 x (3/7)5 aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa d) (4 x 322 + 7 x 321) x 57 : (19 x 274)2
e) (125 x 157 x 243) : (186 x 108 x 64) aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa f) (52 x 65 x 143 x 352) : (23 x 104 x 215)


Lesson 03


1. Solve the following motion problems.
a) An aircraft travels 2000 km from Winnipeg to Montreal in 3.2 h and returns in 4 h. If the wind speed is constant, find the wind speed and the speed of the aircraft in still air.
b) A patrol plane can carry fuel for 8 h flying time, and can fly at 300 km/h in still air. If its outbound patrol is against a 30 km/h headwind and it returns with a 30 km/h tailwind, how far can it fly against the headwind and return safely?
c) During part of a trip from Calgary to Halifax, Monique's average driving speed is 7 km/h faster than Senta's. Monique and Senta each drive for 5 hours. Altogether, they drive 995 km. On average, how fast does each drive?

2. Find the value of the following (do without a calculator and show your work).
1) -3 5/24 - (-2 11/36) aaaaaaaaaaa 2) 1 2/15 - 2 3/10 - 1 1/6 aaaaaaaaaaaaa 3) 2 5/21 - 4 1/7 + 1 1/14 aaaaaaaaaaa 4) 4 2/35 - 2 5/14 - 1 3/10
5) 1 2/9 + 2 5/6 - 5 1/5 aaaaaaaaaa 6) (2/5 + (-0.5)) + (-1 1/4) aaaaaaaaaaa 7) (0.6 + 2/3) + (-2 1/15) aaaaaaaaaaa 8) 1 2/9 - 1 1/3 + 1 5/18

3. Expand.
1) 17a (5a + 6b - 3ab) aaaaaaaaaaaaaaaaaaaaaaaaaaaaa 2) 8ab (2b - 3ac + c2)
3) 3x2y (5x + 6y + 7z) aaaaaaaaaaaaaaaaaaaaaaaaaaaaa 4) xyz (x2 + 2y2 + 3z2)
5) (1/2 a3b2 - 3/4 ab4)(4/3 a3b) aaaaaaaaaaaaaaaaaaaaaa 6) (2/3 a2b4 + 1/2 a3b)(3/2 ab3)
7) (- 26/11 ax6)(14/7 a3x3 - 23/4 a2x3 - 11ax4) aaaaaaaaa 8) (-24/9 b6y + 21/5b3y2 - 11by5)(-21/22 b4y5)

4. Solve the following problems.
a) The shadow of a tree is 14 meters long. The shadow of a person who is 1.8 meters tall is 4 meters. How tall is the tree?
b) The shadow of a telephone relay tower is 32 m long on level ground. At the same time, a boy 1.8 m tall casts a shadow 1.5 m long. What is the height of the tower?

5. aaa 1) What percentage of 6.5 is 1.3? aaaaaaaaaaaaaaa 2) What percent of 19.6 is 14.7?
3) What percentage of 3.2 is 1.2? aaaaaaaaaaaaaaa 4) What percent of 3/4 is 2?

6. Solve the following problems:
a) A pound of water is evaporated from 6 pounds of seawater containing 54% salt. What is the percentage of salt water in the remaining solution?
b) A 15-gallon mixture of 18% alcohol has 3 gallons of water added to it. Find the new strength of the mixture, as a percent.
c) If 2.5% of a number is 6, then what is the number?
d) At a mine in Newfoundland 19% of an ore is iron. How many tonnes of ore are required to yield 15 t of iron?

7. Solve the following word problems:
a) Sue and Nancy wish to buy a snack. They combine their money and find they have $4.00, consisting of quarters, dimes, and nickels. If they have 35 coins and the number of quarters is half the number of nickels, how many quarters do they have?
b) The father earns twice as much an hour than his son. If the two of them combined earns a total of $168 per day, and the father works 8 hours a day and the son works only 5 hours, how much does the father earns per hour?
c) The least of three consecutive integers is divided by 10, the next is divided by 17, and the greatest is divided by 26. What are the numbers if the sum of the quotients is 10?

8. Find the length, to one decimal place, indicated by each letter.

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9. Solve the following Number Theory problems.
a) There is 1 number less than 100 that has exactly 7 different factors. What is it?
b) List all the numbers greater than 50 but less than 100 that have exactly 6 factors.
c) How many numbers less than 100 have exactly 3 factors?

10. Write each expression as a power.
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Lesson 04


1. Solve the following motion problems.
a) The front wheels of a wagon are 7 feet in circumference and the back wheels are 9 feet in circumference. When the front wheels have made 10 more revolutions than the back wheels, what distance, in feet, has the wagon gone?
b) A plane flies 360 miles at one speed and 240 miles at another, and makes the 600-mile trip in 6 hours. By doubling its speed for the 360-mile part of the trip but keeping the original speed for the 240-mile part, it decreases the total time by 1 hour 30 minutes. Find the speeds for the 360-mile trip.
c) Boat A makes a 120-mile trip, and boat B makes a 72-mile trip, moving at uniform speeds. Boat A requires 1 hour more than B to make its trip. If each boat should make the trip of the other, the time of boat B would be 9 hours more than that of boat A. Find the speeds of the boats.

2. Find the value of the following (do without a calculator and show your work).
1) -7 2/15 + 4 1/6 - 1.2 aaaaaaaaaaaaaaaaaaaaaaaaaaa 2) -3.7 + (-5 11/30 + 3 4/15) aaaaaaaaaaaaaaaaaaaa 3) (-3 3/7 + (-2 1/14)) + (-3 1/2) aaaaaaaaaaa
4) (-3 3/8 - (- 4 1/4)) + (-1 5/6 + (-2 7/12)) aaaaaaaaaaa 5) (- 2/3 + (- 2/15)) - (-1.85 - (-1.35)) aaaaaaaaaaa 6) (-3.25 - (-1 3/4)) - (-1 2/3 - (-1 4/9))

3. Simplify and evaluate.
1) b2(5ab) - (5a)(5a2b) for a = 1/5, b = -2; aaaaaaaaaaaaaaaaaaaaaaaa 2) (x2y)(xy) - (xy2)(xy) + 4x2y2 for x = - 3, y = 2;
3) 3ab(4a2 - b2) + 4ab(b2 - 3a2) for a = 8, b = -5; aaaaaaaaaaaaaaaaa 4) 4a2(5a - 3b) - 5a2(4a + b) for a = -2, b = -3.

4. Solve the following problems:
a) Karen is 37.5 m from a church. She finds that a pencil 4.8 cm long, which is held with its base 60 mm from her eye, just blocks the church from her sight. How high is the church?
b) D and E are points on the sides AB and BC of a triangle ABC, DE || AC. If BD = 5 cm, BE = 6 cm. DE = 4 cm, and AC = 6 cm, what is the perimeter of the triangle ABC?

5. Find x if
1) 42% of x is 6.3 aaaaaaaaaaaaaaaaa 2) 375% of x is 525 aaaaaaaaaaaaaaaa 3) 210% of x is 6.3
4) 3.8% of x is 28.5 aaaaaaaaaaaaaaaa 5) 0.3% of x is 0.27 aaaaaaaaaaaaaaaa 6) 12.5% of x is 7.5

6. The table shows the number of students in the grade 7, 8, and 9 classes.

Grade Number of students
Boys
Girls
7
13
15
8
15
11
9
9
17

a) What percent of students are in each grade?
b) What percent of the students are boys?
c) What percent of the students are girls?

7. Solve the following word problems:
a) Find three consecutive even numbers such that 3 times the sum of the first and the last is 4 times the sum of the middle one and 10.
b) Find four consecutive odd numbers such that if the first is increased by 2, the second decreased by 3, the third multiplied by 4, and the forth divided by 5, the sum of the four resulting numbers is 136.
c) One-third of a number increased by 3/5 of a number three larger than the first is 27. Find the number.

8. Find the length, to one decimal place, indicated by each letter.

1)
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9. Solve the following Number Theory problems.
a) What is the smallest multiple of 300 which has all its prime factors to an even power?
b) What is the smallest multiple of 300 which is the square of a positive integer?
c) What is the smallest multiple of 300 which is the cube of a positive integer?

10. Simplify.
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Lesson 05


1. Solve the following motion problems.
a) A car 12 feet long going 70 feet per second passes a truck 38 feet long going 50 feet per second. The front end of the car is initially 30 feet behind the truck and when completely passed, the rear end of the car is 40 feet ahead of the truck. Find the time required for passing and the distance the car goes while passing.
b) As part of her physical fitness training, Brenda cycled at 30 km/h and then jogged at 8 km/h. The total time spent was 2 h, and she covered a total distance of 49 km. How much time did she spent jogging?
c) Marie drove her 18-wheeler 1280 km from Calgary to Winnipeg in 15.2 h. Part of the trip she drove in a snowstorm at an average speed of 60 km/h. The rest of the time she drove at 100 km/h. How far did she drive in the storm?

2. Find the missing number.
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3. Find each product.
1) (a + 2)(a + 3) aaaaaaaaaaaaaaaaaaaaaaaaaaaa 2) (x + 6)(x - 1)
3) (x + y)(x - y) aaaaaaaaaaaaaaaaaaaaaaaaaaaaa 4) (3a + 4b)(3a - 4b)
5) (0.5a + 3b)(0.5a - 3b) aaaaaaaaaaaaaaaaaaaaa 6) (0.2x + 0.7y)(0.2x - 0.7y)
7) (5x2 - 6y2)(6x2 - 5y2) aaaaaaaaaaaaaaaaaaaaaa 8) (x2 - 3x + 2)(x + 3)
9) (x - 4)(x2 - 7x + 5) aaaaaaaaaaaaaaaaaaaaaaa 10) (2x - 3)(x2 - 6x + 4)

4. Solve the following word problems:
1) To determine the height of a tree, Jerry placed a 2 m rod 27 m from the tree. He finds that he can just align the top of the rod with the top of the tree, when he stands 1.8 m from the rod. If Jerry's eyes 1.6 m from the ground, what is the height of the tree?
2) Jerry has a photograph measuring 20 cm by 25 cm. He wishes to get a copy that is three-quarters the area but with the same length-to-width ratio. What will be the dimensions of the copy?

5. Solve the following word problems:
1) If 56% of x is equal to 140% of 25, what is the value of x?
2) If x% of x is 16, what is x?
3) If x% of x% is 0.36, what is x?

6. Solve the following word problems:
1) When water freezes the ice formed has 9% more volume than the water. How much water must freeze to make 872 m3 of ice?
2) If a discount of 35% off the marked price of a jacket results in saving $21, what is the discounted price of the jacket?
3) What would be the marked price of an article if the cost was 302.40 and the gain was 10% of the selling price?

7. Solve the following word problems:
a) The denominator of a fraction exceeds the numerator by 3. If 5 is added to both numerator and denominator, the result is equivalent to 3/4. Find the fraction.
b) Mrs. Richards divided $45 among her four children: Amanda, Betty, Carol, and Dan. When the children complained that the shares were not equal, she instructed Betty to give Amanda $2. Then she doubled Carol's share and cut Dan's share in half. Now all the children have the same amount. How much money do they have in total?
c) Raymond has a box of candy bars. He gave Monique half of what he had plus half a bar. Then he gave Claude half of what he had left plus half a bar. After which he gave Laura half of what he had left plus half a bar. And, finally, he gave Alfred half of what he had left plus half a bar. Then he had no bars left. How many candy bars did Raymond have start?

8. Solve the following problems:
a) In the diagram, AC = 12 cm, BD = 5 cm, CD = 24 cm. Determine the length of AB.

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b) Two vertical poles, 10 meters high, and 15 meters high, stand 12 meters apart. Find the distance between the top of the poles.
c) In the diagram, AC = 14 cm, BD = 11 cm, CD = 65 cm. Determine the length of AB.

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9. The product of the first n natural numbers is called factorial n and is written n!:
n! = 1 x 2 x 3 x ... x n.
a) Calculate 8!
b) If n! = 39 916 800, what is the value of n?
c) Find the prime factorization of 10!
d) Let N = 20! = 1 x 2 x 3 x 4 x 5 x ... x 19 x 20. What is the largest power of 5 of which N is a multiple? How many zeroes come at the end of the numeral for N?

10. Simplify.
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Lesson 06


1. Solve the following motion problems.
a) Hoi Ching drives to Work in 35 min. If she increases her average speed by 5 km/h she saves 5 min. How far from work does she live?
b) Sue and Bill paddle 20 km upstream in 4 h. The return trip takes 3 h. What is the speed of the canoe relative to the water, and what is the speed of the current?
c) A farmer set out to travel on a 170 km stretch of lonely prairie highway at an average speed of 105 km/h. He was more than halfway across when his car broke down and he had to complete the journey on foot. He walked at 6 km/h, and the whole trip took 7 h. How long did he walk? How far did he walk?

2. Find the value of the following (do without a calculator and show your work).

1) (3)(-2) + (-3)(- 4) - (-5)(7) aaaaaaaaaaaaaaa 2) (- 4.5 + 3.8)(2.01 - 3.81) aaaaaaaaaa 3) (- 4.5)(0.1) + (-3.7)(-2.1) - (-5.4)(- 0.2) aaaaaaaaaaa
4) ((2.3)(-1.8) - (1.4)(- 0.8))(-1.5) aaaaaaaaaaa 5) (2 7/15 - 4)(8 16/23 - 10) aaaaaaaaaaaa 6) (1 1/3)(- 3/4) - (-2 1/7)(1 2/5) aaaaaaaaa

3. Find each product.

1) (3a - 2b)(9a2 + 6ab + 4b2) aaaaaaaaaaaaaaaaaaaaaaaaaaa 2) (5x + 3y)(25x2 - 15xy + 9y2)
3) (3a - 3b + 4c)(3a - 5b) aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa 4) (a + 3b - 4c)(a - 3b - 4c)
5) (m + n - 2)(m - n + 2) aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa 6) (5z - 4y)(- 8z - 3x +6y)
7) (1/3 a2b + 2/5 ab2)(15a - 30b) aaaaaaaaaaaaaaaaaaaaaaaaaa 8) (a2 + 3ab + b2)(7a - 5b)

4. Solve the following problems:
a) The legs of a right-angled triangle are 5 cm and 10 cm while the hypotenuse of a similar triangle is 15 cm. What is the area of the larger triangle?
b) The sides of a triangle are 10 cm, 24 cm, and 26 cm. Find the perpendicular distance from the midpoint of the shortist side to the longest side.

5. Find the unknown value in each of the following proportions.

a) 61/2 : 33/4 = 31/4 : x aaaaaaaaaaaaaaaaaa b) 22/3 : 0.24 = 17/9 : (x + 0.06)
c) (2x) : 9 = 21/3 : 51/4 aaaaaaaaaaaaaaaaaa d) x : 4 = 9 : x

6. Solve the following word problems:
1) A rectangular container with base 9 cm by 11 cm has a height of 38.5 cm. Assuming that water expands 10% when it freezes, determine the depth to which the container can be filled so that when the contents freeze, the ice does not go above the top edge of the container.
2) How many ounces of pure acid must be added to 20 ounces of a solution that is 5% acid to strengthen it to a solution that is 24% acid?

7. Solve the following word problems:
a) Before beginning the exam, Gerry calculated that, if she were to spend 10 minutes solving each of the 12 problems, then she would be able to complete the exam in 2 hours. During the exam, Gerry found some problems difficult. They each took her twice as long as she had calculated. She found the remaining problems easy. They took only half as long as she calculated. She completed the 12 problems in exactly 2 hours. How many problems did Gerry find to be difficult?
b) A man's estate valued at $120 000 is left in a will to his widow, his son, and his two grandchildren, as follows. To each of two grandchildren a certain sum, to the son four times as much as to the two grandchildren together, and to the widow $20 000 more than to the son and grandchildren together. How much goes to each?
c) Kevin says to the owner of a store A: "If you lend me as much money as I have with me, I'll spend $10 in your store." The owner agrees. Kevin then goes on to store B and C in turn and makes the same deal. He then has no money left. How much did he start with?

8. Solve the following problems:
a) What is the height of an equilateral triangle with sides 12 cm long? Give your answer to 1 decimal place.
b) In the diagram AP = 9 cm, BQ = 16 cm, PQ = 24 cm. Find PC.

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9. Solve the following Number Theory problems.
a) Find the exponent of the highest power of 2, which will divide 100! = 1 x 2 x 3 x ... x 99 x 100.
b) Find the exponent of the highest power of 3, which will divide 100! = 1 x 2 x 3 x ... x 99 x 100.
c) Find the exponent of the highest power of 6, which will divide 100! = 1 x 2 x 3 x ... x 99 x 100.
d) Find the number of zeros at the end of the product 100! = 1 x 2 x 3 x ... x 99 x 100.

10. Simplify.
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Lesson 07


1. Solve the following motion problems.
a) If Mr. Swan drives at an average speed of 100 km/h he arrives at work at 09:00. If he leaves home at the same time but averages 80 km/h, he arrives at 09:06. How far does he live from work?
b) A train running between two towns arrives at its destination 10 minutes late when it goes 40 miles per hour and 16 minutes late when it goes 30 miles per hour. What is the distance in miles between the towns?
c) Car A leaves Toronto for Montreal, 500 km away, at an average speed of 80 km/h. Car B leaves Montreal for Toronto on the same highway 2 h later at 100 km/h. How far are they from Toronto when they pass?

2. Evaluate mentally and record your answer.
1) - 7 + 15 + (- 23) + 9 + 11 + (- 33) + 26 = aaaaaaaaaaaaa 2) 21 + (- 35) + (- 7) + 19 + 14 + (- 28) + 47 =
3) - 18 + (- 7) + 26 + 41 + (- 57) + (- 9) + 24 = aaaaaaaaaa 4) 10 - (- 3) + 12 - (- 14) - 21 - 37 =
5) - 18 + 46 + (- 16) - (- 29) + (- 27) - (- 13) = aaaaaaaaaaa 6) - 35 - (- 48) + (- 16) - 27 - (- 57) =
7) 89 + (- 26) - 17 - 36 + (- 9) - (- 45) = aaaaaaaaaaaaaaaaa 8) 57 - (- 11) + (- 12) - 27 - (- 18) - 46 - (- 35) + 13 =

3. Expand and simplify.
1) (x + 3)(x - 3) + (4 - x)x - 3x aaaaaaaaaaaaaaaaaaaaaaaaaaaa 2) x(1- 2x) - (x - 3)(x + 3) + 3x2
3) x2(3 - x) - (2 - x2)(x + 1) - 4x2 aaaaaaaaaaaaaaaaaaaaaaaaaa 4) (x + 2)2 - x(5 - x) - 2x2
5) (a - 3)(a2 - 8a + 5) - (a - 8)(a2 - 3a + 5) aaaaaaaaaaaaaaaaaa 6) (x2 - 3x + 2)(2x + 5) - (2x2 + 7x + 17)(x - 4)

4. Solve the following problems.
1) ABC is a right-angled triangle with the angle C = 90o, BC = 4 cm and AC = 3 cm. D is the midpoint of AB. The perpendicular bisector of AB meets BC at point E. Find the length of ED.
2) In a triangle ABC, M and N are the midpoints of AC and BC, respectively. P and Q are points on AB such that MP and NQ are perpendicular to AB. Find the ratio of the area of PMNQ to the area of ABC.

5. Find the unknown value in each of the following proportions.

a) 12.15 : x = (7x) : 4.2 aaaaaaaaaaaaaaaaaaaaaaaaa b) 1.25 : 0.4 = 1.35 : (0.3x)
c) 0.2 : (x + 3) = 0.7 : ( x - 2) aaaaaaaaaaaaaaaaaaaa d) (2x + 1) : x = 2 : 3

6. Solve the following word problems:
1) In each of three successive years, the cost of living increases by 10%. What is the percentage increase in the three years?
2) The selling price of a coat was reduced by 20% during the spring sale. Since the coat still didn't sell, the sale price was reduced by 15%. What was the total reduction from the original price?

7. Solve the following word problems:
a) Your secret club shares its earnings. The president receives half of the money. The vice-president gets a quarter of the remainder. Then, the secretary gets a third of what is left. Finally, the treasurer and you share what is left equally. Your share is three dollars. Calculate the clubs total earnings.
b) In the sequence x, y, z, 25, when any one of the first three terms is subtracted from the term immediately following it, the result equals x - 1. What is the value of x?
c) How much antifreeze should be added to 3 L of water to make a solution that is 80% antifreeze?

8. Solve the following problems:
a) In a right triangle, the ratio of the legs is 11 : 57. If the area of the triangle is 1 254 square units, what is the length of the hypotenuse?
b) The tip of a straight reed growing in the center of a pond 8 feet in diameter reaches one foot above the water. When the reed is pulled over, with its bottom fixed, the tip can just be made to touch the edge of the pond. Find the depth of the pond, in feet, at the center.

9. Solve the following Number Theory problems.
a) Prove that the sum of three consecutive integers is divisible by 3.
b) Prove that the sum of five consecutive integers is divisible by 5.
c) Prove that the sum of four consecutive odd integers is divisible by 8.
d) The digits 1, 2, 3, 4, and 5 are each used once to compose a five-digit number abcde, such that the three digit number abc is divisible by 4, bcd is divisible by 5, and cde is divisible by 3. Find the five digit number abcde.

10. Simplify.

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Lesson 08


1. Solve the following motion problems.
a) In a 24 km race, during which each runner maintains a constant speed throughout, A crosses the finish line while B is still 8 km from finishing and C is 12 km from finishing. How many kilometers will C still have to complete when B crosses the finish line?
b) Amy, Brigitte, and Cindy run at constant rates. In a race of 1000 m, Amy finished 200 m ahead of Brigitte and 400 m ahead of Cindy. When Brigitte finished, find the number of meters she was ahead of Cindy.
c) A man walked from A to B at 4 mph and from B to C at 3 mph. Then he walked from C to B at 6 mph and from B to A at 4 mph. If the total time taken for the walk was 6 hours and AB = BC, how far did he walk altogether?

2. Calculate mentally and record your answer.
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3. Simplify and evaluate.
1) (a - 4)(a - 2) - (a - 1)(a - 3) for a = 13/4 aaaaaaaaaaaaaaaaaaaaaaaaaaaa 2) (m - 5)(m - 1) - (m + 2)(m - 3) for m = -23/5
3) (x + 1)(x + 2) + (x + 3)(x + 4) for x = - 0.4 aaaaaaaaaaaaaaaaaaaaaaaaa 4) (5x - 1)(x + 3) - (x - 2)(5x - 4) for x = 21/7
5) (a + 3)(9a - 8) - (a + 2)(9a - 1) for a = - 3.5 aaaaaaaaaaaaaaaaaaaaaaaa 6) n(n + 1)(n + 2)(n + 3) + 1 - (n2 + 3n + 1)2
7) 2(a + 1)(b + 1) - (a + b)(a + b + 2) if a2 + b2 = 2

4. Solve the following problems.
1) In the adjacent squares shown, the vertices A, B, and C lie in a straight line. Find the value of x.
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2) ABCD is a trapezoid with bases BC and AD, BC : AD = 2 : 3. If the area of the triangle ABC is 36 cm2, find the area of the trapezoid.

5. Solve the following equations for x.
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6. Solve the following word problems:
1) At the beginning of the week, a stock was worth $100. On Monday, its price went up 10%. On Tuesday, its price dropped 10% and on Wednesday its price went back up 10%. What is the percent increase in the price of the stock over the three-day period?
2) The dimensions of a rectangle are 20 cm by 16 cm. The length and width are increased by 25%. What is the percentage increase in area?

7. Solve the following word problems:
a) How much alcohol should be added to 1 L of a 20% solution of alcohol to increase its strength to 50 %?
b) How many gallons of a 90% solution of acid must be added to ten gallons of a 60% solution to obtain an 80% solution?
c) A man has 10 gallons of a solution 60% acid. He wishes to add to it 20 gallons of another solution and get an 80% solution. What should be the percentage strength of the added solution?

8. Solve the following problems:
a) What is the length of each side of a square with diagonals of length 7 m? Round your answer to the nearest hundredth.
b) In the triangle ABC, what is the length of BC?
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9. Solve the following Number Theory problems.
a) Three ex-teenagers find that the product of their ages is 17 710. What is the sum of their ages?
b) A man has four teenager children, each with a different age. The product of their ages is 75 582. What are their ages?
c) Nick is 1.5 times as old as Ann. The product of their ages is 864. How old is Ann?

10. Simplify.
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Answers


1. aaa a) 6 km. aaaaaaaaaaaaaaaaaa b) 250 m. aaaaaaaaaaaaaaa c) 24 miles.
2.
1) - 5/7 aaaaaaaaaaa 2) - 8/9 aaaaaaaaaaaaa 3) - 1/9
4) 1/12 aaaaaaaaaaaa 5) 1/8 aaaaaaaaaaaaaa 6) - 2/15
7) - 45/24 aaaaaaaaa 8) - 8 2/15 aaaaaaaaaaa 9) - 1 1/4
10) - 4 11/16 aaaaaa 11) - 11/14 aaaaaaaaaaa 12) 0
13) 1/2 aaaaaaaaaaa 14) - 1 1/9 aaaaaaaaaaa 15) - 1/3
16) 1/3 aaaaaaaaaaa 17) - 1 5/8 aaaaaaaaaaa 18) - 10/11

3.
1) - 2a + 5 = 1.5. aaaaaaaaaaaaaaaa 2) - 5m + 11 = 24.
3) 2x2 + 10x + 14 = 10.32. aaaaaaa 4) 28x - 11 = 49.
5) 2a - 22 = - 29. aaaaaaaaaaaaaaaa 6) 0.
7) 0.

4. aaa 1) 12.25 cm. aaaaaaaaa 2) 90 cm2.

5. aaa a) 44. aaaaaaaa b) 1 14/17 aaaaaaaa c) 5. aaaaaaaa d) 1.

6. aaa 1) 8.9%. aaaaaaaaa 2)56.25%.

7. aaa a) 0.6 L. aaaaaaaa b) 20 gallons. aaaaaaaa c) 90%.

8. aaa 1) 4.95. aaaaaaaaa b) 9.

9. aaa a) 80.
b) 13, 17, 18, 19.
c) 24.

10. aaa 1) (24xz) / (5y2) aaaaaaaaa 2) (15x6) / (16z5) aaaaaaaaaaaa 3) y / (4x2) aaaaaaaaaa 4) 2 / (9x2)
a 5) 1/3 pg2 aaaaaaaaaaaaaa 6) (3b) / (d2) aaaaaaaaaaaaaaaa 7) 1 / (xy2) aaaaaaaaaa 8) 225/14 xy2



Lesson 09


1. Solve the following motion problems.
a) Bette visits her friend Keith and then returns home by the same route. She always walks 2 km/h when going uphill, 6 km/h when going downhill, and 3 km/h, when on level ground. If her total walking time is 6 hours, what is the total distance she walks, in km?
b) A man has walked two-thirds of the distance across a railroad bridge when he observes a train approaching at 45 miles per hour. What must his rate of speed be if he can just manage to escape by running at the same uniform speed to either end of the bridge?
c) Two trucks left two towns A and B at the same time, and each was driven to the other town at a constant speed, passing each other at point C. The truck from B completed the journey from C to A in 20 minutes. The truck from A completed the journey from C to B in 45 minutes, while maintaining its steady speed of 40 mph. Find the speed of the other truck in miles per hour.

2. Calculate mentally and record your answer.
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3. Find each product.
1) (x + 1)(x + 2)(x + 3) aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa 2) (2x -1)(3x - 1)(x + 1)
3) (x + 3)(2x - 1)(3x + 2) aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa 4) (x - 3)(2x + 1)(3x - 2)
5) (x - 2)(3x + 1)(4x - 3) aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa 6) (a - b)(a + b)(a - 3b)
7) (a + 1)(a - 1)(a2 + 1) aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa 8) x(x + 1)(x - 1)(3x + 2)

4. Solve the following problems.
1) In the diagram, AD = DB = 5 cm, EC = 8 cm, AE = 2 cm. DE is perpendicular to AC. Find the length of BC.


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2) ABC is a right-angled triangle with AC = 2BC. Find the ratio of the area of the inscribed square DEFC to the area of the triangle ABC.

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5. Solve the following equations for x.
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6. Solve the following word problems:
1) The sides of a rectangle are 20 cm and 25 cm. The length is increased by 50% and the width is reduced by 50%. What is the percentage change in area?
2) Kelly's Cleaner raised the price of dry cleaning a jacket from $4.00 to $5.00. The same percentage increase was applied to the price of dry cleaning a coat. The new cost of dry cleaning a coat was $12.50. What is the old cost, in dollars, of dry cleaning a coat?

7. Solve the following word problems:
a) How many gallons of 20% solution of alcohol in a radiator should be replaced by an equal amount of 90% solution to obtain 10 gallons of a 55% solution?
b) A 44% oil-gas mixture is to be formed by mixing 80% and 20% oil-gas mixture. If 35 L are required, how much of each should be used?
c) An operator finds that his five-gallon cooling system is full of a 20% solution of alcohol. How many gallons should be replaced by an equal amount of 80% solution so as to have a 50% solution?

8. Solve the following problems:
a) A lighthouse shows a light through a sector from east to south-east. Steering north-east the captain of a 10-knot vessel first sees the light at 1 a.m. At 1.30 a.m. the light disappears. How far off is it then? (1 knot = 1.87 km/hr)
b) OACB is a rectangle inscribed in a quarter circle. If AC = 12 and BD = 1, find the length of AO.
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9. Solve the following Number Theory problems.
a) Find the smallest natural number by which 132 can be multiplied so that the number created is divisible by 112.
b) Find the GCF of the numbers 465465 and 345345.
c) Find two numbers between 200 and 300, which have a greatest common factor of 54.

10. Which number of each pair is greater?
1) 730, 440 aaaaaaaaaaaaaaaaaaaaaa 2) 620, 330 aaaaaaaaaaaaaaaaaaaaaaaaaa 3) 3600, 5400
4) 1020, 2015 aaaaaaaaaaaaaaaaaaaa 5) 10020, 900010 aaaaaaaaaaaaaaaaaaaa 6) 232, 320


Answers


1. aaa a) 9 km. aaaaaaaaaaaaaaaaaa b) 15 km/hr. aaaaaaaaaaaaaaa c) 60 miles/hr.
2.
1) 5/8 aaaaaaaaaaa 2) - 5/6 aaaaaaaaaaa 3) - 9
4) 10 1/6 aaaaaaaa 5) 1 aaaaaaaaaaaaaa 6) 1/4
7) 1/3 aaaaaaaaaaa 8) 1 6/7 aaaaaaaa 9) 9
10) 1 aaaaaaaaaaa 11) - 1 aaaaaaaaaaa 12) 3 13/15
13) - 6 aaaaaaaaaa 14) 1/3 aaaaaaaaaaa

3.
1) x3 + 6x2 + 11x + 6 aaaaaaaaaaaaaaa 2) 6x3 + x2 - 4x + 1
3) 6x3 + 19x2 + x - 6 aaaaaaaaaaaaaaaa 4) 6x3 - 19x2 + x + 6
5) 12x3 - 29x2 + 7x + 6 aaaaaaaaaaaaaa 6) a3 - 3a2b - ab2 + 3b3
7) a4 - 1 aaaaaaaaaaaaaaaaaaaaaaaaaaa 8) 3x4 + 2x3 - 3x2 - 2x

4. aaa 1) 10.95 cm. aaaaaaaaa 2) 4 : 9.

5. aaa a) 3. aaaaaaaa b) 2.5 aaaaaaaa c) 36/7

6. aaa 1) Reduced by 25%. aaaaaaaaa 2) $10.00.

7. aaa a) 5 gallons. aaaaaaaa b) 32%: 14 L, 20%: 21L. aaaaaaaa c) 2.5 gallons.

8. aaa 1) 49.073 km. aaaaaaaaa b) 5.

9. aaa a) 28. aaa b) 1 515. aaa c) 216, 270.

10. aa 1) 730 > 440. aaaaaaaaa 2) 620 > 330. aaaaaaaaaaaaa 3) 3600 > 5400.
4) 1020 > 2015. aaaaaaa 5) 10020 > 9 00010. aaaaaaa 6) 232 > 320.



Lesson 10


1. Solve the following motion problems.
a) A truck left Huck's Trucks and traveled north at 80 km/h. One hour later, another truck left Huck's Trucks and traveled south at 60 km/h. How many hours had the first truck traveled when they were 150 km apart?
b) A motorcycle and a truck left a roadside diner at the same time. After traveling in the same direction for one and one-quarter hours, the motorcycle had traveled 25 kilometers farther than the truck. If the average speed of the motorcycle was 60 km/h, find the average speed of the truck.
c) A dog and a rabbit are 160 meters apart. The dog chases the rabbit. For every 9 meters that the dog runs, the rabbit jumps 7 meters. Find the distance in meters, that the dog must run in order to overtake the rabbit.

2. Calculate mentally and record your answer.

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3. Divide polynomials by monomials.
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4. ABCD is a trapezoid with bases AD and BC. The diagonals AC and BD meet at point E. The areas of triangles AED and BEC are 25 cm2 and 16 cm2, respectively. Find the area of the trapezoid ABCD.

5. Find the values of x, y, and z in each of the following proportions.

a) 5 : 6 : 10 = 10 : x : y aaaaaaaaaaaaaaa b) 33 : 12 : x = y : 8 : 20
c) x : 6 : y = 24 : 18 : 27 aaaaaaaaaaaaaa d) x : 1/4 : y = 4 : 1 : 3
e) (x - 3) : 2 : 3 = 8 : (y - 1) : 6 aaaaaaaa f) (x + y) : (x - y) : 9 = 4 : 12 : 18
g) x : 3 : 12 : y = 3 : 1 : z : 4

6. Solve the following word problems:
1) Sophia wants to buy a pair of sandals that are on sale for $19.90. She has $20.80. With a 6% sales tax, does she have enough to buy the sandals?
2) The cost of a jacket, including sales tax is $200. Find the selling price of the jacket if the rate of sales tax is 8%.

7. Solve the following word problems:
a) A invests a sum of money at 5% and B invests $400 more at 4.5%. If the incomes from these investments are equal, find the amount of each investment.
b) Jennifer invested $500, part at 7% per annum and the rest at 10% per annum. After one year the total interest earned was $44. How much did she invest at each rate?
c) Mee Ha invested $2500, part at 8% per annum, and the rest at 12% per annum. In one year, the two parts earned equal amounts of interest. How much did she invest at each rate?

8. Solve the following problems:
a) A box has a length of 28 cm, a width of 21 cm, and a height of 12 cm. How long is the body diagonal of the box? Show all work and logic.
b) A circle passes through the vertices A and D of the square ABCD and touches the side BC at M. Find the length of the radius of the circle if AB = 14 cm.
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9. Solve the following Number Theory problems.
a) List all three-digit numbers formed as permutations of digits 2, 4, and 6. Find the greatest common factor of these numbers.
b) The numbers in the sequence 3, 8, 13, 18, 23, ... increase by five. The numbers in the sequence 0, 6, 12, 18, 24, ... increase by six. The number 18 occurs in both sequences. Find the next two numbers occur in both sequences.
c) The numbers in the sequence 2, 7, 12, 17, 22, ... increase by fives. The numbers in the sequence 3, 10, 17, 24, 31, ... increase by sevens. The number 17 occurs in both sequences. Find the next three numbers that occurs in both sequences.

10. Simplify.
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Answers


1. aaa a) 1.5 h. aaaaaaaaaa b) 40 km/h. aaaaaaa c) 720 m.
2.
1) - 1/4 aaaaaaaaaaaa 2) - 1/6 aaaaaaaaaa 3) 4/15
4) 2 aaaaaaaaaaaaaa 5) - 6 aaaaaaaaaaa 6) 22.5
7) - 3.84 aaaaaaaaaa 8) - 2.4 aaaaaaaaa 9) 13 1/3

3.
1) - 7m + 4 aaaaaaaaaaaaa 2) 2 - x aaaaaaaaaaaaaaaaaa 3) 1 - y aaaaaaaaaaaaaaaaaaaaa 4) - c + 1
5) 5m - 6p aaaaaaaaaaaaaa 6) 4a - 3b aaaaaaaaaaaaaaaa 7) - 2x - 3y + 4 aaaaaaaaaaaaa 8) 0.6a2 - 0.8b2
9) 5a - 6b + 4 aaaaaaaaaa 10) a + 3a2b - 2 aaaaaaaaaaa 11) - 0.25ab + 0.75a2 aaaaaaa 12) - 2/3cd - 1
13) 2.7kl3 - 2.1 aaaaaaaaa 14) 1 - 3x2y2 + 6xy3

4. aaa 81 cm2.

5. aaa a) x = 12, y = 20. aaaaaaaa b) x = 30, y = 22. aaaaaaaa c) x = 8, y = 9. aaaaaaaaaa d) x = 1, y = 0.75.
e) x = 7, y = 5. aaaaaaaaaaa f) x = 4, y = - 2. aaaaaaaaa g) x = 9, y = 12, z = 4.

6. aaa 1) No: $21.09 > $ 20.80. aaaaaaaaa 2) $185.19.

7. aaa a) A - $3600, B - $4000. aaaaaaaa b) 7% - $200, 10% - $300. aaaaaaaa c) 8% - $1500, 12% - $1000.

8. aaa a) 37 cm. aaaaaaaaa b) 8.75 cm.

9. aaa a) 6 numbers, GCF = 6. aaaaaaaa b) 18, 48, 78. aaaaaaaa c) 17, 52, 87.

10. aa 1) - 1/6 aaaaaaaaa 2) 127 aaaaaaaaaaaaa 3) 1 23/27 aaaaaaaaaaaaaa 4) 0.3.



Lesson 11


1. Solve the following motion problems.
a) Moe and Joe start together at point A and walk toward point B. Moe walks 1.5 times as fast as Joe. Moe reaches B, and then travels back till he meets Joe. At this point, what is the fraction of the distance AB that Joe has traveled?
b) Two astronauts Pat and Mike were orbiting the earth in separate space capsules. They were orbiting in the same direction and in the same plane. Pat orbits in 3 hours and Mike in 7.5 hours. At 12 noon Toronto time, Mike sees Pat directly below. What will be the next time that Mike's capsule is directly above Pat's capsule?
c) At exactly 12 o'clock, the hour hand of a clock begins to move at twice its normal speed and the minute hand begins to move at half its normal speed. What will be the correct time when the two hands next coincide?

2. Calculate mentally and record your answer.
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3. Simplify.
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4. ABCD is a trapezoid with bases AD = 7 cm and BC = 1 cm. The area of the trapezoid is bisected by the line segment MN parallel to AD and BC. Find the length of MN.
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5. Solve the following ratio problems.
a) If a : b = 2 : 3 and b : c = 4 : 5, find a : c.
b) If x : y = 2 : 5 and x : z = 3 : 2, find y : z.
c) If x : y = 5 : 6 and x : z = 2 : 3, find y : z

6. Solve the following word problems:
1) The cost of a can of wax, including 8% sales tax, is $1.20. If the price of a can of wax, without sales tax, is reduced to 90% of the original cost, what is the new price before the tax is added?
2) An article, which normally sells for $16.00, is on sale at 30% off. If you bought this article at the sale price, what is the amount you would pay, including 7% sales tax?

7. Solve the following word problems:
a) In a provincial mathematics examinations, the average mark of those who passed was 64%, and of those who failed 43%. If the average mark of the 840 students who participated was 58%, how many passed?
b) How much water must evaporate from a 30-gallon barrel full of brine containing 5/6 pound of salt per gallon to have a solution containing 2 pounds of salt per gallon?
c) A car averages 8.5 L/100 km in city driving and 6.3 L/100 km on the highway. In 720 km of driving it used 55.7 L of fuel. How far was the car driven on the highway? How much fuel did it use in highway driving?

8. Solve the following problems:
a) The inside dimensions of a crate are 45 cm by 60 cm by 180 cm. What is the length of the longest item that can be placed in the crate corner to corner?
b) Each side of a rhombus has length 10 cm. Find the sum of the squares of the diagonals.

9. Solve the following Number Theory problems.
a) The least common multiple and the greatest common factor of two numbers are 180 and 6, respectively. Find the product of the numbers.
b) The least common multiple and the greatest common factor of two numbers are 120 and 12, respectively. Find the first number, if the second number is 24.
c) The least common multiple and the greatest common factor of two numbers are 1260 and 36, respectively. Find the numbers.

10. Find the value of the following (do without a calculator and show all your work).
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Answers


1. aaa a) 4/5 aaaaaaaaaa b) 5 p.m. aaaaaaa c) 3 p.m.
2.
1) 7700 aaaaaaaaaaaa 2) - 1 aaaaaaaaaa 3) - 1/8
4) 90 aaaaaaaaaaaaaa 5) - 0.1 aaaaaaaaa 6) 1/4
7) 7

3.
1) 1.5x2 aaaaaaaaaaaaa 2) 9x3 aaaaaaaaaaaaaa 3) 30x3 aaaaaaaaaaaaaa 4) - 9x2 - 3x
5) 10a aaaaaaaaaaaaaaa 6) 1 aaaaaaaaaaaaaaa 7) - 8y aaaaaaaaaaaaaaa 8) - 3y

4. aaa 5 cm.

5. aaa a) a : c = 8 : 15. aaaaaaaa b) y : z = 15 : 4. aaaaaaaa c) y : z = 4 : 5.

6. aaa 1) $1.00 aaaaaaaaa 2) $11.98.

7. aaa a) 600. aaaaaaaa b) 17.5 gallons. aaaaaaaa c) 15.75 L.

8. aaa a) 195 cm. aaaaaaaaa b) 400.

9. aaa a) 1080. aaaaaaaa b) 60. aaaaaaaa c) 180, 252 or 36, 1260.

10. aa 1) 12 aaaaaaaaa 2) 15 aaaaaaaaaaaaa 3) 1 aaaaaaaaaaaaaa 4) 9 aaaaaaaaaaaaaa 5) 144/615 aaaaaaaaaaaaaa 6) 0.8



Lesson 12


1. Solve the following ratio problems.
a) To make concrete, mix 4 shovels of stone, 2 shovels of sand, and 1 shovel of cement. Find the number of shovels of stone required to make 350 shovels of concrete.
b) The ratio of gravel to sand to cement in a mixture is 7 : 4 : 1. How many kilograms of cement do you need for 600 kg of mix?
c) Baking soda is a combination of sodium, hydrogen, carbon, and oxygen in a ratio of 23 : 1 : 12 : 16 by mass. How many grams of each are there in 972 g of baking soda?

2. Calculate mentally and record your answer.
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3. Divide polynomials by monomials.
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4. Find the area of the shaded regions.

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5. Solve the following ratio problems.
a) If x is 2/3 of y and y is 3/4 of z, what is the ratio z : x?
b) If a : b = 3 : 2 and b : c = 4 : 3, find a : b : c.
c) If a : b : c = 6 : 4 : 5, find the value of (3a - b) : (4b + c)

6. Solve the following word problems:
1) A certain radio costs a merchant $72. At what price must the merchant sell the radio in order to make a profit of 20% of the selling price?
2) A suit marked at $89 is sold for $68. What is the rate of discount?

7. Solve the following word problems:
a) A 22-wheeler uses 36 L/100 km in a city driving and 21 L/100 km on the highway. On a 2450 km trip, 585 L of fuel was used. How far was it driven in the city? How much fuel was required for the highway?
b) A can do a piece of work in 4 days, and B in 6 days. How long will it take them both working together?
c) An oil tanker can be filled in 18 h using one pump, and in 15 h using a different pump. If both pumps are used, how long would it take to fill the tanker?

8. Solve the following problems:
a) ABC is a right-angled triangle with the perimeter 8.2 cm and the length of the hypotenuse 4 cm. Find the area of the triangle ABC.
b) In the diagram, AB = 9 cm and AD = 8 cm. Two circles, centers E and F, touch each other and also touch the sides of the rectangle at M, N, K, and L. If the radius of the smaller circle is 2, find the radius of the larger circle.
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9. Solve the following Number Theory problems.
a) Erin's age, when divided by 2, 3, 4, 5, or 6, gives a remainder 1. Find the least age that Erin could be given that she is older than 1.
b) List the smallest 3 numbers, which leave a remainder of 1 when divided by 6, 7 or 8.
c) What is the smallest positive number, other than 2, which, when it is divided by 3, 4, 5 or 6, will leave a remainder of 2?
d) Find the smallest number having remainders 1, 2, 3, 4, and 0, when divided by 2, 3, 4, 5, and 7, respectively.

10. Find the value of the following (do without a calculator and show all your work).
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Answers


1. aaa a) 200. aaaaaa b) 50 kg. aaaaaa c) 429.9 g, aaa 18.7 g, aaa 224.3 g, aaa 229.1 g.
2.
1) - 0.6 aaaaaaaaaa 2) - 0.2 aaaaaaaaaa 3) - 1/3
4) -37 aaaaaaaaaaa 5) - 7 aaaaaaaaaaaa 6) 3/4
7) 2/3 aaaaaaaaaaaa 8) 1 1/7

3.
1) 4a2b3 - 3ab2 + 2b aaaaaaa 2) 3x2y2 - 2xy + 1/x2 aaaaaaa 3) - n3 + 2mn2 - 3m2n aaaaaaa 4) 3a2 - 2ab + b2 aaaaaaa 5) 3yz2 - 2xy2z3 + x2y3z4

4. aaa 1) 25/8 (pi - 2) = 3.57 aaaaaaa 2) 8

5. aaa a) 2 : 1. aaaaaaaa b) 6 : 4 : 3. aaaaaaaa c) 2 : 3.

6. aaa 1) $ 90. aaaaaaaaa 2) 23.6%.

7. aaa a) 470 km, aaa 415.8 L. aaaaaaaa b) 2.4 days. aaaaaaaa c) 8 h 11 min.

8. aaa a) 0.41 cm2. aaaaaaaaa b) 3 cm.

9. aaa a) 61. aaaaaaaa b) 169, aaa 337, aaa 505. aaaaaaaa c) 62 aaaaaaaa d) 119

10. aa 1) 22.75 aaaaaaaaa 2) 31.25 aaaaaaaaa 3) 0.75 aaaaaaaaa 4) 1. aaaaaaaaa 5) 48 10/27. aaaaaaaaa 6) - 7 5/8.



Lesson 13


1. Solve the following ratio problems.
a) Three people invest in a small business with $35 000, $10 500 and $21 000 respectively. They share in the profits in the ratio of the amounts invested. In what ratio should the profits be shared? How much should each receive from a profit of $10 000, to the nearest dollar?
b) The angles of a quadrilateral are in the ratio 3 : 4 : 5 : 6. What are their measures?
c) There are three numbers in the ratio 3 : 4 : 12. The sum of their squares is 1 521. Find the numbers.

2. Divide mentally and record your answer.
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3. Simplify and evaluate.

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4. In terms of the square units, what is the area of the semicircle?

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5. Solve the following ratio problems.
a) If x : y = 3 : 2 and x + 3y = 27, find x and y.
b) If x : y = 3 : 4 and x : (y + z) = 2 : 5, find the value of x : z.
c) If ab = 16, b : c = 1 : 3, and c : a = 12 : 1, find the value of b.

6. Solve the following word problems:
1) What is the rate of discount on a new car that lists for $19 575 and is reduced by $2 500?
2) A ski shop offered a 25% discount on a pair of skies that originally sold for $90.00. The new price was then reduced by 10%. Find the final sale price.

7. Solve the following word problems:
a) A can do a piece of work in 10 days, B in 12 days, and C in 15 days. In how many days can they do the work all three working together?
b) A powerboat has two separate motors. When motor A is used, the boat will run for 5 h on a tank of fuel. When motor B is used, the boat will run for 7 h on the same amount of fuel. How long would a tank of fuel last if both motors were using it?
c) A can do a piece of work in 10 days, B in 12 days, and C in 15 days. If A and B work together for 3 days, how long will it take C working along to finish the work?

8. Solve the following problems:
a) A triangle has sides of length 6, 10, and 11. An equilateral triangle has the same perimeter. Find its area.
b) In a triangle ABC, BD is perpendicular to AC, BD = 15 cm, AD = 12 cm, and DC = 8 cm. Prove that the angle ABC = 90o.

9. Solve the following Number Theory problems.
a) The difference between two-digit number and the number obtained by reversing the order of the digits is 36. Find the number, if the sum of its digits is 10.
b) The sum of the digits of a two-digit number is 9. When the digits are interchanged, the number is decreased by 45. What is the number?
c) Find the least two-digit number whose sum with the number obtained by reversing the order of the digits is 132.

10. Find the value of the following (do without a calculator and show all your work).

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Answers


1. aaa a) 10 : 3 : 6, aaa $5263,   $1579,   $3158. aaaaaaaaa b) 60o,   80o,   100o,   120o. aaaaaaaaa c) 9,   12,   36.
2.
1) 1 2/3 aaaaaaaaaa 2) - 1 2/5 aaaaaaaaaa 3) - 2/9
4) 2 2/3 aaaaaaaaaa 5) - 5/6 aaaaaaaaaaaa 6) 6
7) - 5/6 aaaaaaaaaa 8) - 1/12 aaaaaaaaaaaa 9) - 7
10) - 9 aaaaaaaaaa 11) - 11/48 aaaaaaaaa 12) - 2 1/4
13) - 1 4/5 aaaaaaa 14) 3/4 aaaaaaaaaaaa 15) - 3 5/7
16) - 6.4 aaaaaaaa 17) - 0.51 aaaaaaaaa 18) 8/13
19) - 1/3 aaaaaaaaa 20) - 6

3.
1) - 3a, 24 aaaaaaaaa 2) 6a + 3, - 3 aaaaaaa 3) 2a2 - 4a + 2, 11/8
4) x + y, - 3. aaaaaaa 5) 330 aaaaaaaaaaaaaa 6) 315

4. aaa 32.5 pi.

5. aaa a) x = 9, y = 6. aaaaaaaa b) x : z = 6 : 7. aaaaaaaa c) b = - 8 or b = 8.

6. aaa 1) 12.77%. aaaaaaaaa 2)$60.75.

7. aaa a) 4 days. aaaaaaaa b) 2 h 55 min. aaaaaaaa c) 6.75 days.

8. aaa a) 35.074 units2. aaaaaaaaa b)

9. aaa a) 73. aaaaaaaa b) 72. aaaaaaaa c) 39.

10. aa 1) 5 aaaaaaaaa 2) 1/81 aaaaaaaaa 3) 100.
4) 1. aaaaaaaaa 5) 1/4 aaaaaaaaa 6) 1/3



Lesson 14


1. Solve the following ratio problems.
a) Three positive integers are in the ratio 4 : 5 : 6. If the product of the three integers is 25 920, find the larger integer.
b) Jessie has some nickels, dimes, and quarters in the ratio 5 : 7 : 4. The value of the coins is $11.70. How many nickels, dimes and quarters does Jessie have?
c) The monthly salaries of a plumber, mechanic, and accountant are in the ratio 8 : 5 : 6. The sum of the monthly salaries for 3 plumbers, 6 mechanics, and 4 accountants is $21 645.00. What are the monthly salaries of a plumber, mechanic, and accountant?

2. Simplify.
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3. Find each product:
1) (x - 1)(x + 1)
2) (x - 1)(x2 + x + 1)
3) (x - 1)(x3 + x2 + x + 1)
4) (x - 1)(x4 + x3 + x2 + x + 1)
5) Predict the product of (x - 1)(xn + xn - 1 + ... + x + 1).

4. What is the area of a circle that is inscribed in a square with a diagonal of 8?

5. Charles' Law states that if the pressure is kept constant, the volume of a given amount of gas is directly proportional to its absolute temperature: V ~ T [constant P]. A balloon contains 2 L of helium at 10oC.
a) What is its volume when the temperature is 30oC?
b) What is the temperature if its volume is 4 L?

6. Solve the following word problems:
1) A pair of skis is priced at $185. Find the cost with a 15% discount and a 7% sales tax.
2) At a selling price of $273, a refrigerator yields a 30% profit on the cost. What selling price will yields a 10% profit on the cost?

7. Solve the following word problems:
a) Two candles of the same height were lit at the same time. The first candle was consumed in 4 h. The second candle was consumed in 3 h. How long after the lighting was the first candle twice as high as the second?
b) Pipes A and B run water into a tank, and pipe C removes it. Pipe A operating alone fills the tank in 60 minutes, and pipe B in 80 minutes. Pipe C operating alone empties a full tank in 90 minutes. Initially the tank is empty. How long will it take to fill the tank:
(1) if all pipes are open?
(2) if A, B, and C are opened for 40 minutes and then pipe A is closed?

8. Solve the following problems:
a) If the diagonals of a quadrilateral intersect at right angles, the sum of the squares on one pair of opposite sides is equal to the sum of the squares on the other pair. Prove.
b) In a triangle ABC, the angle C = 90o. On BC a point M is taken. Prove that AM2 + BC2 = AB2 + CM2.

9. Solve the following Number Theory problems.
a) I am thinking of three 4-digit numbers. Each number has four different digits. Also in each number, the two-digit numbers, formed by the first two digits, middle two digits, and last two digits are perfect squares. What are the three numbers?
b) Between the digits of a two-digit square, a third digit is inserted to create a three-digit square. Find the number of three-digit squares that can be obtained by this process.
c) What number subtracted from both 176 and 92 produces perfect squares?

10. Simplify. Writhe the answer with positive exponents.
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Answers


1. aaa a) 36. aaaaaaaaa b) 5 : 7 : 4. aaaaaaaaa c) $2220.00, aaa $1387.50, aaa $1665.
2.
1) - 2/3 aaaaaaa 2) - 9/20 aaaaaaa 3) - 4.5 aaaaaaa 4) 1/13 aaaaaaa 5) 3 aaaaaaa 6) 3

3.
1) x2 - 1 aaaaaaa 2) x3 - 1 aaaaaaa 3) x4 - 1 aaaaaaa 4) x5 - 1. aaaaaaa 5) xn + 1 - 1

4. aaa 8 pi.

5. aaa a) 6 L. aaaaaaaa b) 20o.

6. aaa 1) $168.26. aaaaaaaaa 2) $231.

7. aaa a) 2 hr 24 min. aaaaaaaa b) aaa (i) 55 min 13 sec. aaa (ii) 240 min.

8.
a) By Pythegorean Theorem: AB2 + CD2 = AO2 + BO2 + CO2 + DO2 = AD2 + BC2.
b) By Pythegorean Theorem: AM2 + BC2 = CM2 + AC2 + BC2 = CM2 + AB2.

9. aaa a) 8164, 3649, 1649. aaaaaaaa b) 3: 196, 225, and 591. aaaaaaaa c) 76 and -80.

10. aa 1) 1/3x4y7 aaaaaaaaa 2) 81 ab7 aaaaaaaaa 3) 15x5y5 aaaaaaaaa 4) 1.25p5q10 aaaaaaaaa 5) 4x aaaaaaaaa 6) 27x3y



Lesson 15


1. Solve the following ratio problems.
a) My piggy bank has only nickels, dimes, and dollar bills. The ratio of nickels to dimes is 2 : 3, and the ratio of dimes to dollar bills is 8 : 1. What is the ratio of coins to dollar bills?
b) If the ratio 4a to 3b is 8 to 9, what is the ratio of 3a to 4b?
c) What number must be added to each term of the ratio 5 : 7 so that the result is equivalent to the ratio 10 : 11?
d) In a school the ratio of the number of grade 9 students to the number of grade 8 students is 4 : 3. If there were 5 fewer students in grade 9 and 3 more students in grade 8, the ratio would be 7 : 6. How many students are in each grade?

2. Find the missing value.
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3. Solve equations.
1) 10(1 - 2x) = 5(2x - 3) - 3(11x - 5) aaaaaaaaaaaaaaaaaaaaaaaaaaaa 2) 5(x - 3) - 2(x - 7) + 7(2x + 6) = 7
3) 5(8x - 1) - 7(4x + 1) + 8(7 - 4x) = 9 aaaaaaaaaaaaaaaaaaaaaaaaaa 4) 2(5 - 2x) + 3 = 9(x + 1) - 6(x - 1) - (7x + 2)
5) 28 - 20x = 2(x - 6) - 16(x - 1) - 3(2x - 3) aaaaaaaaaaaaaaaaaaaaaa 6) 1.3(x - 0.7) - 0.12(x + 10) - 5x = - 9.75

4. In the diagram, AB is a circular arc with center O, and both OC and BC one unit in length. What is the area of the shaded region?
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5. Solve the following direct variation problems.
a) Gay-Lussac's Law states that at constant volume, the pressure of a gas is directly proportional to the absolute temperature: P ~ T [constant V].
A tank contains a gas at a pressure of 450 kPa and temperature 0oC. Find the pressure in the tank at 60.7oC. Hint: Any temperature on the Celsius scale can be changed to absolute (on the Kelvin scale) by adding 237.15 to it: T(K) = T(oC) + 273.15 o.
b) The surface area of a sphere is directly proportional to the square of its radius. If a baseball with a radius 3.7 cm has a surface area of 172 cm2, find the surface area of a soccer ball, which has a radius 11.1 cm.

6. Solve the following word problems:
1) What single discount is equivalent to two successive discounts of 10% and 15%?
2) Which result is the greater discount on an item: successive discounts of 5%, 10%, and 10%, or one discount of 25%?

7. Solve the following word problems:
a) Find two numbers that differ by 17 and have a sum of 183.
b) The sum of two numbers is 63. Three times the smaller number is 14 more than twice the larger number. Find the numbers.
c) The sum of two numbers is 136. Four times the smaller is 1 less than the larger. What are the numbers?

8. Solve the following problems:
a) Given: A triangle ABC and any point O in the interior, OS is perpendicular to AB, OU is perpendicular to BC, and OT is perpendicular to AC. Prove: AS2 + BU2 + CT2 = SB2 + UC2 + TA2.
b) ABCD is a rectangle and M is any point. Prove that MA2 + MC2 = MB2 + MD2.

9. Solve the following Number Theory problems.
a) Find all pairs of natural numbers whose difference of squares is 91.
b) The sum of two numbers is 177. The greater number has the quotient 3 and the remainder 9 when divided by the smaller. Find the numbers.
c) Two numbers 90 and 100 have the remainders 18 and 4 when divided by the same divisor. Find the divisor.

10. Simplify. Write the answer with positive exponents.
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Answers


1. aaa a) 40 : 3. aaaaaa b) 1 : 2. aaaaaa c) 15. aaaaaa d) grade 9: 68; aaa grade 8: 51.
2.
1) 25 aaaaaaaaaa 2) 9 aaaaaaaaaa 3) - 380
4) 3 aaaaaaaaaaa 5) - 3/2 aaaaaaaaaaaa 6) - 2/3
7) 8/3 aaaaaaaaaaaa 8) 11/3 aaaaaaaaaaaa 9) - 9.5
10) 2.495 aaaaaaaaaaaa 11) - 0.64 aaaaaaaaaaaa 12) - 2.9
13) 4/15 aaaaaaaaaaaa 14) - 6 2/3 aaaaaaaaaaaa 15) 52.5
16) - 10.6 aaaaaaaaaaaa 17) - 2.5

3.
1) - 3 1/3 aaaaaaa 2) - 2 aaaaaaa 3) 1 1/4 aaaaaaa 4) any real number aaaaaaa 5) no solutions aaaaaaa 6) 2

4. aaa pi/4 - 1/2

5. aaa a) 490.7 kPa. aaaaaaaa b) 1548 cm2.

6. aaa 1) 23.5%. aaaaaaaaa 2) 23.05% < 25%.

7. aaa a) 100, 83. aaaaaaaa b) 28, 35. aaaaaaaa c) 27, 109.

9. aaa a) 46, 45 or 10, 3. aaaaaaaaa b) 135, 42. aaaaaaaaa c) 24.

10. aa 1) b13 / (243a9) aaaaaaaaa 2) (8y5z) / (3x2) aaaaaaaaa 3) 4a4b2c4 aaaaaaaaa 4) (4x16) / (9y18)


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