Vladimir Luzgin

Math Lessons for Gifted Students

Level F

(grade 7)

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 01

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1. Brian ran 2 km less than Tom and 3 km greater than Dan. They ran a total distance of 26 km. Find how far each boy ran.

2. Solve the following problems.

a) A pair of gloves, which regularly sell for $32.50, is sold at the discount of 30%. What is the sale price of the gloves?

b) What is the regular price of a portable television set if it costs $86.25 and the discount rate is 31%?

3. Line segments AE and AF trisect the area of the square ABCD. Find the ratio DF : FC.

4. Although we are not aware of it, we are constantly moving in a circle, along with the Earth, making one complete rotation each day. If you were on the equator, you would complete a circle with a radius equal to the Earth's radius. The radius of the Earth at the equator is 6 378 km. If you were on the equator, find:

a) how far you would travel in one day;

b) your speed in kilometer per hour.

5. A car traveled 414 km on 46 L of gasoline. How far will the car travel on 200 L of gasoline?

6. Calculate the unit price for each of the following:

a) 750 mL of juice for $2.99;

b) 1.5 L of milk for $2.99;

c) 284 g of cheese for $ 2.25.

7. Solve the following problems:

a) One tenth is what part of three-fourths?

b) 12 is ³/₄ of what number?

c) If ⁵/₆ of a number is 60, what is ³/₄ of the number?

8. Solve equations.

1) 0.2x + 2.3 = 0.7x - 3.2; aaaaaaaaaaa 2) 7.9x + 3.21 = 3.2x + 42.69;

3) 0.71x + 1.98 = 0.37x - 1.76; aaaaaaa 4) 11.3x - (9.7x - 0.8x) + 7.4 = 17.

9. In triangles ABC and A'B'C', M and M' are the midpoints of BC and B'C' respectively. Prove that the triangles ABC and A'B'C' are congruent if AC = A'C', BC = B'C', and AM = A'M'.

10. Suppose this pattern were continued.

a) How many shaded squares would there be on the 11-th diagram?

b) If the position of the diagram in the pattern were known, how could the number of shaded squares be found? Write an expression for the number of shaded squares in terms of the position of the diagram.

11. Find the angle measure indicated by each letter.

12. All rows, columns, and diagonals of a magic square have the same sum. Complete the magic square.

7/12		5/18
	5/12

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Answers 1. aa Brian: 9 km, Tom: 11 km, Dan: 6 km.

2. aa a) $22.75 aaaaaaaaaa 2) $125.00.

3. aa 2 : 1

4. aaa a) 40 074 km; aaa b) 1 670 km/h.

5. aa 1 800 km

6. aaa a) $3.99/L; aaa b) $1.99/L; aaa c) $7.92/kg;

7. aaa 1) ²/₁₅ aaa 2) 16; aaa 3) 54.

8. aa 1) 11; aaa 2) 8.4; aaa 3) -11; aaa 4) 4.

9.

a # a	a Statement	a Reason
a 1. aaa	a AC = A'C'.	a Given
a 2. aaa	a BC = B'C'.	a Given
a 3. aaa	a BM = CM.	a Given
a 4. aaa	a B'M' = C'M'.	a Given
a 5. aaa	a AM = A'M'.	a Given
a 6. aaa	a MC = M'C'.	a Statements 2, 3, and 4.
a 7. a	a The triangles AMC and A'M'C' are congruent.	a SSS: statements 1, 5, and 6.
a 8. a	a Angle (ACM) = Angle (A'C'M').	a Statement 7.
a 9. a	a The triangles ABC and A'B'C' are congruent.	a SAS: statements 1, 2, and 8.

10. aa 1) 41 aaa 2) 4n - 3.

11. aa 1) x = 35^o; aa 2) y = 20^o.

12.

7/12	17/36	5/18
5/36	4/9	3/4
11/18	5/12	11/36

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Lesson 02

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1. Marie ran twice as far as Brenda and 4 km less than Lana. They ran a total distance of 19 km. Find how far each girl ran.

2. Solve the following problems.

a) Andy's Sport Shop was selling a sport bag for $24.95. The bags were not selling, so Andy marked them on sale at 20% off. The bags cost him $18 originally. Will Andy still make a profit? How much?

b) A dress that sells for $80 is placed on a sale at "15% off". Another two weeks on the rack, the current selling price is reduced another 10%. What is the new selling price?

3. Find the four numbers in the ratio 3 : 4 : 6 : 9 whose sum is 330.

4. A 500 m track has semicircular ends. If the length of the track is three times its width, what is the width?

5. Seven meters of steel wire has a mass of 11.9 kg. What is the mass of 53 m of the same steel wire?

6. Which is the better value?

a) 750 mL for $1.29 or 500 mL for $0.85;

b) 20 kg for $24 or 50 kg for $65.

7. If ⁵/₃₆ of a number is 73 ¹/₃, what is ¹³/₈₈ of the original number?

8. Solve equations.

1) 7(1 - 3x) - 3.5 = 0.5 - 24x; aaaaaaaaaaaaa 2) 10.9 + 7x = 6(x - 1.4);

3) 1.7x - 0.2 = 1.4x + 2.5; aaaaaaaaaaaaaaaa 4) 0.5(3x - 1.2) = 0.5x.

9. In triangles ABC and A'B'C', M and M' are the midpoints of BC and B'C' respectively. Prove that the triangles ABC and A'B'C' are congruent if AB = A'B', AC = A'C', and AM = A'M'.

Hint. Extand AM and A'M' to points D and D' such that AM = MD and A'M' = M'D'. Prove that the triangles ACD and A'C'D' are congruent.

10. The diagrams show the molecular structure of some fuels. C represents a carbon atom and H represents a hydrogen atom.

a) How can the number of hydrohen atoms in a molecule of octane be found, if the number of carbon atoms is 8?

b) If the number of carbon atoms in the pattern were known, how could the number of hydrogen atoms in a molecule be found? Write an expression for the number of hydrogen atoms in terms of the number of carbon atoms.

11. Find the angle measure indicated by each letter.

12. All rows, columns, and diagonals of a magic square have the same sum. Complete the magic square.

1/14	6/7
1/2

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Answers 1. aa Marie: 6 km, Brenda: 3 km, Lana: 10 km.

2. aa a) $1.96; aaaaaaaaaa b) $61.20.

3. aa 45, 60, 90, 135.

4. aaa 500 / (4 + pi) = 70 m.

5. aa 90.1 kg.

6. aaa a) The second choice is better: $1.72/L > $1.70/L; aaa b) The first choice is better: $1.2/kg < $1.3/kg.

7. aaa 78.

8. aa 1) -1; aaa 2) -19.3; aaa 3) 9; aaa 4) 0.6.

9.

a # a	a Statement	a Reason
a 1. aaa	a AB = A'B'.	a Given
a 2. aaa	a AC = A'C'.	a Given
a 3. aaa	a BM = CM.	a Given
a 4. aaa	a B'M' = C'M'.	a Given
a 5. aaa	a AM = A'M'.	a Given
a 6. aaa	a Extand AM to a point D such that AM = DM.	a By construction.
a 7. a	a Angle (AMB) = Angle (DMC).	a Statement 6, opposite angles.
a 8. a	a The triangles AMB and DMC are congruent.	a SAS: statements 3, 6, and 7.
a 9. aaa	a AB = DC.	a Statement 8.
a 10. aaa	a Extand A'M' to a point D' such that A'M' = D'M'.	a By construction.
a 11. a	a Angle (A'M'B') = Angle (D'M'C').	a Statement 10, opposite angles.
a 12. a	a The triangles A'M'B' and D'M'C' are congruent.	a SAS: statements 4, 10, and 11.
a 13. aaa	a A'B' = D'C'.	a Statement 12.
a 14. aaa	a DC = D'C'.	a Statements 1, 9, and 13.
a 15. aaa	a AD = A'D'.	a Statements 5, 6, and 10.
a 16. a	a The triangles ACD and A'C'D' are congruent.	a SSS: statements 2, 14 and 15.
a 17. a	a Angle (MAC) = Angle (M'A'C').	a Statement 16.
a 18. a	a The triangles MAC and M'A'C' are congruent.	a SAS: statements 2, 5 and 17.
a 19. aaa	a CM = C'M'.	a Statement 18.
a 20. aaa	a BC = B'C'.	a Statements 3, 4, and 19.
a 21. a	a The triangles ABC and A'B'C' are congruent.	a SSS: statements 1, 2, and 20.

10. aa 1) 18 aaa 2) 2n + 2.

11. aa 1) x = 40^o; aa 2) y = 80^o.

12.

1/14	6/7	5/14
5/7	3/7	1/7
1/2	0	11/14

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Lesson 03

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1. The combined mass of a dog and a cat is 24 kg. The dog is three times as heavy as a cat. Find the mass of each animal.

2. Solve the following problems.

a) A real estate agent may charge 6% commission on the sale of your house. If your house sells for $250 000, how much commission do you have to pay, and how much is left from the sale of your house? Show your work in an organized way.

b) A new car which lists at $24 348 was sold for $22 450. Express the discount as a percent to the nearest tenth.

3. A triangle has sides in the ratio 3 : 5 : 6. The perimeter of the triangle is 210 cm. Find the length of each side of the triangle.

4. In one year the Earth travels once around the sun. It follows a circle with a radius approximately 1.5 x 10⁸ km.

a) How far does the Earth travel in one day?

b) What is the Earth's speed in kilometers per hour?

5. An astronaut who has a weight of 72 kg on Earth has a weight of 12 kg on the moon. If another astronaut has a mass of 10.5 kg on the moon, what is her mass on Earth?

6. Fred earns $75.15 for 9 h of work.

a) How much will he earn for 28 h of work?

b) How long must he work to earn $342.35?

7. What part of an hour elapses between 11:50 a.m. and 12:14 p.m.?

8. Solve equations.

9. In a triangle ABC, BE _|_ AC, CD _|_ AB, BD = CE. Prove that AB = AC.

10. The squares along one diagonal of each face of a cube are colored, as shown, including the diagonals of the faces that can't be seen.

a) How many squares are colored on the 6-th diagram?

b) How many squares are plain on the 10-th diagram?

c) If the position of the diagram in the pattern were known, how could the numbers of colored and plain squares be found? Write expressions for the numbers of colored and plain squares in terms of the number of a diagram.

11. Find the angle measure indicated by each letter.

12. All rows, columns, and diagonals of a magic square have the same sum. Complete the magic square.

	1/5
	1/2
1/4

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Answers 1. aa Cat: 6 kg, dog: 18 kg.

2. aa a) $15 000, $ 235 000; aaaaaaaaaa b) 7.8 %.

3. aa 45 cm, 75 cm, 90 cm.

4. aa a) 2.58 x 10⁶ km; aa b) 1.076 x 10⁵ km/h.

5. aa 63 kg.

6. aaa a) $233.80; aaa b) 41 h.

7. aaa 0.4.

8. aa 1) 1.5; aaa 2) 105.94; aaa 3) 25.45; aaa 4) 24.

9.

a # a	a Statement	a Reason
a 1. aaa	a BE _|_ AC.	a Given
a 2. aaa	a CD _|_ AB.	a Given
a 3. aaa	a BD = CE.	a Given
a 4. a	a The triangles BDC and CEB are congruent.	a HS: statements 1, 2, and 3, BC = CB.
a 5. a	a Angle (CBD) = Angle (BCE).	a Statement 4.
a 6. aaa	a AB = AC.	a Statement 5, ITT, part 2.

10. aa 1) 36 aaa 2) 540 aaa 3) 6n, 6n^2w - 6n.

11. aa 1) x = 20^o; aa 2) y = 30^o.

12.

1/20	1/5	3/4
7/10	1/2	3/10
1/4	4/5	9/20

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Lesson 04

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1. A Jaguar traveled 1.2 times as fast as a Mercedes. The difference in their speeds was 24 km/h. Find the speed of each car.

2. Solve the following problems.

a) What is the rate of discount on a $50 shirt that is reduced to $38?

b) What is the rate of discount on a radio with an original price of $149.50 and a sale price $99.95?

3. A microbe 0.002 mm long, seen under a microscope, appears to be 4 mm long. What is the magnifying power of the microscope?

4. A path 2 m wide is to enclose a circular lawn that has a 25 m radius. What will be the total cost of the material for the path if the cost per square meter is $ 3.00?

5. A tree of height 42 m casts a shadow of 24 m. Find the height, in meters, of a tree casting a 20 m shadow.

6. The moon revolves around the earth 4 times in 118 days.

a) How long does it take the moon to revolve 15 times round the earth?

b) How many times does the moon revolve round the earth in nine years?

7. Twenty nine thirty sixths of a number is 6 ¹¹/₁₈ less than 18 ⁷/₂₄. What is the number?

8. Solve equations.

9. ABC is an equilateral triangle and AK = BM = CL. Prove that KML is an equilateral triangle.

10. How many squares are needed to make the 40-th figure in this pattern? Write an expression for the number of squares in terms of the number of diagram.

11. Find the angle measure indicated by each letter.

12. All rows, columns, and diagonals of a magic square have the same sum. Complete the magic square.

11/6		5/6
	2/3

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Answers 1. aa Mercedes: 120 km/h, Jaguar: 144 km/h.

2. aa a) 24%; aaaaaaaaaa b) 33.144 %.

3. aa 2 000 : 1.

4. aa $980.18.

5. aa 35 m.

6. aa a) 442.5 days; aa a) 111 times.

7. aa 14.5.

8. aa 1) 3.3; aaa 2) 10; aaa 3) 4; aaa 4) 0.7.

9.

a # a	a Statement	a Reason
a 1. aaa	a AB = BC = CA.	a Given
a 2. aaa	a AK = BM = CL.	a Given
a 3. aaa	a AL = BK = CM.	a Statements 1 and 2.
a 4. a	a Angle (A) = Angle (B)= Angle (C) = 60o.	a Statement 1.
a 5 a	a The triangles AKL, BMK, and CLM are congruent.	a SAS: statements 2, 3, and 4.
a 6. aaa	a LK = KM = ML.	a Statement 5.

10. aa 121, 3n + 1.

11. aa 1) x = 45^o; aa 2) y = 40^o.

12.

11/6	0	5/6
1/3	2/3	1
1/2	1 1/3	1/6

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Lesson 05

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1. Joan's jump was longer than Enid's jump by 15 cm. Joan's jump was 1.04 times as long as Enid's jump. How far did each person jump?

2. Solve the following problems.

a) Ernst receives a commission of 12.5% of his total sales for selling tapes. How much does he earn if he sells $1 350?

b) Calculate the regular price if 30% discount is $60.

3. A gear 50 inches in diameter turns a small gear 30 inches in diameter. If the larger gear makes 15 revolutions, how many revolutions does the smaller gear make in that time?

4. A bicycle has a diameter of 50 cm. How far does the bicycle travel when the wheel makes 30 turns?

Hint: The distance covered by a wheel in one revolution is equal to the circumference of the wheel.

5. In physics, Hook' law says that the force exerted by a spring is proportional to the amount that the spring is stretched. If a force of 70 N is needed to stretch a spring 4 cm, what force is needed to stretch the same spring 6.8 cm?

6. A drain can lower the water level of a pool 15 cm in 2 h.

a) How long will it take the drain to lower the water level 33 cm?

b) If the drain is opened for 6 h, by how much will the water level be lowered?

7. What is the minimum number of identical square tiles required to completely tile a rectangle having dimensions 1 ¹³/₁₄ cm by 2 ¹/₄ cm?

8. Solve equations.

9. ABC is an isosceles triangle with AB = AC, D is a point inside the triangle such that Angle (DBC) = Angle (DCB). Prove that AD bisects the angle BAC.

10. Use the pattern suggested by the diagram to evaluate this sum: 1 + 3 + 5 + 7 + ... + 15. Find the formular for the sum 1 + 3 + 5 + ... + (2n - 1).

11. Find the angle measure indicated by each letter.

12. Find the following and express your answer in lowest terms.

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Answers 1. aa Joan: 3.90 m, Enid: 3.75 m.

2. aa a) $ 168.75; aaaaaaaaaa b) $ 200.

3. aa 25.

4. aa 47.125 m.

5. aa 119 N.

6. aa a) 4 h 24 min; aa a) 45 cm.

7. aa 42.

8. aa 1) 1; aaa 2) 3; aaa 3) 5; aaa 4) 12. aaa 5) -3. aaa 6) -2.

9.

a # a	a Statement	a Reason
a 1. aaa	a AB = AC.	a Given
a 2. aaa	a Angle (DBC) = Angle (DCB).	a Given
a 3. aaa	a BD = CD.	a Statement 2, ITT, part 2.
a 4 a	a The triangles ABD and ACD are congruent.	a SSS: statements 1 and 3, AD is common.
a 5. aaa	a Angle (BAD) = Angle (CAD).	a Statement 4.

10. aa 1 + 3 + 5 + 7 + ... + 15 = 8² = 64, 1 + 3 + 5 + ... + (2n - 1) = n².

11. aa 1) x = 30^o; aa 2) y = 20^o.

12. aa 1) ¹/₄; aaa 2) 6; aaa 3) ⁴/₉; aaa 4) ¹/₂.

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Lesson 06

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1. The length of a rectangle is 5 cm longer than the width. The perimeter is 78 cm. Find the dimensions of the rectangle.

2. What was the original price of a radio that sold for $70 during a 20%-off sale?

3. A school has enough bread to feed 30 children for 4 days. If 10 more children are added, how many days will the bread last?

4. Sam's bicycle has wheels each of which has a diameter of 60 cm. When Sam goes for a 2 km ride on his bike, find the approximate number of times each wheel will rotate.

5. To estimate the number of trout in a lake, the game warden caught 85 trout, tagged them, and then released them. Later, he caught 95 trout and found tags on 7 of them. Approximately how many trout are in the lake?

6. The amount of energy required to melt 16 g of ice is about 5.3 kJ.

a) How much energy is required to melt 50 g of ice?

b) How many grams of ice can be melted with 30 kJ energy?

7. It requires 1 hour 10 minutes to fill ⁷/₁₂ of a swimming pool. Find the number of hours required to fill the remainder of the pool at this rate.

8. Solve equations.

9. ABC is an isosceles triangle with AB = AC, AM is the median drawn to BC. Find AM if the perimeters of the triangles ABC and ABM are 50 cm and 40 cm respectively.

10. Suppose this pattern were continued.

a) Discuss how the pattern is produced. How many squares would there be on the 10-th diagram?

b) Let x represents the number of a diagram, and y represents the number of squares on the diagram. Write an equation relating x and y.

11. Find the angle measure indicated by each letter.

12. Find the following and express your answer in lowest terms.

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Answers

1. aa Length: 22 cm, width: 17 cm.

2. aa $ 87.50;

3. aa 3 days.

4. aa 1 061.

5. aa About 1150.

6. aa a) 16.56 kJ; aa b) 90.7 g.

7. aa 50 min.

8. aa 1) -3; aaa 2) 0; aaa 3) 2; aaa 4) 5.4. aaa 5) ¹/₃; aaa 6) 7.

9. aa AM = 15 cm.

10. aa a) 28; b) aa y = 3x - 2.

11. aa 1) x = 36^o; aa 2) y = 37.5^o.

12. aa 1) 13 ⁴/₇; aaa 2) ¹/₈; aaa 3) 2 ²/₃; aaa 4) ¹/₂₇.

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Lesson 07

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1. A rectangular table is twice as long as it wide. How long is the table if its perimeter is 15.6 m?

2. Express each reduction as a percent, to 1 decimal place, of the original price.

a) A TV set regularly priced at $540 is selling for $499.

b) An overcoat regularly priced at $195 is selling for $156.

3. If 15 cans of food are needed for seven adults for two days, what is the number of cans needed to feed four adults for seven days?

4. AE is divided into four equal parts and semicircles are drawn on AC, CE, AD, and DE, creating paths from A to E as shown. Determine the ratio of the length of the upper path to the length of the lower path.

5. Two steel spheres have 1-inch and 2-inch radii, respectively. The smaller weighs 9.5 pounds. Find the weight of the larger.

6. A jet travels 2 400 km in 3h.

a) How far does it travel in 7.5 h?

a) How long would it take to fly 5 800 km?

7. The mass of a candy-bar wrapper is ¹/₁₁ the mass of the wrapped bar. If the candy bar alone has a mass of 75 g, what is the mass of the wrapper?

8. Solve equations.

9. In the diagram, AB is parallel to DC. The semicircle AED has the diameter AD of length 4 cm. Find the perimeter and the area of the figure.

10. Determine if it is possible to draw each diagram without lifting your pencil from the paper, and without going any line twice.

11. Find the angle measure indicated by each letter.

12. Find the following and express your answer in lowest terms.

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Answers

1. aa 5.2 m.

2. aa a) 7.593% aa b) 20%.

3. aa 30 cans.

4. aa 1 : 1.

5. aa 76 pounds.

6. aa a) 6 000 km; aa a) 7.25 h.

7. aa 7.5 g.

8. aa 1) 6; aaa 2) 39; aaa 3) 1; aaa 4) 1.

9. aa Perimeter = 2 pi + 20 cm., 2 pi + 30 cm².

10. aa yes, yes, yes, no.

11. aa 1) x = 30^o; aa 2) y = 60^o.

12. aa 1) 0; aaa 2) 1.5; aaa 3) 1; aaa 4) 60 ¹/₃.

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Lesson 08

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1. A rectangular walk is a line of 9 identical square cement tiles. The perimeter of the walk is 40 m. What is the area of each cement tile?

2. Solve the following problems.

a) When water freezes the ice formed has 9% more volume than the water. How much water must freeze to make 872 m3 of ice?

b) 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. In an isosceles triangle, the two different sizes of angles are in the ratio 4 : 7. What are the angles?

4. Four pipes, each of diameter 1 m, are held tightly together by a metal band as shown. How long is the band?

5. If 3 miles are equivalent to 4.83 kilometers, then 11.27 kilometers are equivalent to how many miles?

6. Janet travels 48 km in 45 minutes. Find her speed, in kilometers per hour.

7. Lorie is one-third of the way up a flight of stairs. If she climbs 11 more steps, she will be half way up. Find the number of steps in the flight.

8. Solve equations.

9. In a triangle ABC, AB = 41 cm, BC = 15 cm, BH _|_ AC, BH = 9 cm. Determine the area of the triangle ABC.

10. Staccy has a 3 L bucket and an 8 L bucket. How can she use these two unmarked buckets to obtain exactly 4 L of water?

11. Find the angle measure indicated by each letter.

12. Find the following and express your answer in lowest terms.

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Answers

1. aa 4 m².

2. aa a) 800 m³, aa b) $39.

3. aa 48^o, 48^o, 84^o or 40^o, 70^o, 70^o.

4. aa pi + 4 m.

5. aa 7 miles.

6. aa 64 km/h.

7. aa 66.

8. aa 1) ¹/₁₂; aaa 2) 15; aaa 3) 2.88; aaa 4) 2 ²⁵/₃₂.

9. aa 234 cm².

10.

a # a	a 3 L bucket a	a 8 L bucket a
a 1. aaa	a 0 L	a 0 L
a 2. aaa	a 3 L	a 0 L
a 3. aaa	a 0 L	a 3 L
a 4 a	a 3 L	a 3 L
a 5. aaa	a 0 L	a 6 L
a 6. aaa	a 3 L	a 6 L
a 7. aaa	a 1 L	a 8 L
a 8. aaa	a 1 L	a 0 L
a 9. aaa	a 0 L	a 1 L
a 10. aaa	a 3 L	a 1 L
a 11. aaa	a 0 L	a 4 L

11. aa 1) x = 15^o; aa 2) y = 150^o.

12. aa 1) 5 ⁷/₁₂; aaa 2) 1 ⁵/₆.

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Lesson 09

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1. A milk shake costs twice as much as an order of French fries. If two milk shakes and three orders of French fries cost $4.20, what is the cost of a milk shake?

2. Solve the following problems.

a) What would be the marked price of an article if the cost was 302.40 and the gain was 10% of the selling price?

b) 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.

3. The ratio of nickels to dimes to quarters in a sum of money is 3 : 4 : 5. What is the value of the money if there are 204 coins?

4. Find the area of a circle whose circumference is pi².

5. A map is drawn so that 1 cm represents 30 km (the scale on the map is 1 : 3 000 000). How far apart are two cities if they are 11.4 cm apart on the map?

6. From 9 a.m. to 2 p.m., the temperature rose at a constant rate from -14^o F to +36^o F. What was the temperature at noon?

7. The grand prizewinner of a lottery won ⁷/₁₀ of the total prize money available. Shortly the reafter, she spent ³/₄ of the winnings, and still had $2 100 left. Find the total amount of prize money available in the lottery.

8. Solve equations.

9. What is the area of an equilateral triangle with sides s units long?

10. A train leaves at 7:00 a.m. daily from Toronto bound for Vancouver. Simultaneously, another train leaves Vancouver for Toronto. The journey takes exactly 4 days in each direction. If a passenger boards a train in Vancouver, how many Vancouver bound trains will she pass in route to Toronto?

11. One angle of a triangle is twice the size of the second angle, and the third angle is 66^o. Find the smallest angle.

12. Find the following and express your answer in lowest terms.

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Answers

1. aa $1.20.

2. aa a) $336 aa b) 35 cm.

3. aa $30.6.

4. aa 0.25рi³.

5. aa 342 km.

6. aa 16^oF.

7. aa $12 000.

8. aa 1) -34; aaa 2) 79; aaa 3) ⁴⁵/₉₁; aaa 4) ¹/₇.

9. aa Area = s²sqrt(3)/4.

10. aa 7 trains.

11. aa 38^o.

12. aa 1) 3; aaa 2) ³/₈; aaa 3) 10;

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Lesson 10

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1. Donna's average mark out of three tests was 84 out of 100. Her highest mark was one-and-one-quarter times her lowest mark. The middle mark was 81. What were Donna's marks on the three tests?

2. Solve the following problems.

a) 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?

b) If 56% of x is equal to 140% of 25, what is the value of x?

3. A pulley having a 9-inch diameter is belted to a pulley having a 6-inch diameter, as shown in the figure. If the large pulley runs at 120 rpm, how fast does the small pulley run, in revolutions per minute?

4. A square and a circle have equal perimeters. Find the ratio of the area of the square to the area of the circle.

5. The scale of a map reads 1 : 500 000. Find the distance, in km, between two towns, which are 2.5 cm apart on the map.

6. Village A has a population of 6 800, which is decreasing at a rate of 120 per year. Village B has a population of 4 200, which is increasing at a rate of 80 per year. In how many years will the population of the two villages be equal?

7. At Ungerville High School, the ratio of girls to boys is 2 : 1. If ³/₅ of the boys are on a team and the remaining 40 boys are not, how many girls are in the school?

8. Solve equations.

9. Given that the area if the triangle ABD is 12. 5 cm², AD = 5 cm, DC = 7 cm, determine the area of the triangle ABC.

10. What is the 100-th number in the pattern 2, 5, 8, 11, 14, 17, ...?

11. In a triangle ABC, angle B is 36^o larger than angle A, and angle C is six times angle A. Find the number of degrees in angle A.

12. Find the following and express your answer in lowest terms.

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Answers

1. aa 76, 81, 95.

2. aa a) 5 ounces aa b) 62.5.

3. aa 180 rpm.

4. aa рi : 4³.

5. aa 12.5 km.

6. aa 13 years.

7. aa $12 000.

8. aa 1) -34; aaa 2) 79; aaa 3) ⁴⁵/₉₁; aaa 4) ¹/₇.

9. aa Area = 30 cm².

10. aa 299.

11. aa 18^o.

12. aa 1) 5 ⁵/₉; aaa 2) ³/₈.

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