if you use the laws of arithmetic you can make these...

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Tools we need.

Arithmetic operations. How many arithmetic operations do you know?

1. Addition:+3

7 10

2. To “undo” (invert) addition, we use subtraction:

- 3 7 10

3. To do addition many times, multiplication was created: 7 + 7 + 7 = 7 3

3 7 21

4. To “undo” (invert) multiplication, we use division;

3 7 21

5. To do multiplication many times, raising to a power was created.

7 7 7 7 = 74

7 2,401

6. To “undo” (invert) raising to a power, we use the method of taking the root of a number.

7 2,401

6rade-RSM new- Ch1 Introduction 1 ©Russian School of Mathematics

Of course, multiplication is a multiple addition only when you multiply by a whole number; then the operation of multiplication was expanded for no whole numbers as well.

4ٱ

These are just agreements, not mathematical rules (people decided this to avoid confusion). On the contrary, mathematical rules are not something people decide; they make sense and just have to be this way. We’ll discuss examples later.

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The order of operations. Now, when we write down these operations with numbers, there are some agreements about the order of these operations: You probably know these agreements. They are listed here for convenience only:

First, do the operations in the parentheses. If not specified by parentheses, raise to powers and take roots before all other

operations. Multiply and divide after you take care of the powers and roots. Add and subtract last. When you deal with operations of the same level (addition and subtraction;

multiplying and dividing; raising in powers and taking roots), do them in order from left to right.

All these agreements exist only to make sure a reader reads exactly what a writer has written. After a bit of practice you’ll get used to them.

1. Do the calculations:a) 9 – 8 + 7 – 6 + 5 – 4 + 3 – 2 + 1 b) (9 – 8) + (7 – 6) + (5 – 4) + (3 – 2) + 1 c) 9 – (8 + 7) – (6 + 5) – (4 + 3) – (2 + 1)

2. In the expression 9 – 8 + 7 – 6 + 5 – 4 + 3 – 2 + 1, put parentheses in some places, as it was done in problem 6, to get expressions with as many different values as you can.

3. Do the calculations:a) 7 · 7 – 6 · 8 d) 7 · (7 – 6) · 8 g) 2 · 12 + 8 · 12 – 9 · 12 b) 3 + 4 · 6 – 5 · 5 e) 3 + 4 · (6 – 5) · 5 h) (3 + 4) · (6 – 5) · 5 c) 32 + 23 f) 3 · 42 i) (3 · 4) 2 j) 23 · 24

4. Insert the parentheses to make an expression true:a) 4 2 + 3 2 = 32 b) 4 2 + 3 2 = 22c) 44 11 – 7 55 = 605 d) 4 – 4 4 + 4 4 = 0e) 12 – 4 + 4 5 – 4 = 12 f) 12 – 4 + 4 5 – 4 = 4g) 14 55 – 55 4 = 0 h) 7598 98 –96 3799 = 1i) 1,001 140 – 127 150 – 139 25 – 18 = 1j) 169 – 156 314 – 303 1,000 – 993 = 1,001k) 55 5 – 5 5 – 5 = 11l) 382 292 – 101 + 8 = 10m) 3,333 111 – 100 3 + 91 = 1,000

5. Make as many different answers as possible:a) 1 + 2 3 + 4 5b) 6 7 + 8 + 9 10c) 3,434 17 – 15 3 + 2

The LAWS of Arithmetic. There are also certain RULES, or better to say, LAWS in Arithmetic. These laws will exist even if people wouldn’t want them to. We are going to list here not all the laws of arithmetic, but only those that will be useful for you in your calculations.


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1. CommutativityIf you need to add two numbers, the order you are adding them does not matter:

7 + 3 = 3 + 7 78 + 34 = 34 + 78

And, generally, for any numbers a and b,

a + b = b + a

If you need to multiply two numbers, the order you are multiplying them does matter:7 3 = 3 7

78 34 = 34 78And, generally, for any numbers a and b,

a b = b a

2. Associativity If you need to add three numbers, it does not matter which two numbers you add first. If you need to multiply three numbers, it does not matter which two you are multiplying first. (We are thinking about “more convenient” way of computations!)

(24 + 78) + 322 = 24 + (78 + 322)(63 25) 4 = 63 (25 4)

And, generally, for any numbers a , b, and c,

(a + b) + c = a + (b + c)(a b) c = a (b c)

3. Distributivity The two laws above are probably obvious. This one you need to think about.

Problem 1: Billy’s Mom bought 7 oranges at $2 per orange and 12 apples at $2 per apple. In computing the total bill, the clerk proceeded

7 oranges at $2 each = 7 2 = $1412 apples at $2 each = 12 2 = $24

Total = $38

Or you can do it more simply: (7 + 12) fruit at $2 each = $2 (7 + 12) = $2 19 = $38

Problem 2: Find the area of a rectangle. In this problem you use the geometrical meaning of multiplication, which is actually how do we

compute the area of a rectangle. The following picture will help you:


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6. Using the drawing explain the equalities:

a) (5 + 3) 4 = 5 4 + 3 4

b)(a + b) c = a c + b c

To multiply the sum by a number, you can multiply each of the addends by this number and then add the results up.

7. Using the drawing explain the equalities and solve them. a)

14 6 = (10 + 4) 6 = 10 6 + 4 6 =

b) c)

57 8 49 58. Insert the missing numbers:

a) b) c)

97 4 52 9 18 5

9. Write down as a mathematic expression and compute:d) The sum of 3 and 7 multiplied by the number 8.e) Add 18 to the quotient of 35 and 7.f) Divide 15 by the quotient of 12 and 4.g) Divide the product of 6 and 5 by the quotient of 12 and 4.

10. Using the digits 2, 5, 9 write down all possible three digits numbers. Don’t repeat the digits.11. Use the picture and distributive property of multiplication:

30 5

7 35 7 =


a + b

a b

c (a + b) c = a c + b c

35

4

ba

c

410

6

750

8

940

5

……

…

……

…

……

…

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12. Compute:a) 79 3 b) 32 8 c) 64 9 d) 86 7

Distributive property for division:

13. Ryan’s mother bought 20 lollipops and 6 chocolate candy bars. How is she going to divide them, between Ryan and his brother Alex? Explain the expression below and solve:

(20 + 6) 2 = 20 2 + 6 2 =

To divide the sum of two (or more) numbers, you can divide each addend by the divisor.

14. Solve using the different methods:48 4 = (40 + 8) 4 = 40 4 + 8 4 =48 4 = (28 + 20) 4 =48 4 = (24 + 24) 4 =

Represent the dividend as the sum of two numbers and compute:a) 39 3 d) 65 5 b) 52 4 e) 66 6c) 84 2 f) 91 7

15. Do the calculations:a) 28 · 80 + 22 · 80 f) 33 · 70 + 37 · 70 k) 132 · 120 + 168 · 120 b) 59 · 57 + 41 · 57 g) 22 · 41 – 20 · 41 l) 927 · 56 + 56 · 73 c) 34 · 28 + 66 · 28 h) 43 · 33 – 40 · 33 m) 555 · 8 – 545 · 8 d) 12 · 245 + 12 · 55 i) 335 · 11 – 135 · 11 n) 750 ÷ 90 – 300 ÷ 90 e) 80 ÷ 5 – 30 ÷ 5 j) 223 · 87 – 123 · 87 o) 345 · 17 – 145 · 17

16. Solve the problems.a) 400 · 2 d) 86 · 2 g) 5 · 200 j) 39 · 6 b) 80 4 e) 497 7 h) 300 30 k) 72 24 c) 48 2 f) 60 12 i) 246 6 l) 421 3

17. (CL) It is known that 25 · 38 = 950. Using the result find the answers to these problems.a) 38 · 25 d) 25 · 39 b) 950 38 e) 25 · 37 c) 950 25 f) 38 · 24

18. Explain the solution of the first example, then solve the next two problems.a) Example: 200 80 4284 · 3 = (200 + 80 + 4) · 3 =

200 · 3 + 80 · 3 + 4 · 3 = 3 600 + 240 + 12 = 852


… … ……

(a + b) c = a c + b c

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b) 156 · 5 100 9

c) 109 · 8 = (100 + 9) · 8

19. Solve.a) 429 · 2 b) 270 · 3 c)106 · 7 d) 327 · 3 e)158 · 4 f)140 · 5

20. Compute, using the properties of addition and multiplication (use the 2 5 = 10!):a) (94 + 179) + 21 f) 2 5 2 5 7 2 5b) 287 + (13 + 598) g) 4 19 25c) (356 + 849) + (51 + 644) h) 2 4 25 5 3d) 329 + 994 + 71 + 6 i) 20 9 500e) 2005 + 768 + 32 + 995 +19 j) 7 15 + 7 85

21. Insert parentheses to make the equality correct:a) 1 + 2 · 3 + 4 · 5 = 29 d) 1 + 2 · 3 + 4 · 5 = 55b) 1 + 2 · 3 + 4 · 5 = 27 e) 1 + 2 · 3 + 4 · 5 = 65c) 9 + 8 – 7 – 4 = 6 f) 9 + 8 – 7 – 4 = 14

22. Do the calculations:a) 28 · 70 + 22 · 70 f) 33 · 40 + 37 · 40 k) 132 · 120 + 168 · 120 b) 59 · 157 + 41 · 157 g) 22 · 41 – 20 · 41 l) 627 · 56 + 56 · 373 c) 34 · 33 + 66 · 33 h) 43 · 33 – 40 · 33 m) 855 · 18 – 845 · 18 d) 180 ÷ 5 – 130 ÷ 5 i) 323 · 87 – 223 · 87 n) 345 · 17 – 145 · 17 e) 12 · 345 + 12 · 55 j) 335 · 11 – 135 · 11 o) 650 ÷ 70 – 300 ÷ 70

23. Compute, using the properties of addition and multiplication:1) 51 + 52 + 53 + 54 + 55 + 56 + 57 + 58 + 592) 99 + 99 + 99 + 99 + 99 + 99 + 99 + 99 + 83) 999 + 999 + 999 + 999 + 999 + 74) 82 4 + 18 45) 36 97 + 36 36) 24 128 + 76 128

24. Compute:a). 438 90 – 238 90 b). 603 7 + 603 93

25. Compute using distributive property of multiplication:a). 4 505 b). 25 399 c). 16 403 d). 12 499

26. (CL) Using distributive property of multiplication, simplify the expressions and then compute the value of each:

a) 4a + 36a − 8a + 3a if a = 6 b) 52b − 7b − 6b + b if b = 25

c) 14m + m + 17m − 9m if m = 30d) 31n + 7n − 21 n + 12n if n = 20

27. Compute in most convenient way:a) (972 + 379) – 972 e) 134 – 94 – 2 i) 851 – (831 + 7)b) (382 + 417) – 416 f) 580 – 79 – 21 j) 24 96 – 24 86c) (538 + 245) – 245 g) 83 9 – 73 9 k) 276 – (8 + 176)d) (725 + 158) – 625 h) 7 38 – 7 28 l) 76 52 – 66 52


…

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28. (CL) Solve out loud:a) 450 9 e)

300 5 4 2 i) 160 4 8 – 34

b) 360 90 f) 720 8 + 15 6 j) 19 + 7 80 10c) 700 35 g) 630 9 14 5 k) 3 9 2 10 60d) 600 120 h) 800 ÷ 360 9 15 l) 700 4 100 3

29. * Find expressions identical to the expression 2b – 2a among the following:a) 2 (b – a) b) 2 (a + b) c) 2 (a – b) d) –2a – 2b e) –2a + 2b

30. Calculate:a) (9 20 + 60) 4 – 16 + 4 (20 5) b) 490 7 + (57 + 7) 8 2 − 3 (26 – 6)

31. There are 35 animals at the zoo. They are lions, crocodiles, monkeys and elephants. There are 6 lions; there are 2 less crocodiles than lions; there are 5 times more monkeys than crocodiles. How many elephants are there?

32. Compute(use the most rational way):a) 24 371 – 371 16 + 371 b) 353 26 – 14 353 + 12 147

33. Solve:a) 22,302 –( 9,302 + 7,383 + 4,617) b) 14,375 + 17,718 – (12,449 + 17,644)

34. The first number is 150,096. The second number is 119,388 less than the first one and 20,780 greater than the third number. Find the third number.

35. (CL) Simplify: a) x + x – (x + 1) b) x + 1 – (x – 1)

c) (x + 2) – (x + 1) d) (x + 2) – (x – 1)36. Compute (use the most rational way):

a) 373 26 – 14 373 + 12 251 – 218 12 b) 24 371 – 371 16 + 371 c) 276 35 – 276 + 276 66

37. Write down all two-digit numbers, in which the digit in the tenth place is 3 times more then the digit in the ones place.

38. If you use the laws of arithmetic you can make these problems much easier! a). 69 27 + 31 27 b). 202 87 – 102 87c). 977 49 + 49 23 d). 263 24 – 163 24

39. Compute using distributive property of multiplication:a). 91 8 b). 7 59 c). 6 52 d). 123 4e). 202 3 f). 390 5 g). 24 11 h).35 12

40. (CL) Solve out loud:a) 208 208 f) 15 6 k) 40,000 4b) 890 1 g) 14 5 l) 15,000 1000c) 0 60 h) 25 4 m) 1000 10d) 1 1 – 0 0 i) 25 5 n) 1000 100e) 1 0 – 0 1 j) 85 0 o) 1000 1000

41. Write the expression and solve: a). How many times more is 42 then 7? b). By how much is 5 smaller then 63?


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c). By how much is 56 bigger then 8? d). How many times is 9 smaller then 36?

42. Compute:a) 69 27 + 32 27 – 27 d) 263 24 – 164 24 + 24b) 202 87 – 100 87 – 2 87 e) 162 90 + 238 56 + 238 34c) 977 49 + 49 23 f) 603 7 + 603 93

43. Below, there are some problems for you. While doing these problems, be clever! If you use the laws of arithmetic in a certain way, you can make these problems much easier! I would recommend NOT using a calculator for these problems.

a) · 139 + · 139 e) 1.07 + .88 + 1.93 h) 125 · 17 · 4

b) 164 · 5 – 4 · 5 f) 2.5 · 2.7 · 4 i) 5 · 30 · 1 c) 7 · 21 · 9 ÷ 21 g) 120 · 123 ÷12 j) (132 + 7) · 35 – 35 · 1d) (100 + 12) · 112

44. Insert the parentheses to make the equation true:a) 5 38 – 70 8 – 6 = 60 c) 30 – 49 42 6 8 = 184b) 630 7 2 9 25 = 125 d) 180 300 – 30 9 + 199 = 205

45. How will the product change if:a) One of the factors increased 9 times.b) One of the factors decreases 7 times.c) One of the factors increases 2 times, and second one decreases 8 times.d) One of the factors increases 4 times and second decreases 5 times.e) One of the factors increase n times and second one increase 2 times.f) One of the factors decrease m times and second one decrease 3 times.

46. I thought of a two-digit number. This number is bigger than product of its digits by 52. What was the number?

47. (CL) Using distributive property of multiplication, simplify the expressions and then compute the value of each:

a) 31n + 5 n – n + 19 n if n = 20b) 15 + 3 y + y + 4 + 5 y if y = 7

Integers

Integers are whole numbers and their negatives. Integers may be positive, negative, or zero. We can add, subtract, multiply, and divide integers.


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The dots of the number line mark the integers from –5 through +6.

We use the symbol x to represent the absolute value of x.

Examples:−5 = 5 −12 = 12 −100 = 1005 = 5 12 = 12 100 = 1000 = 0 19 = 19

x = 5 x = 0 x= −3x = 5 or x = −5 x = 0 no solution

Rules for Addition of Integers1. Positive numbers: Add the numbers. The result is positive.2. Negative numbers: Add absolute values. Make the answer negative.3. A positive and a negative number. Subtract the smaller absolute value from the larger. Then:

a) If the positive number has the greater absolute value, make the answer positive.b) If the negative number has the greater absolute value, make the answer negative.c) If the numbers have the same absolute value, make the answer 0.

4. One number is zero. The sum is the other number.F.E.: (−3) + (+5) = 2 (-−3) + (+1) = −2

(−3) + (−5) = −8 (−3) + (0)= −3

SubtractionThe difference a – b is the number that when added to b gives a.

a – b = c a = b + cFor any numbers a and b

a – b = a + (− b)

F.E.: 7 – (−3) = 7 + 3 = 10(−3) – (−8) = −3 + 8 = 5−11 – (8) = −11 – 8 = −19


−5 −4 −3 −2 −1 0 1 2 3 4 5 6

To find absolute value:1) if a number is negative, make it positive2) if a number is positive or zero, leave it alone

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Multiplication and Division

F.E.: a) (−3) (−7) = 21 c) (−3) (7) = −21 e) 3 (−7) = −21 g) 3 7 = 21b) (−48) (8) = −6 d) (−48) (−8) = 6 f) 48 (−8) = −6 h) 48 8 = 6

48. Explain the solutions:

a) (−3) = (−3) (−3) = 9 e)

b) (−3) = (−3) (−3) (−3) = −27 f) doesn’t exist

c) (−1) = 1 (Why?) g) (−3) 0 = 0

d) (−1) = −1 (Why?) h)

49. In case of (− a − b):1. –10 – 5 7. –30 – 45 13. –2 –25t2. –20 – 7 8. –40 – 52 14. –21p – 383. –13 – 5 9. –100 – 88 15. –34q – 18q 4. –15 – 22 10. –10 – 0 16. –12w – 27w 5. –18 – 54 11. –3a – 2a 17. –33x – 22x 6. –20 – 23 12. –14b – 8b 18. –13s – 12s

50. In case of (−a + b):1. –10 + 5 7. –15 + 22 13. –44 + 44 19. –29q + 29q 2. –20 + 7 8. –18 + 15 14. –24b + 34b 20. 0 – 14 3. –7 + 20 9. –20 + 23 15. –54c + 57c 21. –16w – 0 4. –13 – 5 10. 30 – 45 16. –18 + 24 22. 23r –23 5. –5 + 13 11. 40x – 52x 17. –15d + 14d 6. 15 – 22 12. –100a + 88a 18. 14t – 16

51. Find the difference. a) –6 –12 b) –6 – (−12)

52. Simplify.a) x + (−y) – (−4) b) 6 – a – 4 – (-a) – 3 c) –y – (−7a) + – (−7a) + y – 8 – (−y)

Multiplying Integers and Raising Integers to Powers53. Find the product.

a) 8 (−3) b) –15 (+4) c) (−12) (−6) 54. Evaluate. To write an exponential expressions as products and then to evaluate them.

a) (−5) b) −5 c) (−2) d) (−2)55. Simplify. To simplify an algebraic expression containing repeated addition of negatives.


To multiply or divide two numbers:1) Multiply or divide two absolute values.2) If the signs are the same, the answer is positive.3) If the sings are different, the answer is negative.

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–t – t – x –x – x

56. Simplify. To simplify an algebraic expression containing multiplication and exponents.a) x (−1) (−x) (1) c) 3 (x) (y) (x) 0 b) (−2) n (8) n (−n) d) (−3) (−t) (2)

Dividing Integers57. Find the quotient.

a) b) c) d)

58. Write an algebraic expression for “seven less than twice some number.”

59. Calculate:a) 15 + (−12) e) 15 – (−3) i) (−3) m) –19 + 3 q) −8

b) f) –13 + (−7) j) –6 (−14) n) 32 – 49 r) (–3)

c) g) –8 – 7 k) (–8) o) s) –12 (+3)

d) – (−6) + 3 h) l) −(−7) −(− (−9)) p) 20 ∙ (−2)3 t) (−4)3

60. Evaluate the given algebraic expressions replacing x with –2 and y with 1.a) 3x b) (3x) c) x + y d) (x+ y)

61. Simplify.a) T + T + T f) − x + 17 + (−12) + x k) – a – (−b) p) x – 9 – (−x) + (−4)b) 6 – (−2 + 5y) g) –p – (−4) – p – 4 – p l) –(4 – 3x) q) − 5 (−x) (−y)

c) h) m) –5 + (x +4) r) –5 (−a) (−a) (−4) a

d) – 6x (2 − x) i) (6s − 3s ) (2s – 2) n) 12x – 16x s) 3x (x – y) + y ( x –y)e) (−2) (−x) j) 2h – 3h (2 – 3h) o) 13a – (−a) t) 6f(2 – f) – f(f – 6)

62. Write a formula that describes the value of xa) b)

63. Using the formula from the previous exercise, solve for y when x = 19 ft.The Number Line and Integers

64. Find the opposite of an integer.


3y 3y x

10y – 2

y

y

7 ft x

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65. –(18) b) –(−8)66. Find the absolute value of the integer.

a) −9 b) 1467. Place <, >, or = between a pair of integers…

a) –8 … −2 b) –2 … −8 c) −5 … −(−5)

Adding Integers68. Solve.

a) (−13) + (−24) b) 14 + (−18) c) –14 + 1869. Simplify the algebraic expressions containing addition.

a) y + 7 + y + (−2) b) –x + 6 + (−3) + x + (−4)70. A painter charges $12 an hour.

a) While his son charges $6 an hour. If the painter and his son worked the same amount of time together on a job, how many hours did each of them work if their charge for their labor was $108?

b) Painter- $10 son -$5 price-$300c) Painter - $20 son-$15 price- $350

71. An hour- long test has 60 problems. If a student completes 30 problems in 20 minutes, how many seconds does he have on average for completing each of the remaining problems?

72. The total fare for 2 adults and 5 children is $45. a) If each child’s fare is one half of each adult’s fare, what is the adult’s fare?b) Total fare for 5 adults and 5 children is $60c) Total fare for 3 adults and 6 children is $36

Comparing Numbers

73. Which of the two number is greater?1) –4 and –3 5) –41 and 1 9) –100 and 12) –18 and –15 6) 0 and –3 10) 8 and −83) –18 and 0 7) 0 and 3 11) 12 and −14) –18 and 18 8) –31 and –100 12) –181 and –1

74. Which of the following inequalities are true?1) –10 < −1 4) – 49 > −(−7) 7) –(−3) < −(−5)2) 17 < 0 5) 2 < −(−2) 8) −(−11) < 113) 4 17 > −(−10) 6) – (−3) > 5 9) −9<0

75. Arrange the numbers from least to greatest.a) –8, −10, 0, 5, 1, −1, 3, −6, −9, −19, 32b) –100, −98, 1, 2, 9, 11, 32, −5c) 0, −3, 9, 14, −9, −8, 1, 3

76. Arrange the numbers from greatest to least.a) –8, −10, 0, 5, 1, −1, 3, −6, −9, s19, 32b) –100, −98, 1, 2, 9, 11, 32, −5c) 0, −3, 9, 14, −9, −8, 1, 3

77. Write all the integers located on the number line between…a) –8 and 8 b) –7 and 6 c) –3 and 15 d) –5 and 10


Any negative number is less than zero and less than any positive number. The smaller of two negative numbers is the one whose absolute value is greater. The symbol means “less than or equal to” and the symbol “ greater than or equal to.”

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78. Write the five smallest integers greater than…a) –5 b) 7 c) –11 d) –32 e) 0 f) −7

79. Find all integer values for x that satisfy the inequality.a) 0 < x < 5 d) – 4 x −2 g) x < 5 j) x 1b) 0 x 3 e) – 4 < x < −2 h) x 0 k) x < 1c) –5 x 0 f) –1 < x < 1 i) x −3

Equations at a first glance.80. I thought of a number. I added 15, doubled the result, and got 42. What was my number? There are

2 ways to write this down:a) +15 ·2 42 my number

b) (n + 15) · 2 = 42

Solution:a) Using reverse operations

+15 ·2 42 -15 ÷2

b) Solving equation: (n + 15) · 2 = 42 ÷2 ÷2

n + 15 = 21Finish it yourself

81. I thought of a number, took away 10, divided the result by 2, and got 2. What was my number?82. I thought of a number, doubled it and added 7. I got 35. What was the number?83. “I thought of a number…” Nina thought of a number, and that’s what she wrote:

a) 2n + 10 = 361). Tell what Nina did with her number.2). What was her number?

b) 2(n + 5) = 36 c) 2n – 10 = 36 d) 2(n – 5) = 3684. (CL) Solve out loud:

a) 2x + 5 = 15 d) 2 x – 5 = 15 g) 4 x + 20 = 48 j) 4 x – 12 = 48b) 30 – 2 x = 10 e) 30 – 3 x = 21 h) 30 – 4 x = 18 k) 30 – 5 x = 20c) 50 – 2 x = 42 f) 50 – 3 x = 38 i) 50 – 4 x = 30 l) 50 – 5 x = 45

85. Solve for variable:a) 5z + 7 = 32 d) s 3 + 3 = 5 g) a 8 – 3 = 9 j) 4w – 6 = 14b) b 4 + 13 = 17 e) 9c + 12 = 21 h) 10h – 34 = 66 k) x 10 + 8 = 13c) m 6 – 6 = 0f) 16n – 28 = 4 i) 5k – 2 = 43 l) z 3 + 4 = 13

86. Solve:a) 3 · x = 1 c) 4 · x = 1 e) 10 · x = 2


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b) 10 + x = 12 d)10 – x = 8 f) 10 – x = 8

87. 1) I thought of a number, decreased it 7 times, and then added 25. The resulting number was 34. What was my number?

2) I thought of a number, increased it 9 times, and then multiplied by 6. The resulting number was 270. What was my number?

3)* I thought of a number, then I divided 80 by my number, and then added 13, then increased it 5 times. The result was 75. What was my number?

4)* I thought of a number, added 3, and then increased it 5 times. Then the result was subtracted from 70, the final number was 15. Find my number.

88. Solve the following equations:a) 3 + 7a + 7 = 20 + 2 a c) 5 a + 36 = 7 ab) 6 x + 44 + 8 x + 25 – 2 x = 144 d) 4y – 3 y + 14 y – 2 y = 169

89. (Cl) Solve out loud:a) 2 · (x – 5) = 20 d) 3 · (x – 5) = 21 g) 2 · (x + 5) = 20b) 3 · (x + 5) = 21 e) 5 · (10 + x) = 60 h) 5 · (10 – x) = 40 c) 4 · (10 + x) = 60 f) 4 · (10 – x) = 40 i) 3 · (x – 7) = 48

Introduction to Algebra and Expressions

In algebra we use certain letters or variables for numbers and work with algebraic expressions such as…

31 + x 14 t 26 – y and

Example: Evaluate the expression for the given values of variables:1) x – y for x = 83 and y = 49

Solution: x – y = (83) – (49) = 3434 is called the value of the expression

2) for a = 63 and b = –7

Solution:

3) 6 y for y = 15Solution: 6 (15) = 90

Practice: Evaluate the sum of 2x + y and the product xy if:a) x = 12, y = 25; b) x = 3, y = 24; c) x = 8, y = 3; d) x = 14, y = 16.

Translating to Algebraic ExpressionsIn algebra, to solve a word problem, we need to translate the words and sentences in English into Algebraic expressions. Actually translate problems to equations.

Examples: 1) In English: “Eight less than some number”. In Algebra: “t – 8”2) In English: “Twenty-two more than some number”. In Algebra: “y + 22”


An algebraic expression consists of variables numbers and operation signs. When we replace a variable by a number, we say that we are substituting for the variable. When we calculate the results, we get a number. This process is called evaluating the expression.

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3) In English: “Six more than eight times some number”. In Algebra:“8x + 6”

Sentences Algebraic Expressions

a) Nine more than some number m + 9 or 9 + mb) Nine less than some number a – 9c) Seven more than five times some number 5 t + 7 or 7 + 5 td) Three less than a product of two numbers p q – 3e) The difference between two numbers m and n m – n

Practice: Translate:a) The sum of a and b; d) The difference between a and m;b) The sum of x and the product of a and b; e) The difference between m and the quotient of x and y;c) The product of the sum of a and b and x; f) The product of a and the sum of x and y.

Solving EquationsAn equality is a number sentence that says that the expressions on either side of the equals sign “=”represent the same number.An equation is an equality, 7which contain a variable(s).

Consider the following expressions…a) 3 + 4 = 7 1) Which of the expressions are equalities?b) 5 – 1 = 2 2) Which equalities are true?c) 21 + 2 = 24 3) Which equalities are false?d) x – 5 = 12e) 9 – x = xf) 13 + 2 = 15

The replacements of variable by the number, which makes an equation true, are called its solution. To find the solution we need to perform steps strictly following the Main Algebraic Principles:

Addition Principle

In other words the addition to both sides of equation of the same number

Example: x + 6 = –15 – 6 – 6 x = – 21

Multiplication Principle

If an equation a = b is true, then a c = b c is true for any number c, but c ≠ 0!

Example:


If a = b is true, then a + c = b + c is true for any number c.

Name__________________________

3 x = 9

x = 3

Examples:a) 5x – 18 = 17 b) 8x + 6 – 2x = – 4x – 14 + 18 + 18 6x + 6 = – 4x – 14

5x = 35 – 6 – 6 5 5 6x = – 4x – 20 x = 7 + 4x + 4x

10x = – 20 10 10 x = – 2

Here is the summary for the method of solving equations:1) Collect like terms on each side of the equation, if possible.2) Use the addition principle to get all like terms with letters on one side and all other terms on the

other side.3) Collect like terms on each side again, if possible.4) Use the multiplication principle to solve for variable.

Examples: 3 (x – 5) + x = 2 (x – 1) – 1

3x – 15 + x = 2x – 2 – 1 2(5x – 11) = x + 5 4x – 15 = 2x – 3 10x – 22 = x + 5

– 2x – 2x – x – x 2x – 15 = –3 9x – 22 = 5 +15 +15 +22 +22 2x = 12 9x = 27

2 2 9 9 x = 6 x = 3

Exercises

90. Translate to an algebraic expressiona) 7 more than m d) 9 more than t g) 11 less than c j) 47 less than db) 26 greater than q e) 11 greater than z h) b more than a k) c more than dc) x less than y f) c less than b i) Twice x l) Four times p

91. Use algebraic language:a) The sum of a and b The sum of m and nb) The difference between 17 and b The difference between p and qc) 8 more than some number One more than some numberd) 54 less than some number 47 less than some numbere) A number x plus three times y A number a minus 2 times b

92. Solve:a) 3r + 15 = 75 d) b 3 + 17 = 48 g) 8t – 14 = 82 j) 5x + 6 = 41b) v 2 – 4 = 26 e) p 2 – 5 = 9 h) 0.5j + 2 = 6 k) 4y + 15 = 67c) 4v – 22 = 22 f) 4y – 8 = 20 i) c 5 + 7 = 12 l) d 4 + 3 = 33

93. Express in algebraic symbols:


Name__________________________

a) The sum of x and x + 3 b) The product of a and bc) What number exceeds a by b?d) A man traveling r miles per hour for three hours travels how far?e) What number is x more than 2x + 1f) If x + 1 is an integer, what is the next larger consecutive integer?g) What is John’s age 4 years ago, if he will be y years old in 5 years?h) If 2n is an even integer, what is the next larger even integer?i) What number is 3 less than twice x?j) If b is the larger of two numbers a and b, what is their difference?k) The sum of two numbers is s and the larger number is l; what is the smaller number?l) What number is three more than twice x?m) What number is five less than doubled y?n) If x pen cost $20, what is the price of one pen?o) The difference between two numbers is d and the larger number is l. What is the smaller number?p) What is the sum of two numbers a and b decreased by their product?q) What is the average of a and b?r) What is the number of days in w weeks and d days?

94. Solve the problems and check them:a) x · 9 = 720 c) 300 – x = 60 e) x + 9 = 720b) x 90 = 9 d) 300 x = 60 f) x – 90 = 9

95. Compute(use the most rational way):a) 24 371 – 371 16 + 371 b) 353 26 – 14 353 + 12 147

96. A truck has total of a pounds of fruit in each of n boxes. How many pounds of fruit are in the truck?97. Marina bought 4 notebooks, which cost b dollars each. And 3 pens, which cost c dollars each. How much

money did Marina spend?98. Natasha had d dollars. She bought 2 ice creams which cost x dollars each. How much money does she

have left?99. If you use the laws of arithmetic you can make these problems much easier!

a). 69 27 + 31 27 b). 202 87 – 102 87c). 977 49 + 49 23 d). 263 24 – 163 24

100. I bought 8 chairs x dollars each, and a table for y dollars. How much more did I pay for the chairs, than for the table?

101. The first class has a kids in it, the second has b kids in it, and the third class has c kids in it. Kids from all three classes were equally divided between two buses. How many kids are there in each bus?

102. Solve the following equations: a) 40x ÷ 10 = 28 c) 49 ÷ k – 3 = 46 b) y ÷ 10 – 28 = 32 d) (25 – a) ÷ 7 = 3

103. A movie theater has a rows and b seats in each row? How many seats are there in the theater?104. A teacher handed out m notebooks in packs of c notebooks in each pack. How many packs were

there?105. Compute:

a) 69 27 + 32 27 – 27 d) 263 24 – 164 24 + 24b) 202 87 – 100 87 – 2 87 e) 162 90 + 238 56 + 238 34a. c) 977 49 + 49 23 f) 603 7 + 603 93

106. A truck has a total of a pounds of fruit in n boxes. How many pounds of fruit are in each box?


Name__________________________

107. Solve for variable:a) 5z + 7 = 32 d) s 3 + 3 = 5 g) a 8 – 3 = 9 j) 4w – 6 = 14b) b 4 + 13 = 17 e) 9c + 12 = 21 h) 10h – 34 = 66 k) x 10 + 8 = 13c) m 6 – 6 = 0 f) 16n – 28 = 4 i) 5k – 2 = 43 l) z 3 + 4 = 13

108. Solve using the most convenient way!a) 164 · 5 – 4 · 5 d) 25 · 27 · 4 g) 5 · 30 · 1 b) 7 · 21 · 9 ÷ 21 e) 120 · 123 ÷12 h) (132 + 7) · 35 – 35 · 1c) (100 + 12) · 112 f) 107 + 88 + 193 i) 125 · 17 · 4

109. Solve the following equations:a) 4x + 2x + 5 = 3x + 35 b) x + 3 = 2x – 3 c) 7x + 9 = 86 d) 84 x = 6 7e) 3x = 87 – 6 f) 6x = 42 g) 3x + 4 = 37 h) (x – 3) 2 = 30

110. What is the length of a rectangle whose perimeter is P, if the width is 6?111. What number is 7 more than twice as much as x?112. A stick l feet long is broken into two parts, one of which is twice as long as the other. How long is

the shorter piece?113. The larger of two numbers is twice the smaller. If the larger is x, what is the smaller?114. Solve out loud:

a) x – 15 = 22 c) x – 65 = 104 e) x – 87 = 99b) x – 21 = 33 d) x – 15 = 62 f) x – 108 = 1,109

115. Out loud: In algebraic symbols, how many cents are in n nickels and c cents?116. The larger of two numbers is four more than the smaller. If the larger is x, what is the smaller?117. What number exceeds y by 4 less than x?118. Jane is 3 years older than Mary, and Mary is twice as old as Kay. If Kay is x years old, how old is

Jane?119. Out loud: How the product will change if:

a) One of the factors increases 2 times, and second one decreases 8 times.b) One of the factors increases 4 times and second decreases 5 times.c) One of the factors increase n times and second one increase 2 times.d) One of the factors decrease m times and second one decrease 3 times.

120. I thought of a two-digit number. This number is bigger than product of its digits by 52. What was the number?

121. Out loud: Compute. Use the most convenient way.a) ( 382 + 417 ) – 416 c) (725 + 158 ) – 625b) 24 96 – 24 86 d) 716 52 – 61652

122. Solve out loud the following equations:a) 2x – 23 = 57 d) 2x – 75 = 109 g) 2x – 93 = 41b) 2x – 15 = 17 e) 2x – 33 = 105 h) 2x – 105 = 33c) 2x – 234 = 376 f) 2x – 27 = 127 i) 2x –1,897= 1,499

123. Sue sold 26 more papers than Dick, and Dick sold three times as many as Harry. If Harry sold p papers, how many did Sue sell?

124. A has $32 more than B, and B has five times as much money as C. If C has d dollars, how much does A have?

125. Joyce has as much money as George; then they bet 5 cents and George loses. If, after the bet, George has x cents, how much does Joyce have?


Name__________________________

Number ProblemsExample:

There are two numbers whose sum is 68. Three times the smaller number is equal to 12 more than the larger number. What are the numbers?Step 1 (Build up)

Let x be the smaller number3x −12 the larger numberx + (3x −12) the sum of two numbers

Step 2 (Make an equation)x + (3x −12) = 68

Step 3 (Solving the equation)x + (3x −12) = 68 4x − 12 = 68 +12 +124 x = 80÷ 4 ÷ 4

x = 20Step 4 (Solving the problem)

Since x =20, then the smaller number = 203x −12 , then the larger number = 3 ∙ 20 − 12 = 48 ( Check: 20 + 48 = 68)

The answer is: Numbers 20 and 48.

126. There is a number such that three times the number minus 6 is equal to 45. Find the number.127. The sum of two numbers is 41. The larger number is one less than twice the smaller number. Find

the numbers.128. Separate 90 into two parts so that one part is four times the other part.129. The sum of three consecutive integers is 54. Find the integers.130. There are two numbers whose sum is 53. Three times the smaller number is equal to 19 more than

the larger number. What are the numbers?131. There are three consecutive odd integers. Three times the largest is seven times the smallest. What

are the integers?132. The sum of four consecutive even integers is 44. What are the numbers?133. There are three consecutive integers. The sum of the first two is 35 more than the third. Find the

integers.134. A 25- foot long board is to be cut into two parts. The longer part is one foot more than twice the

shorter part. How long is each part?135. Mrs. Mahoney went shopping for the some canned goods, which were on sale. She bought three

times as many cans of tomatoes as cans of peaches. The number of cans of tuna was twice the number of cans of peaches. If Mrs. Mahoney purchased a total of 24 cans, how many of each did she buy?

136. The first side of a triangle is 2 inches shorter than the second side. The third side is 5 inches longer than the second. If the perimeter of the triangle is 33 inches, how long is each side?


Name__________________________

137. Kerrie and Shelly rode their bicycles four more than three times as many miles in the afternoon as in the morning on a trip to the lake. If the entire trip was 112 miles, how far did they ride in the morning and how far in the afternoon?

138. In a 3-digit number, the hundreds digit is four more than the units digit and the tens digit is twice the hundreds digit. If the sum of the digits is 12, find the three digits. Find the number.

139. If the third of 8 consecutive numbers is 12, what is their sum?140. Find the sum of 16 smallest consecutive whole numbers.141. What is the first number of the 4 consecutive number series if the sum of numbers is: 406

142. Solve these equations:a) 16(x + 2) = 2(32 + 5 x) b) 2(x – 1) = 10 c) 12(2 x – 1) = 36

143. Solve:

a) + = + b) 5 (2 x – 1) = 8 x + 1 c) 5x – 3= 2 x – 3

144. A man could arrive on time for an appointment if he drove the car at 40 mph; however, since he left the house 15 minutes late, he drove the car at 50 mph and arrive 3 minutes early for the appointment. How far from his house was his appointment?

145. Jim can do all the work in an hour. Ben can do it in 40 minutes. How long will it take them to do all the work if they do it together?

146. Fill in the gaps, such that the answer is the whole number:a) 3 x + 12 = 4 b) 3 x – 12 = 4 c) 3 x + 20 = 4d) 3 x + 20 = 4 e) 3 x – 20 = 4 f) 5 x + 12 = 4g) 5 x + 28 = 5 h) 5 x – 12 = 4 i) 5 x + 104 = 19

147. Prove that:1) Sum of two even numbers is an even umber.2) Sum of any two consecutive numbers in an odd number.3) Difference of an even and an odd numbers is an odd number.4) Product of any two consecutive numbers is an even number.

148. Simplify by combining like terms:a) 13a + 2b – 2a – b c) – 51a – 4b – 49a + bb) 41x – 58x + 6y – y d) 75x + y – 85x – 35y

149. Simplify by combining like terms:a) 8x – 6y + 7x – 2y c) 35b – 24c – 6c – 7bb) 27p + 14q – 16p – 3q d) 16a + 4x – 28a – 75x

150. True or False?1) If the difference of two numbers is even, then their sum is also even.2) If the difference of two numbers is odd, then their sum is even.3) If the sum of two numbers is even, then both of them are even too.4) If the sum of two numbers is even then at least one of them is even.5) If the sum of two numbers is even then at least one of them is odd.

Common Fractions

151. Reduce the following fractions to

- fifteenths: . , , ,


Name__________________________

- thirty-seconds: , , , ,

- sixteenths: , , , ,

152. Write the following mixed numbers as an improper fraction.

a) 3 b) 8 c) 7 d) 12

e) 15 f) 11 g) 2 h) 20

153. Write the following improper fractions as a mixed number or a whole number.

a) b) c) d)

e) f) g) h)

154. Write the reciprocal for each of the following numbers.

a. b. 1 c. 5 d. e.

155. Determine which one of the two fractions with different denominators is greater?

a. b. c. d.

156. Convert each pair into equivalent fraction with LCD and compare:

a. b. c. d. e. f.

157. Which one of the following fractions is the greatest?

a. b.

158. Convert each of the following to the equivalent fraction with denominator 48:

a. b. c. d. e.

159. Compare the fractions:

a) and 10 f) and k) and p) and

b) and g) and l) and q) and c)

and h) and m) and r) and

d) 1 and i) and n) 2 and s) and

e) 1 and j) and o) 1 and t) and

160. Instead of … write a digit to be able to reduce the fraction. (Find all possible …s)

a) c) e) g)

b) d) f) h)


Name__________________________

Coin ProblemsNotice: in coin problems we are counting the number of coins, and amount of money in the set of coins. For example: we can have three nickels, then it is 15 cents, three quarters is 75 cents, etc.

Example: A clerk at the Dior Department Store receives $15 in change for her cash drawer at the start of each day. She receives twice as many dimes as fifty-cents pieces, and the same number of quarters as dimes. She has twice as many nickels as dimes and a dollar’s worth of pennies. How many of each kind of coin does she receive?Step 1: (Fill in the chart)

Let x be the amount of half-dollarsNomination of

coinsNumber of coins Amount of money The answer

Dimes 2 x 2 x ∙ 10 = 20 xHalf-dollar x x ∙ 50 = 50 x

Quarters 2 x 2 x ∙ 25 = 50 xNickels 2 ∙ 2 x 4 x ∙ 5 = 20 xPennies 100 100TOTAL 20v +50 x +50 x +20 x +100

Step 2: (Make an equation) 20 x +50 x +50 x +20 x +100 = 1,500 (Important: in the chart we

used cents, so $15 = 1,500 cents)

Step 3: (Solve the equation) 20 x + 50 x + 50 x + 20 x + 100 = 1,500 140 x +100 = 1,500

−100 −100140 x = 1,400÷140 ÷140 x = 10

Step 4: (Solve the problem) (It is OK to fill in the last column of the chart, just be sure that you calculate the number of coins, not amount of money)

161. A collection of coins has value of 64 cents. There two more nickels than dimes and three times as many pennies as dimes. How many of each kind of coin are there?

162. Tanya has ten bills in her wallet. She has a total of $40. If she has one more $5 bill than $10 bills, and two more $1 bills than $5 bills, how many of each does she have?

163. Mario bought $5 worth of stamps at the post office. He bought ten more 6 cents stamps than 10 cents stamps. The number of 8 cents stamps was three times the number of 10 cents stamps. He also bought two 20 cents stamps. How many of each kind of stamp did he purchase?

164. Mr. Abernathy purchases a selection of wrenches for his shop. His bill was $78. He buys the same number of $1.50 and $2.50 wrenches, and half that many $4 wrenches. The number of $3 wrenches is one more than the number of $4 wrenches. How many of each did he purchase?

165. A collection of 36 coins consists of nickels, dimes and quarters. There are three fewer quarters than nickels and six more dimes than quarters. How many of each kind of coin is there?


Name__________________________

166. The cash drawer of the market contains $227 bills. There are six more $5 bills than $10 bills. The number of $1 bills is two more than 24 times the number of $10 bills. How many bills of each kind are there?

167. Jennifer went to the post office for stamps. She bought the same number of 8 cents stamps and 10 cents stamps. She also bought as many 2 cents stamps as both of the other two combined. How many of each kind did she get if she paid a total of $4.40 for them all?

168. Terry bought some gum and candy. The number of packages of chewing gum was one more than the number of mints. The number of mints was three times the number of candy bars. If gum was 6 cents a package, mints were 3 cents each, and candy bars were 10 cents each, how many of each did he get for 80 cents?

169. Clem Colfax had $10 to buy groceries. He needed milk at 70 cents a carton, bread at 60 cents a loaf, breakfast cereal at 50 cents a box, and meat at 1.50 a pound. He bought twice as many cartons of milk as loafs of bread, the number of packages of cereal was one more than the number of loafs of bread, and the number of pounds of meat was the same as the number of packages of cereal. How many of each did he purchase if the total was exactly $10?

Operations with Common Fractions

Addition of Fractions and Mixed Numberso Fractions that have same denominator may be added

directly by simply adding the numerators and placing this sum of the numerators over the common denominator

o We may not add two fractions with unlike denominators by the same method since we would then be adding two unlike quantities. For example, we may not add fractions

and directly. Before this addition is performed, we

must 1) Change each of these fractions to an equivalent

fraction with the same denominator (least common denominator LCD)

2) Add the numerators and place this sum of the numerators over the common denominator.

o If you add mixed numbers, you 1) Change the fraction parts of mixed numbers to

an equivalent fractions with LCD2) Add the whole parts.3) Add the fraction parts.


Example:

4341

41

42

21

Sum

36116

36475

36323

983

36152

1252

Sum

Example:

Example:

Name__________________________

o In subtracting fractions we again make use of the general principle that only like quantities may be combined. Sometimes in subtracting from mixed number we need to borrow from the whole part 1 and express it in fraction form.

o In multiplying two or more fractions, we multiply the numerators of the fractions to obtain the new numerator, and multiply the denominators of two fractions to obtain the new denominator

o In multiplying a fraction by the whole number we may regard the whole number as a fraction whose denominator is 1. Then proceed as we did in multiplying two fractions.

o In multiplying mixed numbers we change the mixed numbers to improper fractions and proceed as we did in multiplying two fractions.

o In dividing one fraction by another we invert the divisor (reciprocal) and multiply.

170. Add or subtract (out –loud), explain each step.

a) + d) + g) + j) + m) + p) +

b) + e) + h) – k) – n) – q) –

c) – f) – i) – l) – o) – r) +

171.Subtract (out –loud), explaining each step.

a) 1 – c) 1 – e) 1 – g) 1 –

b) 1 – d) 1 – f) 1 – h) 1 –

172. Subtract (out-loud), explaining each step.

a) 3 – c) 5 – e) 2 – g) 6 –

b) 6 – d) 7 – f) 12 – h) 13 –


434

865

83

83

894

8114

815

Difference

Example:

Name__________________________

173.Subtract.

a) c) e) g)

b) d) f) h)

174. Add or subtract.

a) + + d) – + g) + –

b) + + e) – – h) – –

c) + – f) – – i) + +

175. Add.

a) c) e) g)

b) d) f) h)

176. Add.

a) d) g) g)

b) e) h) k)

c) f) i) l)

177. Add:

a) b)

178. Find the total of

179. What number must replace a question mark to get the correct equation?

a) b) c) d) .

180. Add

a) b) c) d)

181. Find each sum.

a) 5 + b) 0 + c) + 2 d) 2 + +

182. Add the following mixed numbers.

a) b) c) d)

183. Find the difference and check your answer by addition.


Name__________________________

a) b) c) d)

e) f) g) h)

184. An empty jar weighs kg. If the same jar weight kg when filled with honey, how many

kilograms of honey does the jar hold?185. A road between three town A, B, C is being paved with asphalt.

The distance between A and B is km while the distance between

B and C is less than that between A and B. How many

kilometers of road need to be paved?

186. A cross-country skier covered of her race in one hour, of the race the next hour, and the

remainder of her race in the third hour. What fraction of her race did the skier cover during her third hour?

187. A cow gave liters of milk in the morning, liters less than this at noon milking, and liters

in the evening. How many liters of milk did the cow give for the entire day?

188. Two bags hold kg of flour while one of the bags holds kg. Which bag holds more flour

and by how much?

189. One side of a triangle is cm, another side is cm shorter than the first, and the third side is

cm longer than the first. What is the perimeter of the triangle?

190. Evaluate the expression a + for: a) a = ; b) ; c) ; d)

191. Evaluate the expression x – for: a) x = 3; b) ; c) ; d)

192. Solve each equation.

a) d) g)

b) e) h)

c) f) i)

193. Find the value of x. Check your answer.

a) + x = b) x + = c) – x = d) + x =

e) + x = f) – x = g) + = x h) – x =


A

B

C

Name__________________________

194. In the figure to the right, AE + DC = cm, AB =

cm, DE = cm, and BC = cm. Find the

perimeter of the figure.

195. In the figure to the right, AB = m, BC = m,

and DE = m. Find the perimeter of the figure.

196. Evaluate the expression a – + b for:

a) and b) and

197. Multiply. Use cancellation if it is possible.

a) 20 b) 16 c) 4 d) 42 e) 48 f) 15

198. Calculate.

a) of 12 b) of 12 c) of 12 d) of 20 e) of 24 f) of 49

199. One faucet fills up a bathtub in 15 minutes while the second fills it up in 10 minutes. If both faucets are turned on, what fraction of the bathtub will be filled in:

a) 1 minute; b) 2 minutes; c) 5 minutes

200. One faucet fills a barrel in 6 minutes while another fills it in 12 minutes. If both faucets are turned on for one minute, what fraction of the barrel will be left to fill?

201. Find each product. Cancel common factors if it is possible.

a) b) c) d) e) f)

202. Calculate.

a) b) of of c) d) of of

203. Divide each fraction or mixed number by a whole number.

a) 5 b) 5 c) 15 d) 7

e) 5 f) 41 g) 4 h) 9

204. Divide each whole number by a fraction and then simplify.

a) 81 b) 16 c) 23 d) 36 e) 54 f) 29

205. Evaluate the expression for: a) x= ; b) ; c) ; d) ; e) .


C

A B

E D

C

A B

E

D

F

GH

Name__________________________

206. Evaluate the expression for: a) a = 0; b) a = 1; c) ; d) .

207. Solve each equation, checking your answer with multiplication.

a) d) g) j) b)

e) h) k)

c) f) i) l)

208. Evaluate the expression for a = and b= .

209. Simplify.

a) d) g)

b) e) h)

c) f) i)

210. Solve each equation:

a) c) e)

b) d) f)

211. Simplify, using cancellation.

a) c)

b) c) d)

212. Perform division with mixed numbers. Write each quotient in simplest form.

a) 72 7 b) 7 1 c) 1 7 d) 6 5

e) 8 2 f) 1 g) 18 3 h) 21 3

213. How far can a group of hikers walk in 1 hour if they go 2.25 km in:

a) h; b) h; c)

h

214. How much does 1 m of cloth cost if m costs $5.25?

215. A farmer harvested 1476 bushels of wheat from a 30 acre field. What was the farmer’s average

yield per acre?


Name__________________________

216. The volume of one room is 60 meters cubed while another room is 1 times smaller. Find the

volume of the other room.217. Matthew constructed two angles. It turned out that one angle measured 180° while the other was 2

times smaller. Construct these angles yourself.

218. Two cities are 210 km apart. A train covered this distance in 4 and hours and then made the

return trip at a speed of 50 and km/h. How many times greater (or less) was the train’s speed on

the return trip?

219. A sheet of plywood with an area of 2 and meters squared can be cut into how many pieces if the

area of each piece must be 0.3 meters squared?220. A machine puts meat seasoning into packages of 0.03 kg each. How many packages are needed for

22 kg of seasoning?

221. You pay $2.70 for n kilograms of candy at a store. Write an equation for finding the cost of one kilogram of candy. Evaluate this equation for:

a) n = b) n = 1 c) n = 1.2

222. Review the rule for finding a number when a fraction of it is known. Finally, explain how to find

the number of which equal 12.

223. Find the number (out-loud):

a) of which equals 2 of which equals 15 0.6 of which equals 12

b) of which equals 16 of which equals 1.8 0.7 of which equals 1.4

224. Find the number:

a) of which equals 7 0.25 of which equals 12 of which equals 2

b) 0.45 of which equals 3.6; 0.04 of which equals 0.5; 1 and of which equals 0.8

225. Jocelyn has read 150 pages in a book. This is of the entire book. How many pages does the book

have?

226. How many students are there in a class if 12 boys make up of the class?

227. A rectangular field is 240 m wide. The width of the field is the length. What is the area of the

field in square kilometers rounded to the nearest thousandth?

228. After a drought, the Great Salt Lake in Utah covered an area of 2450 kilometers squared or the

area of the lake under normal weather conditions. Find the area of Great Salt Lake under normal weather conditions.


Name__________________________

229. One car went 60 km in of an hour while the second car went 54km in h. Which car was faster?

How many times faster?

230. A farmer has of his land under cultivation, of which is in soybeans. How much land does the

farmer have if he is growing 270 acres of soybeans?

231. Brian had 2 dollars left over after he spent of his money on a book. How much money did Brian

have originally?

Age ProblemsExample. Mrs. Smythe is twice as old as her daughter Samantha. Ten years ago the sum of their ages was 46 years. How old is Mrs. Smythe?Step 1: (Fill in the chart) Let x be Samantha’s age

Step 2: (Make an equation)(2x − 10) + (x − 10) = 46

Step 3: (Solving the equation)(2x − 10) + (x − 10) = 46

3x − 20 = 46 +20 +20 3x = 66÷3 ÷3 x = 22

Step 4: (Solve the problem) (It is OK to fill in the last column of the chart, just be sure that you calculate the age now, not ten years ago)

232. A man is four times as old as his son. In 3 years, the father will be three times as old as the son. How old is each now?


now 10 years ago AnswersMrs. Smythe 2x 2x − 10

Samantha x x − 10TOTAL (2x − 10) + (x − 10)

Name__________________________

233. Abigail is 8 years older than Cynthia. Twenty years ago Abigail was three times as old as Cynthia. How old is each now?

234. Seymour is twice as old as Cassandra. If 16 is added to Cassandra’s age and 16 is subtracted from Seymour’s age, their ages will be equal. What are their present ages?

235. In 4 years Cranston’s age will be the same as Terrill’s age now. In 2 years, Terrill will be twice as old as Cranston. Find their ages now.

236. A Roman statue is three times as old as Florentine statue. One hundred years from now the Roman statue will be twice as old. How old is the Roman statue?

237. Sheri’s age in 20 years will be the same as Terry’s age is now. Ten years from now, Terry’s age will be twice Sheri’s. What are their present ages?

238. Bettina’s age is three times Melvina’s. If 20 is added to Melvina’s age and 20 is subtracted from Bettina’s age, their ages will be equal. How old is each now?

Decimals The decimal or the Decimal Fraction is a special way to write a common fraction whose

denominator is 10, 100, 1000,…, etc. In other words, Decimal is another notation of the common fraction with a power of ten in its denominator.

According to the rule, the common fractions

, n= 1, 2, 3, … can be written in “decimal” form as

0.1, 0.01, 0.001,… 0.00…1 “n –1” zeros

Addition of Decimalso In writing the decimals to be added, arrange them in a column

so that the decimal points are placed directly under each other. Then add the decimals in the same manner as whole numbers. The decimal point in the answer is in line with the decimal points of the numbers that were added.

Subtraction of Decimalso In subtracting decimals, we arrange them in a column,

keeping the decimal points in a straight line. Then we subtract as with whole numbers.

Multiplication of Decimalso In multiplying decimals, follow the procedure used in multiplying whole numbers. It is

not necessary to arrange the numbers so that the decimal points are in a line. The number of decimal places (the places to the right of decimal point) in the result is equal to the sum of the decimal places in the two numbers that were multiplied.


7.091+ 56.748 63.839

Example:

3.82· 5.7

2674+ 1910_

21921.774

← 2 decimal places← 1 decimal place

← 3 decimal places in the answer

Example:

720.72– 69.58

651.14

Example:

Name__________________________

Division of Decimalso In dividing a decimal by the whole number, divide as in

division of whole numbers, placing decimal point of the result in the line with the decimal point in the divident.

o In dividing by a decimal we make the divisor a whole number by moving the decimal point to the right as many places as necessary. In order to avoid changing the result of the division we also move the decimal point in the divident the same number of places to the right.

Multiplying by 10, 100, 1,000, etc.o In multiplying a number by 10 we move decimal point in the number one place to the

right. In some cases it may be necessary to annex a zero to complete the operation. Similarly, in multiplying a number by 100 we move decimal point in the number two places to the right. In some cases it may be necessary to annex two zeros to complete the operation. Similar procedures are followed in multiplying a number by 1,000, 10,000, etc.

Dividing by 10, 100, 1,000, etc.o In dividing a number by 10 we move decimal point in the number one place to the left.

In some cases, it may be necessary to insert a zero between the decimal point and the first digit on the left, to complete the operation. Similarly, in dividing a number by 100 we move decimal point in the number two places to the left. In some cases, it may be necessary to insert two zeros between the decimal point and the first digit on the left, to complete the operation. Similar procedures are followed in dividing a number by 1,000, 10,000, etc.

239. Compute:a) 12.75 ÷ 102 c) 0.378 · 104 e) 0.00075 · 103 g) 413.012 ÷ 103

b) 43.063 · 104 d) 7.65 ÷ 105 f) 7.120 ÷ 102 h) 0.61 · 103

240. Add the following decimals (show your job. Don’s use a calculator).a) 2.346 + 0.595 + 72.07 c) 78.086 + 1206.15 + 0.804b) 1.9 + 39.1 + 234.3 + 0.8 d) 100.1 + 0.052 + 1.05

241. The sum of $0.69, $ 0.47 and $0.05 is(A) $1.01 (B) $1.10 (C) $1.21 (D) $1.11 (E) $1.20

b) The sum of $38.09, $ 0.57 and $7.44 is(A) $56.10 (B) $46.00 (C) $51.23 (D) $36.10 (E) $46.10

242. Find difference and check your answers using addition.a) 7.21 – 3.964 d) 3541 – 832.8 g) 2960.506 – 3.49b) 400 – 16.82 e) 20 – 0.724 h) 3.2005 – 2.399c) 0.67 – 0.0999 f) 2960.506 – 3.49 i) 0.907 – 0.0864

243. Multiply the following pairs of decimals (show your job)a) 0.32 · 0.4 c) 2.3 · 1.52 e) 3.8 · 0.002 g) 16.5 · 0.5


Example:

Example:

Name__________________________

b) 18.5 · 0.02 d) 32.7 · 0.03 f) 1.76 · 0.172 h) 1.36 · 0.04244. Compute:

a) 6.965 + 23.3; e) 6.5 · 1.22; i) 53.4 ÷ 15;b) 76.73 + 3.27; f) 0.48 · 2.5; j) 16.94 ÷ 2.8;c) 50.4 – 6.98; g) 3.725 · 3.2; k) 75 ÷ 1.25;d) 88 – 9.804; h) 0.016 · 0.25; l) 123.12 ÷ 30.4.

245. Compute:a) 489.92 ÷ 12 – 20.16; c) 1.08 · 30.5 – 9.72 ÷ 2.4;b) 6.05 · (53.8 + 50.2); d) 44.69 + 0.5 · 25.5 ÷ 3.75.

246. Compute:a) 155.5 – 5.5 · 20.7; c) 3.6 ÷ 0.08 + 5.2 · 2.5;b) 85.68 ÷ (4.138 + 2.162); d) (9.885 – 0.365) ÷ 1.7 + 4.4.

247. Compute in the fastest way possible:a) 50 · 1.34 · 0.2; c) 25 · (–15.8) · 4;b) – 75.7 · 0.5 · 20; d) 0.47 · 0.4 · 25.

248. Find the value of the expression:a) 3.5 · 6.8 + 3.5 · 3.2; b) 12.4 · 14.3 – 12.4 · 4.3.c) 15.7 · 3.9 + 15.7 · 2.91; d) 4.03 · 27.9 – 17.9 · 4.03

249. Convert the decimals to fractions and divide.

a) ÷ 0.1 d) 0.9 ÷ g) 1.2 ÷ j) ÷ 0.4

b) 0.24 ÷ e) ÷ 0.75 h) ÷ 0.5 k) ÷ 1.25

c) 0.36 ÷ f) 3.5 ÷ i) 4.25 ÷ l) 0.105 ÷

250. Open the parenthesis and simplify like terms:a) 3 (6 – 5x) + 17x – 10; d) 2 (7.3 – 1.6a) + 3.2a – 9.6;b) 8 (3y + 4) – 29y + 14; e) – 5 (0.3b + 1.7) + 12.5 – 8.5b;c) 7 (2z – 3) + 6z – 12; f) – 4 (3.3 – 8c) + 4.8c + 5.2.

251. Simplify the expression:a) 3 (2m + 1) + 4m – 7; d) 0.2 (3a – 1) + 0.3 – 0.6a;b) – 6 (3n + 1) + 12n + 9; e) 0.9 (2b – 1) – 0.5b + 1;c) 5 (0.6 – 1.5p) + 8 – 3.5p; f) – 2.6 (5 – c) – c + 8.

252. Divide.

a) d) g)

b) e) h)

c) f) i)

253. Simplify the expression and find its value:a) (5x – 1) – (2 – 8x) if x = 0.75; c) 12 + 7x – (1 – 3x) if x = –1.7;b) (6 – 2x) + (15 – 3x) if x = – 0.2; d) 37 – (x – 16) + (11x – 53) if x = – 0.03.

254. Simplify the expression:a) (x – 1) + (12 – 7.5x); d) b – (4 – 2b) + (3b – 1);b) (2p + 1.9) – (7 – p); e) y – (y + 4) + (y – 4);


Name__________________________

c) (3 – 0.4a) – (10 – 0.8a); f) 4x – (1 – 2x) + (2x – 7).255. *Solve for x:

1) 55 – 8x = 7; 5) ; 2) ; 6) ;

3) ; 7) ; 4) ; 8) .

Digit ProblemsIn these problems the expanded form of a number is very helpful: 345 = 3∙100 + 4∙10 + 5. The reversed number is the number in which digits go backwards: for 35, the reversed number is 53. “Units” and “ones” are the same.

The tens digit of a two-digit number is three times the units digit. If the digits are reversed, the new number is 36 less than the original number. Find the number.

Step 1: (Build up)Let x be the units digit

3x the tens digit3x∙10 + x the numberx∙10 + 3x the reversed number

Step 2: (Making an equation)(3x∙10 + x) − (x∙10 + 3x) = 36

Step 3: (Solving the equation)(3x∙10 + x) − (x∙10 + 3x) = 36

31x − 13x = 36 18x = 36 x = 2

Step 4: (Solving the problem)Since units digit is 2, tens digit is 6, the number is 62. Check: the reversed number is 26, their difference is 36.

256. The tens digit of a certain number is five more, than the units digit. The sum of the digits is 9. Find the number.

257. The tens digit of a two-digit number is twice the units digit. If the digits are reversed, the new number is 36 less than the original number. Find the number.

258. The sum of the digits of a two-digit number is 13. The units digit is one more than twice the tens digit. Find the number.

259. The sum of the digits of a three-digit number is 6. The hundreds digit is twice the units digit, and the tens digit equals to the sum of the other two. Find the number.

260. The units digit is twice the tens digit. If the number is doubled, it will be 12 more than the reversed number. Find the number.


Name__________________________

261. Eight times the sum of the digits of a certain two-digit number exceeds the number by 19. The tens digit is two less than the units digit. Find the number.

262. The ratio of the units digit to the tens digit of a two-digit number is one-half. The tens digit is two more than the units digit. Find the number.

263. There is a two-digit number whose units digit is six less than the tens digit. Four times the tens digit plus five times the units digit equal 51. Find the digits.

264. The tens digit is two less than the units digit. If the digits are reversed, the sum of the reversed number and the original number is 154. Find the original number.

265. A three-digit number has tens digit two greater than the units digit and the hundreds digit one greater than the tens digit. The sum of the tens and the hundreds digits is three times the units digit. What is the number?


if you use the laws of arithmetic you can make these...

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