Vladimir Luzgin

Math Lessons for Gifted Students

Level H

(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 01

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

2. Roberta is three years younger than Rebecca. Eight year ago, Roberta was one-half of Rebecca's age. How old is each girl now?

3. Elsie can line the football field in 2 h, while Carey can do the job in 1.5 h. How long will it take them to line the field working together?

4. If all integers from 1 to 1 000 are printed, find the number of times the numerical 5 will appear.

5. A man borrowed $3 500 and a year later paid back the loan plus interest with a cheque for $4 200. Find the annual rate of interest, in percent, paid for the loan.

6. My house has a three-digit number less than 200. This number has exactly 15 different factors. What is my house number?

7. Find the value of the sum:

¹/₂ + ¹/₄ + ¹/₈ + ¹/₁₆ + ¹/₃₂ + ¹/₆₄ + ¹/₁₂₈ + ¹/₂₅₆ + ¹/₅₁₂ + ¹/₁₀₂₄.

8. List all common two-digit multiples of 6 and 9. List all common multiples of 36 and 54 between 500 and 1 000.

9. Two trucks left Buck's Trucks traveling in opposite directions. One truck traveled at a rate of 70 km/h, the other at 80 km/h. After how many hours were the trucks 900 km apart?

10. A dog is tethered to the side of a house as shown. The rope is 30 m long and is tied to the house 10 m from a corner. The dimensions of the house are0 m by 30 m. Find the dog's running area to the nearest square meter. 1884 m².

11. In the diagram, BC || MM. Find x.

12. aa a) Prove that in a parallelogramm ABCD,

(i) opposite sides are equal: AB = DC, BC = AD;

(ii) opposite angles are equal: Angle (A) = Angle (C), Angle (B) = Angle (D);

(iii) the giagonals bisect each other AM = CM, BM = DM, where M is the point of intersection of the diagonals AC and BD.

b) A quadrilateral ABCD has Angle (A) = Angle (C), Angle (B) = Angle (D). Prove that ABCD is a parallelogramm.

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Lesson 02

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1. Evaluate (do without a calculator and show your work).

2. Sophia's age is four years less than twice Beryl's age. In two years, Beryl's age will be three-quarters of Sophia's age. How old is each girl now?

3. Drain A will empty a tank in 5 h. Drain B will empty the tank in 6 h. How long will it take to empty the tank using both drains?

4. A box is designed to hold a cube marked with the digits 1 through 6 as shown. In how many different ways can the cube be placed in the box?

5. To help pay college expenses, Marc borrowed $6 000 from his mother for five years. This interest rate was 3% annual interest. How much did Marc owe his mother after five years?

6. Find the greatest common factor and the least common multiple of each set of numbers.

1) 594 and 180 aaaa 2) 594 and 7 920 aaaaaaaa 3) 34, 51, and 68 aaaaaaa 4) 30, 42, and 70 aaaa

5) 48, 60, 80 aaaaaa 6) 60, 168, and 231 aaaaaa 7) 320, 640, and 960

7. A student receives a set of four marks. If the average of the first two marks is 50, the average of the second and third is 75, and the average of the third and fourth is 70, then what is the average of the first and forth?

8. Find the sum of the GCF and LCM of the numbers 4, 20, and 28. Find the difference between the LCM and GCF of the numbers 20, 36, and 60.

9. After lunch, two women drive in opposite directions. Erica drives east at 80 km/h. Willa drives west at 90 km/h.

a) How far apart are they after 45 minutes?

b) How long would it take them to be 204 km apart?

10. Determine the area of the triangle ABC if AB = 41 cm, BC = 15 cm, BH _|_ AC, Bh = 9 cm.

11. ABC is an ososceles triangle, AB = AC. Ae bisects the angle CAD. Prove that AE || BC.

12. In a parallelogramm ABCD, M and N are points on BC and AD respectively, such that BM = DN. Prove that AM || NC.

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Lesson 03

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1. Evaluate (do without a calculator and show your work).

2. Jack is now 4 times as old as his dog. In 6 years he will be only twice as old as his dog. How old is Jack? How old is his dog?

3. Jerome takes 2.5 h to paint a trailer. Ivan can paint the trailer in 2 h. How long will they take to paint the same trailer if they work together?

4. Each letter represents a different digit. Find the values of each letter.

5. How much money must be invested at 5% per annum to amount to $2 250 at the end of five years?

6. Find the common factors of each set of numbers.

a) 36, 54, 126 aaaa b) 56, 72, 1 110 aaaa c) 936, 1 170, 1 404

7. A man travels 100 km at 50 km/h, 150 km at 100 km/h, and 200 km at 150 km/h. For the entire trip, what is this average speed in kilometers per hour?

8. Find the product of the LCM and GCF of the numbers 168 and 198.

9. Two girls, 60 km apart, start cycling toward each other at the same time. One girl cycles at 18 km/h. How fast must the other girl cycle if they are to meet in 1.5 h?

10. Determine the area of the triangle ABC if

1) AB = 14 cm, BC = 13 cm, AC = 15 cm.

2) AB = 15 cm, BC = 10 cm, AC = 17 cm.

11. In the diagram, KM and KM bisects the angles AKL and CLK, respectively. Prove that Angle (KML) = 90^o.

12. aa a) In a quadrilateral ABCD, BC = AD and BC || AD. Prove that ABCD is a parallelogramm.

b) In a quadrilateral ABCD, AB = DC and BC = AD. Prove that ABCD is a parallelogramm.

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Lesson 04

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1. Evaluate (do without a calculator and show your work).

2. Mike has $2.75 in dimes and quarters. There are 14 coins altogether. How many of each does he have?

3. It takes 10 min to fill a bathtub with the tap turned on full. It takes 15 min to fill a bathtub with the plug pulled out. How long will it take to fill the bathtub with the tap turned on full and the plug pulled out?

4. Each letter represents a different digit. Find the values of each letter.

5. After two years, Bonita had earned $900 interest of money invested at 8% annual interest. How much was her investment?

6. Maria paid $3.36 for carnival buttons; Teresa paid $3.78. What is the greatest possible price of a button?

7. An aircraft flies from city A to city B against the wind at an average speed of 600 km/h. On the return trip, the average speed is 1000 km/h. What is the average speed for the round trip?

8. Find the number of common multiples of 24, 42, and 54 between 1 000 and 2 000.

9. Car A and car B leave Halifax on the same road 1 h apart. Car A leaves first and travels at a steady 80 km/h. How fast must car B travel to overtake car A in 4 h?

10. Determine the area of the parallelogramm ABCD.

11. A billiard ball reflects off two adjacent banks of a billiard table. Prove that the lines of approach and reflection are parallel, that is, BN || AM.

12. In a parallelogramm ABCD, M and N are the midpoints of BC and AD respectively. AM and CN intersect the diagonal BD at P and Q respectively. Prove that BP = PQ = QD.

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Lesson 05

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1. Evaluate (do without a calculator and show your work).

2. A piggy bank contains 91 coins, which are nickels, dimes, and quarters. There are twice as many quarters as dimes, and half as many nickels as dimes. How much is in the piggy bank?

3. A bathtub will empty at a uniform rate in 15 minutes. With the plug in, it will fill at a uniform rate in 12 minutes. How long will it take to fill if the plug is removed and the tap turned on?

4. Each letter represents a different digit. Find the values of each letter.

5. A student invests $4 000, part at 6% and part at 7%. The income from these investments in one year is $250. Find the amount invested at 7%.

6. A 70 cm by 84 cm rectangle is entirely covered by identical square tiles Find the least number of the tiles required.

7. A car is driven up a one-mile long hill at 30 mph, and continues down the other side, which is also one mile in length. What is the speed the car must be driven on the down slope, in mph, in order to average 60 mph for the whole trip?

8. When 62 is divided by a certain number, the quotient is 7 and the remainder is 6. Find the number.

9. Train A leaves a station traveling at 80 km/h. Eight hours later, train B leaves the same station traveling in the same direction at 80 km/h. How long does it take train B to catch up to train A?

10. Determine the area of the trapezoid ABCD, BC || AD, if AB = CD = 25 cm, BC = 18 cm, and AD = 32 cm.

11. Find the sum of the angles A, B, C, D, E, and F in the diagram below.

12. ABCD is an isosceles trapezoid with BC || AD and AB = DC. Prove that Angle (A) = Angle (D) and AC = BD.

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Lesson 06

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1. Evaluate (do without a calculator and show your work).

2. A collection of nickels and dimes has a total value of $8.50. How many coins are there if there are 3 times as many nickels as dimes?

3. Two runners start running laps from the same place at the same time. One runner takes 70 s to run a lap. The other runner takes 80 s to run a lap. When will the two runners be even with each other?

4. Each letter represents a different digit. Find the values of each letter.

5. Barbara invests $2 400 in the National Bank at 5%. How much additional money must she invest at 8% so that the total annual income will be equal to 6% of her entire investment?

6. Use the formula dividend = divisor x quotient + remainder to complete the table:

dividend		458	273
divisor	15		10
quotient	8	10
remainder	4	8

7. Make the statements true by inserting grouping symbols (parenthesis) and any of the four arithmetic operations (+, -, x, :).

1)   5   6   7   8   = ⁵/₁₃

2) 3   4   1   6   = ⁷/₁₂

3) 5   6   7   8   = 8 ¹/₈

8. The time is now 10:00 a.m. What the time will be in 12 291 hours?

9. On an 800 km trip, a family traveled at 25 m/s for the first 600 km and 120 km/h for the remainder of the journey. If they had traveled 96 km/h for the entire journey, how much longer would the trip have taken?

10. Determine the area of the trapezoid ABCD, BC || AD, if AB = CD = 26 cm, BC = 50 cm, and AD = 70 cm.

11. In the diagram, BE bisects the angle aBC and EC bisects the angle ACD. If the angle A is 58o, find the angle E.

12. ABCD is a trapezoid with BC || AD and Angle (A) = Angle (D). Prove that AB = DC.

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Lesson 07

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1. Evaluate (do without a calculator and show your work).

2. Erin collected $6.05 in nickels and dimes. She has 8 more dimes than nickels. How many coins does she have?

3. The average of five numbers is 12. When a sixth number is included, the average is 14. What is the sixth number?

4. Each letter represents a different digit. Find the values of each letter.

5. Beads are placed in the following sequence: 1 red, 1 green, 2 red, 2 green, 3 red, 3 green, with the number of each color increasing by one every time a new group of beads is placed. How many of the first 100 beads are red?

6. An uptown bus leaves the terminal every 45 minutes and a downtown bus leaves the same terminal every 54 minutes. If an uptown bus and downtown bus both leave the terminal at 14:00 hours, find the next time the two busses leave together.

7. Using only odd digits, all possible three-digit numbers are formed. What is the sum of all such numbers?

8. At Discount Dave's, the cost of 13 cassette tapes is more than $20 while the cost of 11 of these tapes is less than $17. Find the cost of one tape.

9. Constable Nancy is driving along a highway at 100 km/h. Pat who is driving in the same direction at a constant speed passes her. Ten seconds after Pat passes Nancy, their cars are 100 m apart. Determine the speed of Pat's car, in km/h.

10. Determine the area of the trapezoid ABCD, BC || AD, if AB = 10 cm, BC = 7 cm, CD = 17 cm, and AD = 28 cm.

11. ABC is an equilateral triangle, and BD = CE. Prove that AD = BE and Angle (BFD) = 60^o.

12. ABCD is a quadrilateral with AB = DC and Angle (A) = Angle (D). Prove that BC || AD.

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Lesson 08

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1. Evaluate (do without a calculator and show your work).

2. A cyclist rode 410 km in five days. Each day he traveled 15 km more than he rode the previous day. Find the distance traveled on the first day.

3. In her latest game, Mary bowled 199 and raised her average from 177 to 178. How much must she bowl to raise her average to 179 with the next game?

4. Find the missing digits and the value of each letter in the problem below:

5. Ten points are spaced equally around a circle. How many different chords can be formed by joining any two of these points? (A chord is a line segment joining two points on the circumference of a circle).

6. A boy is given a supply of rectangles each measuring 39 mm by 42 mm. He wants to lay them out to form the smallest possible square. Each rectangle is laid down with the long side horizontal. Find the dimensions of the square and the number of the rectangle will be required.

7. Make the statements true by inserting grouping symbols (parenthesis) and any of the four arithmetic operations (+, -, x, :).

1)   1   2   5   6   = ¹⁴/₁₅

2)   8   2   1   3   = 3 ²/₃

3)   2   2   2   2   2   = ¹/₃

8. Two runners start at the same point on an oval track. The first runner takes 80 s to go around the track once. The second runner takes only 48 s.

a) How long will it be until the runners are again together?

b) How long will it be until the runners are again together at the start point?

9. A motorist drives 60 miles to her destination at an average speed of 45 miles per hour and makes the return trip at an average rate of 55 miles per hour. What is her average speed in miles per hour for the entire trip?

10. Determine the area of the trapezoid ABCD, BC || AD, if AB = 15 cm, BC = 17 cm, CD = 13 cm, BH _|_ AD, BH = 12 cm.

11. In the diagram, AB = AC = CD. Prove that Angle (DCE) = 3 Angle (ABC).

12. In a triangle ABC, M and N are the midpoints of AB and AC respectively. Prove that MN || BC.

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Lesson 09

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1. Evaluate (do without a calculator and show your work).

2. Pietro had 20 problems for homework. His mother paid him 25 cents for each one he solved and deducted 35 cents for each one he couldn't solve. Pietro earned 80 cents. How many problems was he able to solve?

3. A football team scored a total of 97 points in its first four games. After the fifth game, the team had averaged 27 points per game. How many points did the team score in its fifth game.

4. A 3 x 3 x 3 cube is painted red and then cut into 27 unit cubes. How many of these small cubes will have paint on

a) exactly three faces?

b) exactly two faces?

c) exactly one face?

d) zero faces?

5. Make the statements true by inserting grouping symbols (parenthesis) and any of the four arithmetic operations (+, -, x, :).

1)   3   4   5   6   = 3 ¹/₆

2)   2   3   4   5   = 1 ³/₇

3)   5   6   2   4   = 1 ¹/₃

6. Three comets, X, Y, and Z, return to our solar system at these times: X returns every 42 years, Y returns every 45 years, and Z returns every 36 years. All 3 comets appeared in the year 1913. In what year will all 3 comets next appear?

7. Determine which of the fractions ⁷/₉, ³/₅, ⁸/₁₁, ¹⁰/₁₃, ¹¹/₁₅, and ¹²/₁₉ is greater than ²/₃ and less than ³/₄.

8. What is the unit's digit of aaa a) 2¹⁰⁰ aaa b) 3⁷⁵ aaa c)9⁵⁵. aaa

Hint: a) Expand 21, 22, 23, 24, 25... and look for a pattern in the last digits.

9. A race driver drove one circuit of a six km track. For the first 3 km his speed was 150 km/h, for the next 2 km his speed was 200 km/h, and for the final kilometer his speed was 300 km/h. What was his average speed for the complete circuit in km/h?

10. In the diagram, AC = 7 cm, CD = 3 cm, the area of the triangle ABC is 21 cm². Determine the area of the triangle BCD.

11. Find the sum of the angles at A, B, C, D, and E.

12. A quadrilateral ABCD has E, F, G, and H as the midpoints of AB, BC, CD, and AD respectively. Prove that EFGH is a parallelogram.

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Lesson 10

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1. Evaluate (do without a calculator and show your work).

2. On Old MacDonald's farm, every two horses share a trough, every three cows share a trough, and every eight pigs share a trough. Old MacDonald has the same number of each animal, and he has a total of 69 troughs. How many animals does Old MacDonald have on his farm?

3. In a group of men and women, the average age is 31. If the men's ages average 35 years and the women's ages average 25, find the ratio of the number of men to the number of women.

4. A rectangular 5 x 4 x 3 block has its surface painted red, and then is cut into cubes with each edge 1 unit. What is the number of cubes having exactly

a) 0 of its faces painted red?

b) 1 of its faces painted red?

c) 2 of its faces painted red?

d) 3 of its faces painted red?

5. Make the statements true by inserting grouping symbols (parenthesis) and any of the four arithmetic operations (+, -, x, :).

1)   5   6   4   2   = 1 ¹/₃

2)   3   6   8   2   = 2 ¹/₂

3)   4   3   2   1   = 4 ¹/₃

6. The first bus leaves the terminal every 45 minutes, the second bus leaves this terminal every 36 minutes, and the third bus leaves the same terminal every 54 minutes. If all three busses leave the terminal at 9:00 a.m., what will be the time be the next time the three busses leave together?

7. Arrange these fractions from greatest to least: ⁶/₇, ⁵/₈, ⁹/₁₁, ¹⁰/₁₃, ¹³/₁₅.

8. When each of the following is expressed as an integer, what is the last digit of this integer?

a) 3¹³ + 4¹³ +5¹³ aaa b) 23¹³ + 17¹³ + 18¹³ aaa c) 21¹⁴ + 34¹⁴ + 46¹⁴.

9. Joe Fireball, in his 1935 Moonbeam Racer, averages 60 mph, 40 mph, and 30 mph on 3 successive runs over a 120-mile course. If he completes the entire 120 miles each time, find his average speed for the three runs.

10. In a triangle ABC, Angle (ACB) = 90^o, CD _|_ AB, AD = 4 cm and BD = 2 cm.

1) Prove that Angle (A) = Angle (BCD), Angle (B) = Angle (ACD),

2) Prove that the triangle ABC, ACD, and CBD are similar.

3) Prove that each leg is the mean proportional between the hypotenuse and the prodjection of this leg onto the hypotenuse:

AB : AC = AC : AD,       AB : BC = BC : BD.

4) Find AC and BD.

5) Prove that the hight dropped onto the hipotenuse is the mean proportional between the prodjections of the legs onto the hipotenuse:

AD : CD = CD : BD.

6) Find CD.

11. Prove that if two opposite angles of a quadrilateral are equal, then the bisectors of the other two opposite angles are parallel.

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