Vladimir Luzgin

Math Lessons for Gifted Students

Level G

(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
[email protected]

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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. A piece of string, 60 cm long, is cut into three pieces. The middle-sized piece is 2 cm longer than the shortest piece and 5 cm shorter than the longest piece. What is the length of each piece?

2. When 14 is subtracted from three-eights of a number the result is 19. Find the number.

3. A tank is ¹/₆ full of gasoline. If 2 gallons are added, then the tank is ¹/₄ full. What is the total capacity of the tank, in gallons?

4. The distance between two towns 1 300 km. How far apart are they on the map, if the scale is 1 : 10 000 000.

5. A rectangle with the dimensions 32 cm by 72 cm is to be cut into the largest possible squares of equal size. What will be the number of the squares?

6. Write the first five common multiples of 8 and 6.

7. Find the average of ¹/₂, ²/₃, and ³/₄. What is the average of ²/₃, ³/₄, and ⁵/₆?

8. A bag contains 5 yellow marbles, 10 red marbles and 15 white marbles. If one marble is selected at random, what is the probability that it is yellow?

9. A cable is formed by combining 3 circular wires, touching each other as shown. These wires are held in place by a taut band around the system. If each wire has a radius of 2 units, what is the length of the band?

10. If one angle of a triangle is three times a second angle, and the third angle is 20^o more than the second angle, what is the second angle, in degrees?

11. Find the values of x and y.

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

13. A car is traveling 90 km/h. Find the distance, in meters, it travels in 10 seconds.

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Answers

1. aa 17 cm, 19 cm, and 24 cm.

2. aa 88.

3. aa 24 gallons

4. aa 13 cm.

5. aa 36.

6. aa 24, 48. 72, 96, 120.

7. aa $12 000.

8. aa ²³/₃₆, ³/₄.

9. aa 4pi + 12 units.

10. aa 32^o.

11. aa 1) x = 7.5, y = 16; aa 2) x = 9, y = 10.

12. aa 1) 8 ³/₄; aaa 2) 9.

13. aa 250 m.

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Lesson 02

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1. Find two consecutive numbers with a sum of 367.

2. When 28 is added to seven times a number, the result is the same as if 16 were subtracted from eleven times that number. What is the number?

3. A gasoline tank is ¹/₄ full. After adding 10 gallons of gasoline, the gauge indicates that the tank is ²/₃ full. Find the capacity of the tank in gallons.

4. The distance between Moscow and Nizniy Novgorod is 440 km. If these two cities are 8.8 cm apart on a map, what is the scale? Express your answer as a ratio in lowest terms.

5. List all four-digit numbers whose units digit is 5 and which are divisible by 423.

6. List in increasing order all the factors of each number.

1) 32 aaaa 2) 243 aaaa 3) 24

4) 54 aaaa 5) 30 aaaaa 6) 100

7) 72 aaaa 8) 216 aaaa 9) 180

7. Solve the following problems:

a) In a family of five, the heights of the members are 5 feet 1 inch, 5 feet 7 inches, 5 feet 2 inches, 5 feet, and 4 feet 7 inches. What is the average height?

b) The heights of the five starters on the Redwood High's basketball team are 5"11', 6"3', 6", 6"6', and 6"2'. What is the average height of these players?

Hint: 1 foot = 12 inches (1" = 12').

8. A jar contains 4 red candies, 6 orange candies, 3 purple candies, and 7 yellow candies. If you select one candy from the jar without looking, what is the probability that you will select an orange candy? What is the probability that you will select either a yellow or red candy?

9. Find the area of each shaded region, to one decimal place.

10. Find the sum of the interior angles of

1) a quadrilateral;

2) a pentagon;

3) a hexagon.

11. Find the values of x and y.

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

13. A plane traveling 600 miles per hour is 30 miles from Kennedy Airport at 4:58 p.m. At what time will it arrive at the airport?

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Answers

1. aa 183 and 184.

2. aa 11.

3. aa 24 gallons

4. aa 1 : 5 000 000.

5. aa 2115, 6345.

6.

aa 1) 1, 2, 4, 8, 16, 32;

aa 2) 1, 3, 9, 27, 81, 243;

aa 3) 1, 2, 3, 4, 6, 8, 12, 24;

aa 4) 1, 2, 3, 6, 9, 18, 27, 54;

aa 5) 1, 2, 3, 5, 6, 10, 15, 30;

aa 6) 1, 2, 4, 5, 10, 20, 25, 50, 100;

aa 7) 1, 2, 3, 4, 6, 8, 8, 12, 18, 24, 36, 72;

aa 8) 1, 2, 3, 4, 6, 8, 12, 18, 24, 27, 36, 54, 72, 108, 216;

aa 9) 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180.

7. aa a) 5 feet 1 inch; aa a) 6'' 2'.

8. aa 0.3, 0.55.

9. aa 1) 1.25pi cm² or 3.9 cm², aa 2) 9pi cm² or 28.3. cm², aa 3) 25pi + 150 cm² or 228.5 cm².

10. aa 1) 360^o, aa 2) 540^o, aa 3) 720^o.

11. aa 1) x = 13 ¹/₃, y = 12; aa 2) x = 9, y = 7.5.

12. aa 1) 24 ³/₄; aaa 2) 2.

13. aa 5:01 p.m..

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Lesson 03

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1. Find four consecutive numbers with a sum of 234.

2. ³/₄ of a certain number is 8 more than ²/₃ of the same number. What is the number?

3. When 16 liters are added to a gas tank, which is ¹/₂ full, the tank is ²/₃ full. Find the capacity of the tank.

4. The scale of drawing of a rectangle is 2 : 5. The length of the drawing is 38 mm and the width is 26 mm. Find the area of this rectangle.

5. Prove that the difference between any two-digit number and the number obtained by reversing the order of the digits is divisible by 9.

6. A number is divisible by 11 if the difference between the sums of alternating digits is divisible by 11. For example, 7381902 is divisible by 11: 7 - 3 + 8 - 1 + 9 - 0 + 2 = 22, and 22 is divisible by 11. Place the missing digits to get numbers that are divisible by 11.

a) 75_36 aaaaaaa b) 183_ aaaaaaa c) 34_1

7. In a competition, the average score of Pat's first 4 gams was 6.5; the average of her next 5 games was 6.4. If she scored 9 on her tenth game, what was her overall average?

8. Given two 5's and two 6's, calculate the probability of randomly drawing two numbers from a box to form

a) the two-digit number 55;

b) the number 56.

9. Find the area of each shaded region, to one decimal place.

10. Find the sum of the exterior angles of a quadrilateral.

11. Find the values of x and y.

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

13. A plane flies over Denver at 11 : 20 a.m. It passes over Coolidge, 120 miles from Denver, at 11 : 32 a.m. Find the rate of the plane in miles per hour.

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Answers

1. aa 57, 58, 59, 60.

2. aa 96.

3. aa 96 L

4. aa 6 175 mm².

5. aa Hint: (10x + y) - (10y + x) = 9(x - y).

6. aa a) 75636; aa b) 1837; aa c) 3421.

7. aa 6.7.

8. aa a) ¹/₆, aa b) ²/₃.

9.

1) 4.5pi cm² or 14.1 cm², aaaaaaaa 2) 25 - 6.25pi cm² or 5.4. cm²,

3) 3.125(pi - 2) cm² or 3.6 cm², aa 4) 6pi - 9sqrt(3) cm² or 2.2 cm².

10. aa 360^o,

11. aa 1) x = 26 ²/₃, y = 24; aa 2) x = 14.4, y = 10.

12. aa 1) 1.25; aaa 2) 2.16.

13. aa 600 miles per hour.

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Lesson 04

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1. Find two consecutive even numbers with a sum of 270.

2. Millie is four times as old as Marty. Five times Marty's age is 13 more than Millie's age. Find how old they are.

3. A pool is filled to ³/₄ of its capacity. ¹/₁₂ of the water in the pool evaporates. If the poll can hold 24 000 gallons when it is full, how many gallons of water will have to be added in order to fill the pool?

4. If 10 calculators cost $179.50, how much do 17 calculators cost?

5. Among grandfather's papers an old bill was found: "72 Turkeys $_ 67.9 _". The first and the last digits of the number that represented the total price of the turkeys are replaced here with blanks as they had faded and are now illegible. What are the missing digits?

6. Only one of the following numbers is a perfect square. Without using a calculator, determine which it is.

a) 9 415 aaa b) 5 408 aaa c) 9 626 aaa d) 9 204

e) 8 652 aaa f) 7 569 aaaa g ) 3 547 aaa h) 7 383

7. Jane traveled for 2 hours at a rate of 90 kilometers per hour and for 5 hours at a rate of 76 kilometers per hour. What was her average speed for the 7-hour period?

8. An integer is chosen at random from 1 to 60 inclusive. What is the probability the integer chosen contains the digit 4?

9. Three equal circles, with centers A, B, and C, are inscribed in a rectangle, as shown. If AC = 12 cm, find the area of the shaded region.

10. In a quadrilateral ABCD, angle D is 120^o, angle B is 90^o, and angle A is four times angle C. Find the number of degrees in angle C.

11. Find the angle measure indicated by each letter if:

1) KL || MN and EF || GH aaaaaaaaaaaaaaaaaaaaaaaaaa 2) RS || TU

x = ______, aaa y = _______ aaaaaaaaaaaaaaaa x = ______, aaa y = _______.

z = ______, aaa w = _______.

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

13. A car travels the 80 miles between Hamilton and London in 2 hours. It continues at the same rate to Windsor, which is 290 miles from Hamilton. What is the total time, in hours, it takes to drive from Hamilton to Windsor?

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Answers

1. aa 134, 136.

2. aa Millie: 52, Marty: 13.

3. aa 8 000 gallons.

4. aa $ 305.15.

5. aa $ 367.92.

6. aa 7 569 = 87².

7. aa 80 km/h.

8. aa 0.25.

9. aa 108 - 27pi cm² or 23.2 cm².

10. aa 30^o.

11. aa 1) x = 120^o, y = 60^o, z = 60^o, w = 60^o; aa 2) x = 55^o, y = 75^o.

12. aa 1) 16; aaa 2) 2.4.

13. aa 7 h 15 min.

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Lesson 05

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1. The sum of five consecutive odd integers is 165. What is the largest of these integers?

2. Susan is twice as old as Lana. The sum of their ages 4 years ago was 37. Find how old they are now.

3. When a pitcher is ¹/₂ full it contains exactly enough water to fill three identical glasses. How full would the pitcher be if it had exactly enough water to fill four of the same glasses?

4. Jessie cut a lawn in 4 h. It takes Sheila only 2 h to cut the same lawn. How long will it take Jessie and Sheila to cut the same lawn working together?

5. Find a two-digit number, which equals to the triple sum of its digits.

6. Find the number of positive integers that are less than 1 000 that are divisible by 3 and are not divisible by 5.

7. The chart shows the distribution of $1, $2, $3, $4, and $5 door prizes. Find the average value of all the door prizes?

Value of Prize	Number of Prizes
$ 1	25
$ 2	20
$ 3	8
$ 4	4
$ 5	3

8. Pierre has two sets of twelve cards each numbered 1 to 12. One set of cards is red and the other is blue. If one card selected from each deck, what is the probability of the two cards summing to 15?

9. The radius of a circle increased by 50%. What is the percentage change in area of a circle?

10. The largest angle of a given triangle is 35^o more than the smallest angle, and the smallest angle is 10^o less than the third angle. Find the number of degrees in the smallest angle.

11. Find the value of x if:

1) AB || DE aaaaaaaa 2) KL || HG aaaaaaaaaaaa 3) MN || QP

x = ______ aaaaaaaaaaa x = ______ aaaaaaaaaaaaaaa x = ______

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

13. Two towns are 80 km apart. Sylvia wants to drive from one town to the other in exactly one hour. For the first 30 minutes she drives at a rate 60 km/h. At what constant rate must she drive for the next 30 minutes if she is to accomplish her goal?

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Answers

1. aa 31, 32, 33, 34, 35.

2. aa Susan: 30, Lana: 15.

3. aa ²/₃.

4. aa 1 h 20 min.

5. aa 27.

6. aa 267.

7. aa $ 2.00.

8. aa ⁵/₇₂.

9. aa Increased by 125%.

10. aa 45^o.

11. aa 1) x = 9; aa 2) x = 10; aa 3) x = 15.

12. aa 1) 308; aaa 2) ¹³/₁₂₈.

13. aa 100 km/h.

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Lesson 06

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1. One number is three-fifths of another number. The two numbers total 104. Find the numbers.

2. The ages of John and Mary total 27 years. Twice Mary's age plus John's age is 40. How old is each person?

3. One-half of the female students in a certain college eat in the cafeteria and one third of the male students eat there. If the ratio of the female students to the male students in the college is 3 : 4, what fractional part of the student body eats in the cafeteria?

4. How far will a car travel on 18 L of gasoline if it travels 300 km on 40 L of gasoline?

5. Find the number of positive integer divisors of 60, including 1 and 60.

6. List the multiples of 24 between 200 and 300.

7. A class of 20 students averaged 66% on an examination; another class of 30 students averaged 56%. Find the average percentage for all students.

8. Using each of the digits 4, 5, 6, 7, and 8 only once, create a three- digit number. What is the probability that it has only even digits?

9. Three concentric circles are such that the radii of the inner and outer circles are 9 and 15, and the middle circle bisects area between these two. Find the radius of the middle circle.

10. In the figure below, Angle (A) = (9x - 40)^o, Angle (B) = (4x + 30)^o, and Angle (BCD) = (8x + 40)^o. Find the measure is angle ABC.

11. Find the value of xif:

1) BC || DE aaaaaaaaaaaaaaaaaaaa 2) AB || DC aaaaaaaaaaa 3) BC || DE

x = ______ aaaaaaaaaaaaaaa x = ______ aaaaaaaaaaaaaaaaaa x = ______

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

13. Dianne rode her dirt bike up a hill and down the same distance on the other side. She rode down the hill at 4 times the speed she rode up the hill. If the entire trip took 20 min, how many minutes did it take her to ride down the hill?

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Lesson 07

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1. The sum of three numbers is 20. The second number is twice the first, and the third number is 9¹/₆ less than the second. Find the numbers.

2. A mother is three times as old as her daughter. Six years ago, she was five times as old. How old are the mother and the daughter now?

3. A piece of fabric is cut into three sections so that the first is 2 ²/₃ times as long as the second and the second section is 2 ¹/₄ times as long as the third. What part of the entire piece is the largest section?

4. In driving from New York to Boston, Mr. Portney drove for 3 hours at 40 miles per hour and 1 hour at 48 miles per hour. What was his average rate for this portion of the trip?

5. Find the number of all the factors of each number.

a) 1 024 aaaa b) 1 296 aaaa c) 36 000

6. Write the greatest multiple of 54, which is less than 999. Write the least multiple of 72 which is greater than 2 000.

7. In a set of ten numbers, the average of the first four numbers is 12, and the average of the last six numbers is 17. What is the average of all ten numbers?

8. If two dice are rolled, what is the probability that the sum of the spots showing is 9?

9. The arc lengths of three semicircles are as shown. Find the area of the shaded region.

10, In a triangle ABC, Angle (BAC) = 70^o, and the bisectors of the angles B and C meet in D. Find the measure of the angle BDC.

11. Find the values of x and y.

x = ______, aaa y = _______ aaaaaaaa x = ______, aaa y = _______.

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

13. A man makes a trip in 3 hours. Two-thirds of the distance traveled was by car and the remainder by boat. If his rate by car was three times his rate by boat, what was the time spent in the boat?

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Lesson 08

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1. The sum of three numbers is 75. The second number is 5 more than the first, and the third is three times the second. Find the numbers.

2. Tanya is 12 years older than Leah. Three years ago, Tanya was five times as old as Leah. How old is Leah?.

3. The value of a fraction is ³/₄. If 3 is subtracted from the numerator and added to the denominator, the value of the fraction is ²/₅. Find the original fraction.

4. A man travels a distance of 20 miles at 60 miles per hour and then returns over the same route at 40 miles per hour. What is his average rate for round trip in miles per hour?

5. Solve the followinh problems.

1) is the sum of all the divisors of 24?

2) What is the sum of all the even divisors of 512?

3) What prime numbers divide 2 176 and 2 159?

6. List all the two-digit multiples of 13.

7. Five boys wrote a mathematics test. The average mark was 68. If the marks of four boys were 75, 62, 84, and 53, find the mark of the fifth boy.

8. What is the probability of rolling a sum of 11 with two dice? What is the probability of getting any sum other than 11?

9. Two circles, each of radius one unit, touch as shown at point A. BC is the tangent to each circle. Find the area of the shaded region.

10. Find the measure of the angle BAC if Angle (ABE) + Angle (ACD) = 240^o.

11. Find the values of x and y.

x = ______, aaa y = _______ aaaaaaaaaaaaaaaa x = ______.

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

13. Kim lives at the bottom of a mountain. It took her 6 h to ride to the top of the mountain and back. Her average speed up the mountain was 6 km/h and her average speed down the mountain was 24 km/h. How long did it take Kim to ride down the mountain?

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Lesson 09

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1. For two consecutive integers, the sum of twice the larger and three times the smaller is 242. Find the integers.

2. A grocer buys oranges at 3 for 25 cents. He plans to sell them at 6 for 55 cents. In order to make a profit of $3.00, how many must he sell?

3. Find the value of the product:

(1 - ¹/₂) x (1 - ¹/₃) x (1 - ¹/₄) x (1 - ¹/₅) x (1 - ¹/₆) x (1 - ¹/₇) x (1 - ¹/₈).

4. A boat has an average speed of 50 km/h on the first lap of a two-lap race. On the second lap it averages only 30 km/h. What is its average speed for the hole race?

5. Find common factors of each set of numbers.

a) 8, 12 aaa b) 36, 90 aaa c) 40, 60 aaa d) 30, 42, 70

6. I am an odd number between 200 and 300. Two of my factors are 7 and 13. What number am I?

7. The average of five different numbers is 4. When the greatest number is removed from the set, the average of the remaining numbers is 2. What number is removed?

8. You are rolling two 6-sided dice, but one die is numbered from 3 to 8 and the other die is numbered from 4 to 9. What is the probability of rolling a sum of 14 with these two dice?

9. A sector of a circle of radius 14.5 cm has a sector angle of 75^o. Calculate:

a) the arc length of the sector;

b) the area of the sector.

10. In a triangle ABC, M is the midpoint of AB. Find the measure of the angle ACB if AM = BM = CM.

11. In the diagram, AB || CD, Angle (BME) = 60^o, Angle (CNE) = 130^o. Find the number of degrees in angle MEN.

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

13. Two snails are 3 feet apart and directly facing each other. If one snail moves forward continuously at 0.04 inches per second and the other moves forward continuously at 0.05 inches per second, how many minutes will it take for the snails to touch?

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Lesson 10

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1. When five-sixths of a number is added to one-fourth of the number the result is 39. Find the number.

2. Bill is twice as old as his brother Dan. In 7 years, Bill will be only one and one-half times as old as Dan. How old is Bill now?

3. Machine X makes 200 boxes in 3 min and machine Y makes 200 boxes in 2 min 24 sec. With both machines working, how long will it take to make 200 boxes?

4. If p pencils cost d dollars, how many pencils can be bought for c cents?

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

a) 6, 8 aaaaaaa b) 12, 16 aaaaa c) 12, 18

d) 50, 175 aaa e) 675, 875 aaa f) 612, 680

6. My house has a two-digit number. This number has exactly 7 different factors. What is my house number? What are its factors?

7. The average of a set of 10 numbers is 20. If one of the numbers is removed from the set, the average of the remaining numbers is 19. What number was removed?

8. Given one each of 3, 4, 5, 6, and 7, you are asked to create a two-digit number. What is the probability that it is odd?

9. One sector of a circle graph has a sector angle of 80^o. The graph has a radius of 12.5 cm.

a) Find the arc length of the sector.

b) Find the area of the sector.

10. In a triangle ABC, Angle (BAC) = 80^o, Angle (ABC) = 70^o, AP _|_ BC, BQ _|_ AC, M is the point of intersection of AP and BQ. Find the measure of the angle AMB.

11. Given: AB || CD. Find: Angle (BAE) + Angle (AEC) + Angle (DCE).

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

13. Frank and Ernest start jogging on a 110 m circular track. They begin at the same time and from the same point but jog in opposite directions, one at ⁸/₃ m per second and the other at ⁷/₃ m per second. How many times will they meet during the first 15 minutes of jogging?

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