If You Use the Laws of Arithmetic You Can Make These Problems Much Easier
Name______
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
710
3. To do addition many times, multiplication was created:
7 + 7 + 7 = 7 3
3
721
4. To “undo” (invert) multiplication, we use division;
3
721
5. To do multiplication many times, raising to a power was created.
7 7 7 7 = 74
72,401
6. To “undo” (invert) raising to a power, we use the method of taking the root of a number.
72,401
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.
- 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)
- In theexpression 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.
- 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
- Insert the parentheses to make an expression true:
a) 4 2 + 3 2 = 32b) 4 2 + 3 2 = 22
c) 44 11 – 7 55 = 605d) 4 – 4 4 + 4 4 = 0
e) 12 – 4 + 4 5 – 4 = 12f) 12 – 4 + 4 5 – 4 = 4
g) 14 55 – 55 4 = 0h) 7598 98 –96 3799 = 1
i) 1,001 140 – 127 150 – 139 25 – 18 = 1
j)169 – 156 314 – 303 1,000 – 993 = 1,001
k)55 5 – 5 5 – 5 = 11
l)382 292 – 101 + 8 = 10
m)3,333 111 – 100 3 + 91 = 1,000
- Make as many different answers as possible:
a)1 + 2 3 + 4 5
b)6 7 + 8 + 9 10
c)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.
1. Commutativity
If 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 78
And, 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)
(ab) c = a (bc)
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 = $14
12 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:
- Using the drawing explain the equalities:
a)
(5 + 3) 4 = 5 4 + 3 4
b)
(a + b) c = ac + bc
To multiply the sum by a number, you can multiply each of the addends by this number and then add the results up.
- Using the drawing explain the equalities and solve them.
a)
14 6 = (10 + 4) 6 = 10 6 + 4 6 =
b)c)
57 8 49 5
- Insert the missing numbers:
a) b) c)
97 4 52 9 18 5
- 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.
- Using the digits 2, 5, 9 write down all possible three digits numbers. Don’t repeat the digits.
- Use the picture and distributive property of multiplication:
30 5
735 7 =
- Compute:
a) 79 3 b) 32 8
c) 64 9 d) 86 7
Distributive property for division:
- 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.
- 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 3d) 65 5
b) 52 4e) 66 6
c) 84 2f) 91 7
- Do the calculations:
a) 28 · 80 + 22 · 80 f) 33 · 70 + 37 · 70k) 132 · 120 + 168 · 120
b) 59 · 57 + 41 · 57g) 22 · 41 – 20 · 41l) 927 · 56 + 56 · 73
c) 34 · 28 + 66 · 28h) 43 · 33 – 40 · 33m) 555 · 8 – 545 · 8
d) 12 · 245 + 12 · 55i) 335 · 11 – 135 · 11n) 750 ÷ 90 – 300 ÷ 90
e) 80 ÷ 5 – 30 ÷ 5j) 223 · 87 – 123 · 87o) 345 · 17 – 145 · 17
- 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 2f) 60 12 i) 246 6l) 421 3
- (CL) It is known that 25 · 38 = 950. Using the result find the answers to these problems.
a) 38 · 25d) 25 · 39 b) 950 38
e) 25 · 37c) 950 25 f) 38 · 24
- Explain the solution of the first example, then solve the next two problems.
a) Example: 200 80 4
284 · 3 = (200 + 80 + 4) · 3 =
200 · 3 + 80 · 3 + 4 · 3 = 3
600 + 240 + 12 = 852
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
- Compute, using the properties of addition and multiplication (use the 2 5 = 10!):
a) (94 + 179) + 21f) 2 5 2 5 7 2 5
b) 287 + (13 + 598)g) 4 19 25
c) (356 + 849) + (51 + 644)h) 2 4 25 5 3
d) 329 + 994 + 71 + 6i) 20 9 500
e) 2005 + 768 + 32 + 995 +19j) 7 15 + 7 85
- Insert parentheses to make the equality correct:
a) 1 + 2 · 3 + 4 · 5 = 29d) 1 + 2 · 3 + 4 · 5 = 55
b) 1 + 2 · 3 + 4 · 5 = 27e) 1 + 2 · 3 + 4 · 5 = 65
c) 9 + 8 – 7 – 4 = 6f) 9 + 8 – 7 – 4 = 14
- Do the calculations:
a) 28 · 70 + 22 · 70 f) 33 · 40 + 37 · 40k) 132 · 120 + 168 · 120
b) 59 · 157 + 41 · 157g) 22 · 41 – 20 · 41l) 627 · 56 + 56 · 373
c) 34 · 33 + 66 · 33h) 43 · 33 – 40 · 33m) 855 · 18 – 845 · 18
d) 180 ÷ 5 – 130 ÷ 5i) 323 · 87 – 223 · 87n) 345 · 17 – 145 · 17
e) 12 · 345 + 12 · 55j) 335 · 11 – 135 · 11o) 650 ÷ 70 – 300 ÷ 70
- Compute, using the properties of addition and multiplication:
1)51 + 52 + 53 + 54 + 55 + 56 + 57 + 58 + 59
2)99 + 99 + 99 + 99 + 99 + 99 + 99 + 99 + 8
3)999 + 999 + 999 + 999 + 999 + 7
4)82 4 + 18 4
5)36 97 + 36 3
6)24 128 + 76 128
- Compute:
a).438 90 – 238 90b).603 7 + 603 93
- Compute using distributive property of multiplication:
a). 4 505b). 25 399c). 16 403d). 12 499
- (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 = 30
d) 31n + 7n − 21 n + 12n if n = 20
- Compute in most convenient way:
a) (972 + 379) – 972e) 134 – 94 – 2i) 851 – (831 + 7)
b) (382 + 417) – 416f) 580 – 79 – 21j) 24 96 – 24 86
c) (538 + 245) – 245g) 83 9 – 73 9k) 276 – (8 + 176)
d) (725 + 158) – 625h) 7 38 – 7 28l) 76 52 – 66 52
- (CL) Solve out loud:
a) 450 9e)
300 5 4 2 i)
160 4 8 – 34
b) 360 90f) 720 8 + 15 6j) 19 + 7 80 10
c) 700 35g) 630 9 14 5k) 3 9 2 10 60
d) 600 120h) 800 ÷ 360 9 15l) 700 4 100 3
- * 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
- Calculate:
a) (9 20 + 60) 4 – 16 + 4 (20 5) b) 490 7 + (57 + 7) 8 2 − 3 (26 – 6)
- 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?
- Compute(use the most rational way):
a) 24 371 – 371 16 + 371 b) 353 26 – 14 353 + 12 147
- Solve:
a) 22,302 –( 9,302 + 7,383 + 4,617)b) 14,375 + 17,718 – (12,449 + 17,644)
- The first number is 150,096.The second number is 119,388less than the first one and 20,780greater than the third number. Find the third number.
- (CL) Simplify:
a) x + x – (x + 1)b) x + 1 – (x – 1)
c) (x + 2) – (x + 1)d) (x + 2) – (x – 1)
- 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
- 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.
- If you use the laws of arithmetic you can make these problems much easier!
a).69 27 + 31 27b).202 87 – 102 87
c).977 49 + 49 23d).263 24 – 163 24
- Compute using distributive property of multiplication:
a).91 8b).7 59c).6 52d).123 4
e).202 3f).390 5g).24 11h).35 12
- (CL) Solve out loud:
a) 208 208 f) 15 6k) 40,000 4
b) 890 1 g) 14 5l) 15,000 1000
c) 0 60h) 25 4m) 1000 10
d) 1 1 – 0 0i) 25 5n) 1000 100
e) 1 0 – 0 1j) 85 0o) 1000 1000
- Write the expression and solve:
a).How many times more is 42 then 7?
b).By how much is 5 smaller then 63?
c). By how much is 56 bigger then 8?
d).How many times is 9 smaller then 36?
- Compute:
a) 69 27 + 32 27 – 27d) 263 24 – 164 24 + 24
b) 202 87 – 100 87 – 2 87 e) 162 90 + 238 56 + 238 34
c) 977 49 + 49 23 f) 603 7 + 603 93
- 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 · 1
d)(100 + 12) · 112
- Insert the parentheses to make the equation true:
a)5 38 – 70 8 – 6 = 60 c) 30 – 49 42 6 8 = 184
b)630 7 2 9 25 = 125 d) 180 300 – 30 9 + 199 = 205
- 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.
- I thought of a two-digit number. This number is bigger than product of its digits by 52. What was the number?
- (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 = 20
b) 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.
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 = 100
5 = 512 = 12100 = 100
0 = 019 = 19
x = 5x = 0x= −3
x = 5 or x = −5x = 0no solution
Rules for Addition of Integers
- Positive numbers: Add the numbers. The result is positive.
- Negative numbers: Add absolute values. Make the answer negative.
- 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.
- One number is zero. The sum is the other number.
F.E.:(−3) + (+5) = 2(-−3) + (+1) = −2
(−3) + (−5) = −8(−3) + (0)= −3
Subtraction
The difference a – b is the number that when added to b gives a.
a – b = ca = b + c
For 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
Multiplication and Division
F.E.:a) (−3) (−7) = 21c) (−3) (7) = −21e) 3 (−7) = −21g) 3 7 = 21
b) (−48) (8) = −6d) (−48) (−8) = 6f) 48 (−8) = −6h) 48 8 = 6
- Explain the solutions:
a)(−3)= (−3) (−3) = 9e)
b)(−3)= (−3) (−3) (−3) = −27f) doesn’t exist
c)(−1)= 1 (Why?)g) (−3) 0 = 0
d)(−1)= −1 (Why?)h)
- In case of (− a − b):
- –10 – 5 7. –30 – 45 13. –2 –25t
- –20 – 7 8. –40 – 52 14. –21p – 38
- –13 – 5 9. –100 – 88 15. –34q – 18q
- –15 – 22 10. –10 – 0 16. –12w – 27w
- –18 – 54 11. –3a – 2a17. –33x – 22x
- –20 – 23 12. –14b – 8b18. –13s – 12s
- In case of (−a + b):
- –10 + 5 7. –15 + 22 13. –44 + 44 19. –29q + 29q
- –20 + 7 8. –18 + 15 14. –24b + 34b20. 0 – 14
- –7 + 20 9. –20 + 23 15. –54c + 57c21. –16w – 0
- –13 – 5 10. 30 – 45 16. –18 + 24 22. 23r –23
- –5 + 13 11. 40x – 52x17. –15d + 14d
- 15 – 22 12. –100a + 88a18. 14t – 16
- Find the difference.
a) –6 –12 b) –6 – (−12)
- Simplify.
a) x + (−y) – (−4) b) 6 – a – 4 – (-a) – 3 c) –y – (−7a) + – (−7a) + y – 8 – (−y)
Multiplying Integers and Raising Integers to Powers
- Find the product.
a) 8 (−3) b) –15 (+4) c) (−12) (−6)
- Evaluate. To write an exponential expressions as products and then to evaluate them.
a) (−5)b) −5c) (−2)d) (−2)
- Simplify. To simplify an algebraic expression containing repeated addition of negatives.
–t – t – x –x – x
- 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 Integers
- Find the quotient.
a) b) c) d)
- Write an algebraic expression for “seven less than twice some number.”
- Calculate:
a) 15 + (−12)e) 15 – (−3)i) (−3)m) –19 + 3q) −8
b) f) –13 + (−7)j) –6 (−14)n)32 – 49r) (–3)
c) g) –8 – 7k) (–8)o)s) –12 (+3)
d) – (−6) + 3h) l) −(−7) −(− (−9))p) 20 ∙ (−2)3t) (−4)3
- Evaluate the given algebraic expressions replacing x with –2 and y with 1.
a) 3xb) (3x)c) x+ yd) (x+ y)
- Simplify.
a) T + T + Tf) − x + 17 + (−12) + xk) – a – (−b)p) x – 9 – (−x) + (−4)
b) 6 – (−2 + 5y)g) –p – (−4) – p – 4 – pl)–(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 – 16xs) 3x (x – y) + y ( x –y)
e) (−2) (−x)j) 2h – 3h (2 – 3h)o) 13a – (−a)t) 6f(2 – f) – f(f – 6)
- Write a formula that describes the value ofx
a) b)
- Using the formula from the previous exercise, solve for y when x = 19 ft.
The Number Line and Integers
- Find the opposite of an integer.
- –(18)b) –(−8)
- Find the absolute value of the integer.
a) −9b) 14
- Place <, >, or = between a pair of integers…
a) –8 … −2b) –2 … −8c) −5 … −(−5)
Adding Integers
- Solve.
a) (−13) + (−24)b) 14 + (−18)c) –14 + 18
- Simplify the algebraic expressions containing addition.
a) y + 7 + y + (−2)b) –x + 6 + (−3) + x + (−4)
- 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-$300
c)Painter - $20 son-$15 price- $350
- 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?
- 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 $60
c)Total fare for 3 adults and 6 children is $36
Comparing Numbers
- Which of the two number is greater?
1)–4 and –35) –41 and 19) –100 and 1
2)–18 and –156) 0 and –310) 8 and −8
3)–18 and 07) 0 and 311) 12 and −1
4)–18 and 188) –31 and –10012) –181 and –1
- Which of the following inequalities are true?
1)–10 < −14) – 49 > −(−7)7) –(−3) < −(−5)
2)17 < 05) 2 < −(−2)8) −(−11) < 11
3)4 17 > −(−10)6) – (−3) > 59) −9<0
- Arrange the numbers from least to greatest.
a)–8, −10, 0, 5, 1, −1, 3, −6, −9, −19, 32
b)–100, −98, 1, 2, 9, 11, 32, −5
c)0, −3, 9, 14, −9, −8, 1, 3
- Arrange the numbers from greatest to least.
a)–8, −10, 0, 5, 1, −1, 3, −6, −9, s19, 32
b)–100, −98, 1, 2, 9, 11, 32, −5
c)0, −3, 9, 14, −9, −8, 1, 3
- Write all the integers located on the number line between…
a)–8 and 8b) –7 and 6c) –3 and 15d) –5 and 10
- Write the five smallest integers greater than…
a) –5b) 7c) –11d) –32e) 0f) −7
- Find all integer values for x that satisfy the inequality.
a)0 < x < 5d) – 4 x −2g) x < 5j) x 1
b)0 x 3e) – 4 < x < −2h) x 0k) x < 1
c)–5 x 0f) –1 < x < 1i) x −3
Equations at a first glance.
- 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 = 21
Finish it yourself
- I thought of a number, took away 10, divided the result by 2, and got 2. What was my number?
- I thought of a number, doubled it and added 7. I got 35. What was the number?
- “I thought of a number…” Nina thought of a number, and that’s what she wrote:
a)2n + 10 = 36
1). Tell what Nina did with her number.
2). What was her number?
b)2(n + 5) = 36c) 2n – 10 = 36d) 2(n – 5) = 36
- (CL) Solve out loud:
a)2x + 5 = 15 d) 2 x – 5 = 15g) 4 x + 20 = 48j) 4 x – 12 = 48
b)30 – 2 x = 10 e) 30 – 3 x = 21h) 30 – 4 x = 18k) 30 – 5 x = 20
c)50 – 2 x = 42 f) 50 – 3 x = 38i) 50 – 4 x = 30l) 50 – 5 x = 45
- Solve for variable:
a)5z + 7 = 32d) s 3 + 3 = 5g) a 8 – 3 = 9j) 4w – 6 = 14
b)b 4 + 13 = 17e) 9c + 12 = 21h) 10h – 34 = 66k) x 10 + 8 = 13
c)m 6 – 6 = 0f) 16n – 28 = 4i) 5k – 2 = 43l) z 3 + 4 = 13
- Solve:
a)3 · x = 1 c) 4 · x = 1e) 10 · x = 2
b)10 + x = 12d)10 – x = 8f) 10 – x = 8
- 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.
- Solve the following equations:
a)3 + 7a + 7 = 20 + 2 ac) 5 a + 36 = 7 a
b)6 x + 44 + 8 x + 25 – 2 x = 144 d) 4y – 3 y + 14 y – 2 y = 169
- (Cl) Solve out loud:
a)2 · (x – 5) = 20d) 3 · (x – 5) = 21g) 2 · (x + 5) = 20
b)3 · (x + 5) = 21e) 5 · (10 + x) = 60 h) 5 · (10 – x) = 40
c)4 · (10 + x) = 60 f) 4 · (10 – x) = 40i) 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 + x14 t26 – yand
Example: Evaluate the expression for the given values of variables:
1)x – y for x = 83 and y = 49
Solution: x – y = (83) – (49) = 34
34 is called the value of the expression
2) for a = 63 and b = –7
Solution:
3)6 y for y = 15
Solution: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 Expressions
In 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”
3)In English: “Six more than eight times some number”. In Algebra:“8x + 6”
SentencesAlgebraic Expressions
a) Nine more than some numberm + 9or 9 + m
b) Nine less than some numbera – 9
c) Seven more than five times some number5 t + 7 or7 + 5 t
d) Three less than a product oftwo numbersp q – 3
e) The difference between twonumbers m and nm – 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 Equations
An 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 = 71) Which of the expressions are equalities?
b)5 – 1 = 22) Which equalities are true?
c)21 + 2 = 243) Which equalities are false?
d)x – 5 = 12
e)9 – x = x
f)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:
3 x = 9
x = 3
Examples:
a) 5x – 18 = 17b) 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 – 12(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 = 129x = 27
2 29 9
x = 6 x = 3
Exercises
- Translate to an algebraic expression
a) 7 more than md) 9 more than tg) 11 less than cj) 47 less than d
b) 26 greater than qe) 11 greater than zh) b more than ak) c more than d
c) x less than yf) c less than bi) Twice xl) Four times p
- Use algebraic language:
a) The sum of a and bThe sum of m and n
b) The difference between 17 and bThe difference between p and q
c) 8 more than some numberOne more than some number
d) 54 less than some number47 less than some number
e) A number x plus three times yA number a minus 2 times b
- Solve:
a) 3r + 15 = 75d) b 3 + 17 = 48g) 8t – 14 = 82j) 5x + 6 = 41
b) v 2 – 4 = 26e) p 2 – 5 = 9h) 0.5j + 2 = 6k) 4y + 15 = 67
c) 4v – 22 = 22f) 4y – 8 = 20i) c 5 + 7 = 12l) d 4 + 3 = 33
- Express in algebraic symbols:
a) The sum of x and x + 3b) The product of a and b
c) 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 + 1
f) 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?
- Solve the problems and check them:
a) x · 9 = 720c) 300 – x = 60e) x + 9 = 720 b) x 90 = 9 d) 300 x = 60 f) x – 90 = 9
- Compute(use the most rational way):
a) 24 371 – 371 16 + 371 b) 353 26 – 14 353 + 12 147
- A truck has total of a pounds of fruit in each of n boxes. How many pounds of fruit are in the truck?
- Marina bought 4 notebooks, which cost b dollars each. And 3 pens, which cost c dollars each. How much money did Marina spend?
- Natasha had d dollars. She bought 2 ice creams which cost x dollars each. How much money does she have left?
- If you use the laws of arithmetic you can make these problems much easier!
a).69 27 + 31 27b).202 87 – 102 87
c).977 49 + 49 23d).263 24 – 163 24
- 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?
- 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?
- Solve the following equations:
a) 40x ÷ 10 = 28 c) 49 ÷ k – 3 = 46
b) y ÷ 10 – 28 = 32 d) (25 – a) ÷ 7 = 3
- A movie theater has a rows and b seats in each row? How many seats are there in the theater?
- A teacher handed out m notebooks in packs of c notebooks in each pack. How many packs were there?
- Compute:
a) 69 27 + 32 27 – 27d) 263 24 – 164 24 + 24
b) 202 87 – 100 87 – 2 87 e) 162 90 + 238 56 + 238 34
- c) 977 49 + 49 23 f) 603 7 + 603 93
- A truck has a total of a pounds of fruit in nboxes. How many pounds of fruit are in each box?
- Solve for variable:
a)5z + 7 = 32 d) s 3 + 3 = 5g) a 8 – 3 = 9j) 4w – 6 = 14
b)b 4 + 13 = 17e) 9c + 12 = 21h) 10h – 34 = 66k) x 10 + 8 = 13
c)m 6 – 6 = 0f) 16n – 28 = 4i) 5k – 2 = 43l) z 3 + 4 = 13
- 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 · 1
c) (100 + 12) · 112 f) 107 + 88 + 193 i) 125 · 17 · 4
- Solve the following equations:
a) 4x + 2x + 5 = 3x + 35b) x + 3 = 2x – 3c) 7x + 9 = 86d) 84 x = 6 7
e) 3x = 87 – 6f) 6x = 42g) 3x + 4 = 37h) (x – 3) 2 = 30
- What is the length of a rectangle whose perimeter is P, if the width is 6?
- What number is 7 more than twice as much as x?
- 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?
- The larger of two numbers is twice the smaller. If the larger is x, what is the smaller?
- Solve out loud:
a) x – 15 = 22c) x – 65 = 104e) x – 87 = 99
b) x – 21 = 33d) x – 15 = 62f) x – 108 = 1,109
- Out loud: In algebraic symbols, how many cents are in n nickels and c cents?
- The larger of two numbers is four more than the smaller. If the larger is x, what is the smaller?
- What number exceeds y by 4 less than x?
- 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?
- 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.
- I thought of a two-digit number. This number is bigger than product of its digits by 52. What was the number?
- Out loud:Compute. Use the most convenient way.
a) ( 382 + 417 ) – 416c) (725 + 158 ) – 625