four basic arithmetic processes
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FOUR BASIC ARITHMETIC PROCESSES. ADDITION is the processes of combining numbers of the same kind. Measures like dollars and miles cannot be added because they are two different things. - PowerPoint PPT PresentationTRANSCRIPT
FOUR BASICARITHMETIC PROCESSES• ADDITION is the processes of combining numbers of
the same kind. Measures like dollars and miles cannot be added because they are two different things.
• SUBTRACTION is the process of finding the difference between two like numbers by taking one (the subtrahend) from the other (the minuend). The answer is the remainder.
• MULTIPLICATION is the process of adding one number (the multiplicand) to itself as many times as there are units in another number (the multiplier). The answer is called the product. The multiplicand and multiplier are also called the factors of the product.
Arithmetic processes (cont…)• DIVISION is the process of finding how many times
one number (the divisor) is contained in another number (the dividend). The result is the quotient.
• Examples7 – 3 = 4 5 * 3 = 15
Minuend
Subtrahend
Remainder
Multiplicand
Multiplier
Product
4
024246
DivisorDividend
Quotient
Remainder
Common Fraction• A FRACTION is a portion of the whole; it signifies
division. The top number is called the numerator (same as the dividend), and the bottom number is called the denominator (same as the divisor). The line between the numbers is the division sign.
• A PROPER FRACTION has a numerator that is smaller than the denominator, as in ¾ or 2/7.
• An IMPROPER FRACTION has a numerator that is equal to or greater than the denominator, as in 6/6 or 9/4.
65 Numerator (or dividend)
Denominator (or divisor)
For convenience, this isoften written as 65
Common Fraction (cont…)• An improper fraction can be converted to a whole number
(6/6 =1) or a mixed number ( ) by dividing the denominator into the numerator. A mixed number contains both a whole number and a fraction.
• A complex fraction has one or more fractions in either the numerator or the denominator, or both. A complex fraction may be converted to a simple fraction by dividing the numerator by the denominator (remember the line in a fraction is a division sign). To divide by a fraction invert the denominator fraction (divisor) and multiply:
61
67 1
Common Fraction (Cont…)
NOTE: The answer to any example or problem should not be left as an improper fraction. It should be converted to a proper fraction or a mixed number, then reduce to lowest terms.
1514
1521
32
2115
32
211532
2117
235
257
527
527
1123
141
8314
83
1483
7
1
ADDITION OF FRACTIONS• To add fractions with the same denominators, add the
numerators.
• To add fractions with different denominators, select a common denominator into which all the denominators will divide evenly. The smallest number into which all denominators divide evenly is called the least (or lowest) common denominator (L.C.D) or least common multiple (L.C.M).
53
512
51
52
ADDITION OF FRACTIONS (cont…)• To find the L.C.M, spread out all the denominators in a short-
division box. Divide the denominators by a prime number (a number divisible only by itself and 1) or a multiple of a prime number that goes into at least two of the denominators. When a denominator cannot be divided by a divisor (prime number), it is brought down into the quotient. Repeat the process until no two parts of the quotient are divisible by the same prime number. Then multiply all divisors and all parts fo the quotient together. The result is the L.C.M.
• To find the L.C.M. of 207&
53,
85
4011245112418420585
40231
4063
40142425
4027835540
2040754038405207
53
85
L.C.M
PROJECT 1 FRACTIONSChange to the improper fractions.
75140
151320
3216
2916
8326
2717
1619
4381. 2. 3.
4. 5. 6.
7. 8.
PROJECT 1 (Cont…)Add the fractions and reduce the answers to the lowest terms.
3236,
2114,
5223
325,
658,
876
16728,
3291,
8364,
6537
214,
10719,
5226,
438
6112,
835,
327
413,
1654,
836
53,
107,
32,
65
95,
2411,
87,
61
1. 2.
3. 4.
5.
6.
7. 8.
SUBTRACTION OF FRACTIONS• To subtract fractions that have the same denominator,
simply subtract the numerators.
• To subtract fractions with different denominators, select the least common denominator, and proceed with the operation.
1711
17415
174
1715
256
5012
50152750
5312750
)1050(15)5050(27103
5027
25
6
MULTIPLICATION OF FRACTIONS• There are three steps in multiplying fractions: (a) Multiply the
numerators. (b) Multiply the denominators. (c) Reduce to the lowest terms.
• “Cancelling” saves time and work. In the above example:
• When multiplying by a whole number, think of the whole number as a fraction with a denominator of 1.
3 5 3 5 15 57 9 7 9 63 21
5
21
215
3751
35
71
95
73
1
3
123443
116
4316
4
1
DIVISION OF FRACTIONS• To divide fractions, invert the divisor, as explained earlier,
and multiply. An inverted fraction is a fraction with the numerator and denominator interchanged.
NOTE:When addition, subtraction, multiplication and division involves a mixed number, convert all mixed numbers to improper fractions. Then proceed with the operations.
65
3215
34
85
43
85
12
PROJECT 2 FRACTIONS• Perform the indicated operations. Reduce answers to the
lowest terms where necessary.
2127
5425
28516
7417
1636
217
4318
2156
652
836
655
43
181
513
3014
2013
2515
6548
3216
433
73
542
72
1613
4327
7341
323
878
617
4313
1. 2. 3.
4. 5. 6.
7. 8.
9. 10. 11.
12. 13.
Project 3 Fractions• Compute the total hours worked by each employee and on each day, and verify the totals.
434
417
216
439
41710
215
419
217
438
438
415
2110
416
217
217
418
6219
219
219
415
418
217
4325
215
416
219
433
214
2128
437
419
218
4138
415
2177
736
Mon. Tues. Wed. Thurs. Fri. Sat. Total.
Total.
Ali
Saeed
Aina
Sofia
Jawad
Iqrar
Azhar
Jamila
Sana
THE REAL NUMBER SYSTEM• A set is a collection of elements listed within braces. The
set {a, b, c, d, e} consists of five elements, namely a, b, c, d, and e. a set that contains no element is called an empty set (or null set). The symbol { } or the symbol Ø is used to represent the empty set.
• There are many different sets of real numbers, two important sets are the natural numbers and the whole numbers.
Natural numbers: {1, 2, 3, 4, 5, …. }Whole numbers: {0, 1, 2, 3, 4, 5, … }
• To understand the sets of numbers we introduce the real number line.
-4 -3 -2 -1 0 1 2 3 4
• The real number line continues indefinitely in both directions.
SETS OF REAL NUMBERS (cont…)• Another important set of numbers is the integers.
Integers: {… , -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, … }
Negative integers Positive integers• The set of rational numbers consists of all numbers that
can be expressed as the quotient of two integers, with denominator not zero.
• Some numbers are not rational. Numbers such as square root of 2, written as , are not rational numbers. Any number that can be represented on the real number line that is not rational is called irrational number.
2
REAL NUMBERS• Any numbers that can be represented on the real number
line is real number.Real numbers
Rational numbers Irrational numbers
Integers Non-Integers
Negative 0 Positive numbers numbers
PROJECT 4 NUMBER SYSTEM1. List the elements of the set that are:a) Natural numbers b) Whole numbersc) Integers d) Rational numberse) Irrational numbers f) Real numbers
2. List the numbers that area) Positive integers b) Whole numbersc) Negative Integers d) Integerse) Rational numbers f) Irrational numbersg) Real numbers
,5,7,9.2,
74,9,0,3,96,
214,5.0,6
219,300,5,67.1,
125,
412,2,2,
21
ALGEBRA• Some Basic Definitions• Variable
A characteristic which changes from one individual to the other, e.g. the height of a student in your class, the temperature of different cities in Afghanistan, etc.
Variable is denoted by the lower case letters, e.g. x, y, z, etc.
Constant
A characteristic which does not change, e.g. the dimension of your class room, the height of the chair you are setting on, the number of papers in BBA 1st semester exam, etc.
A constant is denoted by the alphabets like a, b, c, d, etc.
Expression It is the combination of operands and operators,
e.g. x + y, 4 - 17, etc. Here x, y, 4 and 17 are called operands, while
the symbols +, - are called operators. The other operators are *and ÷.
The expression x + y is called algebraic expression and 4 - 17 is called the arithmetical expression.
Equation
An expression which involves the sign of equality is called equation.
Examples2x - 5y = 12
x2 + 3x - 5 = 0 x2 + y2 = r2(Equation of circle centered at the origin having radius r) 1/x + 7 = x/3
Continue….• Subtraction Subtract 9x – 2.5y from 4x + 5y
4x + 5y +9x – 2.5y
-5x + 7.5y In addition and subtraction we combine the similar terms of
the expressions.
+__________________
Continue….
Example:Add 7x + 3y + z, -3x + y, and 5x – 4z 7x + 3y + z
-3x + y + 0z5x + 0y – 4z
_____________9x + 4y – 3z
Distributive property• For any real numbers a, b, c
a (b + c) = ab + acExamples1. 2 (3 + 7) = 2 * 3 + 2 * 7 = 202. 4 (2x - 4y) = 4 * 2x – 4 * 4y = 8x - 16y3. -7 (3p - 5q) = -7(3p)-7(-5q)=-21p+35q4. x (2.3y + 1.2z) = 2.3xy + 1.2xz5. 3 (x + y + z) = 3x + 3y + 3z
Group Activity
Simplify1. 2x2 – 4x +8x2 – 3 (x + 2) – x2 - 2Sol: 2x2 -4x + 8x2 - 3x – 6 - x2 - 2 =(2x2 + 8x2 – x2) + (-4x - 3x) +(-6 - 2) = 9x2 - 7x - 82. x2 + 2y - y2 + 3x + 5x2 + 6y2 + 5y3. 2 [ 3 + 4 (x - 5) ] - [ 2 - (x - 3) ]
Linear Equation• An equation of the form
ax + b = c (1) where a, b and c, a≠0 are constants (real numbers) is called a linear
equation.
Examples1. x + 4 = 7 (a = 1, b = 4, c = 7)2. 2x – 3 = 5.4 (a = 2, b = -3, c = 5.4)3. -(x + 3) - (x - 6) = 3x - 4.5
Though this equation is not of the form (1), but it is still a linear equation, since it can be rewritten as:
5x - 4.5 = 3 (a = 5, b = -4.5, c = 3)
Solution of a linear equation
By the solution of a linear equation we mean to find that value of the unknown ‘x’ which satisfies the equation.
Ex1. x = 3, is the solution of x – 2 = 12. x = -6, is the solution of 2x + 1 = x - 53. x = 11/2 is the solution of 3 (x + 2) = 5 (x-1)
4. y = 9 is not a solution of 2y – 3 = 5
How to solve a linear equationWe can solve a linear equation by1. Substitution method2. Graphical method3. Algebraic method
(explanation by examples)
The substitution method In this method we make a guess for the solution of the
linear equation, we then put the guessed value in the equation and check whether the guess is correct or not?
Continue….
Examples Let us try to solve the equation ‘4x+2=4’, An
accurate guess for the solution is “x = ½”, we substitute this value in the given equation4 * ½ + 2 = 4 2 + 2 = 4 4 = 4
⁄⁄
Continue…. The substitution method is generally a time
consuming method, i.e. sometimes it takes very long time to find an accurate guess.
The graphical methodIn this method we sketch the graph of the linear equation, and then find the point where the graph cuts the x-axis, such a point is called the ‘x-intercept ’. This method is also not a very good method.
The algebraic methodIn this method we use the algebraic operations to find the solution. The method is described in the following examples.
Examples (Exercise Set 2.3, p#103)
Solve the following equations and check your solution.1. 2x=6Sol: Given the equation
2x=6Dividing both sides by 2
—x=22
—62
⁄⁄
⁄⁄3
Continue….Thus the solution is x=3.
Q16. x/3 = -2Sol: Multiplying both sides by 3, we get x = -6, which is the desired solution.Q19. -32x = -96Sol: Dividing both sides by -32, we get x=3 as the
required solution.Q24. -x = 9Sol: Multiplying both sides by -1, we get the solution x = -
9.
Continue….Q25. -2 = -ySol: Multiplying both sides by -1, we get
y = 2 as solution.Q33. 13x = 10Sol: Dividing both sides by 13, we get
x = 10/13 the required solution.Q52. -2x = 3/5Sol: Dividing both sides by -2, we get x = -3/10,
which is the required equation.
Continue….Q66. When solving the equation 3x = 5. Would you divide both
sides of the equation by 3 or by 5? Explain.Q67. When solving the equation -2x = 5. Would you add 2 to both
sides of the equation or divide both sides of the equation by -2? Explain.Q69. Consider the equation 4x = 3/5. Would it be easier to solve this
equation by dividing both sides of the equation by 4 or by multiplying both sides of the equation by ¼, the reciprocal of 4? Explain your answer. Find the solution to the problem.
Exercise Set 2.4 (9-65), p# 110Solve each equation.11. -2x-5=7Sol: Given
-2x-5=7Adding 5 to both sides-2x-5+5=7+5
-2x=12Dividing both sides by -2
-2x/2=12/-2 => x=-6The desired solution.
Continue….25. -4.2 = 2x + 1.6Sol: We have
-4.2 = 2x + 1.6Subtracting 1.6 from both sides -4.2 - 1.6 = 2x + 1.6 – 1.6
-5.8 = 2xNow dividing both sides by 2
-5.8/2 = 2x/2 => -2.9 = x (Using calculator)
Or x = -2.9 as desired.
Continue….Q33. x + 0.05x = 21Sol: 1.05x = 21
Dividing both sides by 1.05 (the coefficient of x)
x = 21/1.05 => x = 20 as required.Q36. -2.3 = -1.4 + 0.6xSol: Adding 1.4 to both sides
0.6x = -2.3 + 1.40.6x = -0.9Dividing through out by 0.6 x = -0.9/0.6 => x = -1.5
Continue….
Q38. 32.76 = 2.45x – 8.75xSol: Given
32.76 = 2.45x – 8.75x=> 32.76 = -6.30x
Dividing both sides by -6.30x = -32.76/6.30x = -5.2 the answer.
Continue….Q46. -2(x+4) + 5 = 1Sol: Using distributive property
-2x – 8 + 5 = 1 or -2x – 3 = 1
Adding 3 to both sides -2x -3 + 3 = 1 + 3
i.e. -2x = 4Now dividing both sides by -2
x = -4/2or x = -2 as desired.
Continue….Q58. 0.1(2.4x + 5) = 1.7Sol: given the equation
0.1(2.4x + 5) = 1.7Using distributive property0.24x + 0.5 = 1.7Subtracting 0.5 from both sides0.24x = 1.7 – 0.5
Þ 0.24x = 1.2Finally dividing both sides by 1.2x = 1.2/0.24x = 5the required answer.
Continue….Q65. 5.76 – 4.24x – 1.9x = 27.864Sol: Given that
5.76 – 4.24x – 1.9x = 27.864Þ 5.76 – 6.14x = 27.864
Now subtracting 5.76 from both sides 5.76 - 5.76 – 6.14x = 27.864 – 5.76
-6.14x = 22.104Finally dividing both sides by -6.14
-6.14x/-6.14 = -22.104/6.14x = 3.454
Which is the desired result.
Challenging problems
Solve each equation.1. 3(x-2) – (x+5) – 2(3-2x) = 182. -6 = -(x-5) – 3(5+2x) – 4(2x-4)3. 4[3 – 2(x+4)] – (x + 3) = 13
Solving linear equations with the variable on both sides of the equation
Hints:1. Use the distributive property to remove the parentheses.2. Combine like terms on the same side of the equal sign.3. Rewrite the equation with all terms containing the variable on one side of the
equation and all the terms not containing the variables on the other side of the equation.
4. Isolate the variable using the multiplication property, this gives the solution.5. Check your answer, by putting the value of ‘x’ in the original equation.
Exercise Set 2.5(9-54), p#118/119
Solve each equation.9. 4x = 3x + 5Sol: Given the equation
4x = 3x + 5Combining the terms involving variables on one side
if the equation4x – 3x = 5 x = 5
Which is the required answer.
Continue….15. 15 – 3x = 4x – 2xSol: Given that
15 – 3x = 4x – 2xShifting 3x to the right side of the equation.15 = 4x – 2x + 3x
Þ 15 = 5xIsolating ‘x’ using the multiplication property
15/5 = x or x = 3 as required.
Continue….25. x – 25 = 12x + 9 + 3xSol: We have
x – 25 = 12x + 9 + 3x Shifting ‘x’ to the right & 9 to left of the equation -25 – 9 = 12x + 3x – x -34 = 14x
Using multiplicative property-34/14 = x
or x = 2.428 as required.
Continue….28. 4r = 10 – 2(r-4)Sol: Given that
4r = 10 – 2(r-4)Using distributive property4r = 10 – 2r + 8Shifting 2r to the left of the equation4r + 2r = 10 + 8 6r = 18Dividing both sides by 6
r = 18/6or r = 3
as required.
Continue….34. 3y – 6y + 2 = 8y + 6 – 5ySol: Given the equation
3y – 6y + 2 = 8y + 6 – 5yÞ -3y + 2 = 3y + 6
Shifting -3y to the right & 6 to the left of the equation 2 – 6 = 3y + 3y -4 = 6yDividing both sides by 6 -4/6 = y
or y = -2/3 or y = -0.667
Continue….33. 0.1(x + 10) = 0.3x -4Sol: Given
0.1(x + 10) = 0.3x -4Using the distributive property 0.1x + 1 = 0.3x - 4Combining the terms involving the variable on one side of the
equation & constant on the other side0.3x – 0.1x = 4 + 1
0.2x = 5Using multiplicative property to isolate ‘x’
x = 5/0.2or x = 25
Which is the required solution.
Continue….36. 5(2.9x - 3) = 2(x +4)Sol: We are given that
5(2.9x - 3) = 2(x +4)By the use of distributive property, we have14.5x – 15 = 2x + 8
Shifting 2x to the left & -15 to the right of the equation14.5x – 2x = 8 + 1512.5x = 23 Isolating ‘x’ x = 23/12.5
or x = 1.84 as desired.
Continue….37. 9(-y + 3) = -6y + 15 – 3y + 12Sol: Given
9(-y + 3) = -6y + 15 – 3y + 12Using the distributive property-9y + 27 = -9y + 27Since the same expression appears on both
sides of the equation, therefore the statement is true for all real values of y. If we continue to solve this equation further, we arrive at
0 = 0.
A difference
Note that an equation is true only for a specific value(s) of the variable, while the identity is true for all values of the variable. In the previous question the given equation is an identity as it is true for all values of ‘x’. Further, every identity is an equation but not every equation is an identity.
Continue….39. -(3 - p) = -(2p + 3)Sol: Given that
-(3 - p) = -(2p + 3)Using distributive property-3 + p = -2p – 3Shifting 2p to the left(???) & -3 to the right of the
equationp + 2p = -3 + 3 3p = 0
=> p = 0Which was required.
Continue….51. 5 + 2x = 6(x + 1) – 5(x - 3)Sol: We have
5 + 2x = 6(x + 1) – 5(x - 3)Using distributive property, we have5 + 2x = 6x + 6 – 5x + 15
Or 5 + 2x = x + 21Shifting x to the left & 5 to the right2x – x = 21 – 5 x = 16 as required.
Challenging Problems
1. Solve-2(x+3)+5x=-3(x-3)+4x-(4-x)
2. Solve4(2x-3)-(x+7)-4x+6=5(x-2)-3x+7(2x+2)
3. Solve4 - [5 - 3(x + 3)]= x - 3
Ratio and ProportionDefinition
A Ratio is a quotient of two quantities. Ratios provide a way of comparing two numbers or quantities. The ratio of ‘a’ and ‘b’ is written as
a to b, a:b or a/ba and b are called the terms of the ratio.
EXAMPLES
1. If an algebra class consists of 32 males and 13 females, find
(a) The ratio of males to females.(b) The ratio of females to the entire class.Sol: (a) 32:13
(b) 13:45
Exercise Set 2.6, p#127The results of an English examination are 5 A’s, 6 B’s, 8 C’s, 4
D’s and 2 F’s. Write the following ratios.1. A’s to C’s.2. A’s to total grades.3. D’s to F’s.4. Grades better than C to total grades.5. Total grades to D’s.6. Grades better than C to grades less than C.
(Try these!)
Continue….Determine the following ratios. Write each ratio in lowest term.
7. 5 feet to 3 feet.8. 60 dollars to 80 dollars.9. 20 hours to 60 hours.10. 100 people to 80 people.11. 4 hours to 40 minutes.12. 6 feet to 4 yards. (1yard=3feet)13. 26 ounces to 4 pounds. (1pound=16 ounces)14. 7 dimes to 12 nickels. (1dime=10cents,
1nickle=5cents, 1dime=2nickle)
ProportionA proportion is a special type of equation. It is a statement of equality between two ratios.
How to denote a proportion?A proportion is denoted as a:b=c:d (read as ‘a is to b as c is to d’). We can also denote a proportion asa/b = c/d, a and d are referred as the extremes, and the b and c are referred as the means of the proportion.
Cross MultiplicationIf a/b = c/d then ad = bcNote that the product of means equal to the product of extremes.If any three of the four quantities of a proportion are known, the fourth can easily be found.
ExampleSolve for ‘x’ using cross multiplying x/3 = 25/15.
SolutionGiven that
x/3 = 25/15Using cross multiplication
x.15 = 3*25x = 75/15x = 5 (Check the answer!!!!!!!!)
Exercise Set 2.6 (21-32) p#128
Solve for the variable by cross multiplying.21. 4/x = 5/2022. x/4 = 12/4823. 5/3 = 75/x24. x/32 = -5/425. 90/x = -9/1026. -3/8 = x/4027. 1/9 = x/4528. y/6 = 7/4229. 3/z = 2/-2030. 3/12 = -14/z31. 15/20 = x/832. 12/3 = x/-100
To Solve Problem Using Proportion
1. Represent the unknown quantity by a variable (a letter).2. Set up the proportion by listing the given ratio on the left
side of the equal sign, and the unknown and other given quantities on the right side of the equal sign. When setting up the right side of the proportion, the same respective quantities should occupy the same respective positions on the left and right.
Exercise Set 2.6, P#127
18. In 1970 in the United States, 72,700 metric tons of aluminum was used for soft-drink and beer containers. In 1990 this amount had increased to 1,251,900 metric tons. Find the ratio of the amount of aluminum used for beer and soft-drink containers in 1990 to the amount used in 1970.
Sol: (Try!!!!!!!!).
Continue….33. A car can travel 32 miles on 1 gallon of gasoline. How far it can travel
on 12 gallon of gasoline?Ans: Let the distance covered in 12 gallon gasoline be x miles.
Nowmiles/gallon = miles/gallon
Or 32/1 = x/12Using cross multiplication
x . 1 = 32 * 12Or x = 384 as required.
Inequalities in One VariableAn inequality in one variable is the Mathematical statement in which one or more of the following symbol are used<, less than symbol>, greater than symbol≤, less than or equal to symbol≥, greater than or equal to symbolThe direction of the symbol is sometimes called the sense or order of the inequality.
Examples of inequalities in one variable
1. x+3<5 (x+3 is less than 5)2. x+4≥2x-6(x+4 is greater than or equal to 2x-6)3. 4>-x+3 (4 is greater than-x+3)4. 2x+6≤-2 (2x+6 is less than or equal to -2)Properties Used to Solve Inequalities
For real numbers a, b, and c5. If a>b, then a+c>b+c.6. If a>b, then a-c>b-c.7. If a>b and c>0, then ac>bc8. If a>b and c>0, then a/c>b/c9. If a>b and c<0, then ac<bc10. If a>b and c<0, then a/c<b/c
Similarly we can state the properties for the symbol ‘<‘
Exercise Set 2.7 (1-40), p#137
Solve the inequality and graph the solution on the real line.8. -4≤-x-3Sol: Given
-4≤-x-3Adding 3 to both sides-4+3≤-x-3+3 -1 ≤-xMultiplying both sides by -1
or x ≤ 1 (as -1<0, the inequality reverses the order).
Continue….12.6≥-2xSol: This inequality can also be written as
-2x≤6Dividing both sides by -2-2x/-2 ≥ 6/-3
Or x ≥ -3Which is the desired answer.
Continue….15. 12x + 24 < -1217. 4 – 6x > -519. 15 > -9x + 5024. -2x - 4 ≤ -5x + 1228. 2(x - 3) < 4x + 1029. -3(2x - 4) > 2(6x - 12)32. x + 5 ≥ x – 233. 6(3 - x) < 2x +1235. -21(2-x) + 3x > 4x + 438. -2(-5-x) > 3(x+2) +4 –x39. 5(2x + 3) ≥ 6 + (x + 2) - 2x40. -3(-2x +12) < -4(x+2) - 6
Think!!!!!!!!!!!!!41. When solving an inequality, if you obtain the result 3 < 5, what is the
solution?42. When solving an inequality, if you obtain the result 4 ≥ 2, what is the
solution?43. When solving an inequality, if you obtain the result 5 < 2, what is the
solution?44. When solving an inequality, if you obtain the result -4 ≥ -2, what is
the solution?45. When solving an inequality, under what conditions will it be
necessary to change the direction of the inequality symbol?
Practice Test (1-20), p#141
3.1 FormulasDefinition:
A formula is an equation commonly used to express a specific relationship mathematically.
Examples:1. The formula for the area of a rectangle is
area = length . Width or A = lw2. The formula for the perimeter of a square is
perimeter = 4. one side or P=4s3. The formula for the area of a triangle is
area = ½.base . Height or A=½bh
Simple Interest FormulaA formula used in banking is the simple interest formula, which is given byinterest = principal . rate . Time
Or i = prtHere ‘p’ is the principal (the amount invested or borrowed), ‘r’ is the interest rate, and ‘t’ is the amount of time of the investment or loan.This formula is used to determine the simple interest, i , earned on some savings accounts, or the simple interest an individual must pay on certain loan.
How to use the simple interest formula?
Example1: Avery borrows $2000 from a bank for 3 years. The bank charges 12% simple interest per year for the loan. How much the interest will Avery owe the bank?
Sol: Given thatThe principal, p, is $2000The rate, r , is 12% = 12/100=0.12
and the time, t , is 3 yearsUsing the formula i = prtPutting the corresponding values
i = 2000(0.12)(3)i = 720
The simple interest is $720. Thus Avery will pay $2720, after 3 years. (the principal, $2000 + the interest, $720).
Continue….Example: Amber invests $5000 in savings account which earns simple interest for 2
years. If the interest earned from the account is $800, find the rate.Sol: Here
Principal (investment) =p=$5000Time =t =2yearsInterest =i=$800rate =r=?Using the simple interest formula
i=prtSolving for ‘r’, we get r = i/ptPutting the values
r = 800/[(5000)(2)]r = 800/10000r = 0.08
Thus the simple interest rate is 0.08, or 8% per year.
Exercise Set 3.1, p#153In Exercise 73 through 76, use the simple interest formula.73. Mr. Thongsophapporr, borrowed $4000 for 3 years at 12% simple interest
rate per year. How much interest did he pay?74. Ms. Rodriguez lent her brother $4000 for a period of 2 years. At the end
of the 2 years, her brother repaid the $4000 plus $640 interest. What simple interest rate did her brother pay?
75. Ms. Levy invested a certain amount of money in a savings account paying 7% interest per year. When she withdrew her money at the end of 3 years, she received $1050 in interest. How much money did Ms. Levy place in the savings account?
76. Mr. O’Connor borrowed $6000 at 7½% simple interest per year. When he withdrew his money, he received $1800 in interest. How long had he left his money in the account?
Continue….Solve each equation for y; then find the value of y for the given value of x.
25. 2x + y = 8, when x = 228. -3x – 5y = -10, when x = 030. 15 = 3y – x, when x = 334. -12 = -2x – 3y, when x = -436. 2x + 5y = 20, when x = -5
Solve for the variable indicated.38. d = rt, for r42. V = lwh for w47. 4n + 3 = m, for n50. Y = mx + b, for x57. ax + by = c, for y
Continue….
Use the formula d = (1/2)n2 – (3/2)n, to find the number of diagonals in a figure with the given number of sides.
61. 10 sides62. 6 sides
Use the formula C = (5/9)(F - 32) to find the Celsius temperature (C) equivalent the given Fahrenheit temperature (F).
63. F = 500
64. F = 860
Continue….In chemistry the ideal gas law is P = KT/V, where P is the pressure, T is the temperature, V is the volume, and K is a constant. Find the missing quantity.
67. T = 10, K = 1, V = 168. T= 30, P = 3, K = 0.569. P = 80, T = 100, V = 570. P = 30, K = 2, V = 6
The sum of the first n even numbers can be found by the formula S = n2 + n. Find the sum of the numbers indicated.
71. First 5 even numbers.72. First 10 even numbers.
Changing Application Problems into Equations
Verbal Algebraic5 more than a number x+5A number increased by 3 x+37 less than a number x-7A number decreased by 12 x-12Twice a number 2xThe product of 6 and a number 6xOne-eighth of a number (1/8)x or x/8A number divided by 3 (1/3)x or x/34 more than twice a number 2x+45 less than three times a number 3x-53 times the sum of a number and 8 3(x+8)Twice the difference of a number and 4 2(x-4)
Express Relationships between Two Related Quantities
Verbal One number Second numberTwo Numbers differ by 3 x x + 3John’s age now and john’s age in 6 years
x x + 6
One number is six times the other
x 6x
One number is 12% less than the other
x x – 0.12x
A 25 foot length of wood cut in two pieces
x 25 - x
The sum of two numbers is 10
x 10 - x
84
Exercise Set 3.2, p#162/163Write as an algebraic expression.1. Five more than a number.2. Seven less than a number.3. Four times a number.4. The product of a number and eight.5. 70% of a number x.6. 8% of a number y.8. A 7½% sales tax on a car costing p dollars.9. The 16% of the U.S population, p, who do not receive adequate nourishment.10. Only 7% of all U.S tires, t, are recycled.11. Three less than six times a number.12. Six times the difference of a number and 3.13. Seven plus three-fourths of a number.14. Four times a number decreased by two.15. Twice the sum of a number and 8.16. Seventeen decreased by x.
Continue….17. The cost of purchasing x rolls of electrical tape at $4 each.18. The rental fee for subscribing to home box office for x months at $12 per month.19. The cost in dollars of traveling x miles at 23 cents per mile.20. The distance covered in t hours when traveling 30 miles per hour.22. The cost of waste disposal for y months at $16 per month.23. The population growth of a city in n years if the city is growing at a rate of 300
persons per year.24. The number of calories in x gram of carbohydrates if each gram of carbohydrates
contains 4 calories.25. The number of cents in x quarters. (Quarter=a fourth of a dollar)26. The number of cents in x quarters and y dimes.27. The number of inches in x feet.28. The number of inches in x feet and y inches.29. The number of ounces in e pounds.31. An average chicken egg contains 275 milligram of cholesterol and an ounce of
chicken contains about 25 mg of cholesterol. Write the amount of cholesterol in x chicken eggs and y ounces of chicken.
Continue….Express as a verbal statement. (There are many acceptable answers.)
33. x – 6 (six less than a number)34. x + 3 (three more than a number)35. 4x + 1 (one more than four times a number)36. 3x – 4 (four less than three times a number)37. 5x – 7 (seven less than five times a number)38. 2x – 3 (three less than twice a number)39. 4x – 2 (two less than four times a number)40. 5 – x (a number subtracted from five)41. 2 – 3x (three times a number subtracted from
two)42. 4 + 6x (four more than six times a number)43. 2(x - 1) (twice the difference of a number and one)44. 3(x + 2)(three times the sum of a number and two)
Continue….Select a variable to represent one quantity and state what that variable represents. Then express the second quantity in terms of the variable.
45. Eileen’s salary is $45 more than Martin’s salary.46. A boy is 12 years older than his brother.47. A number is one-third of another.48. Two consecutive integers.49. Two consecutive even integers.50. One hundred dollars divided between two people.51. Two numbers differ by 12.52. A number is 5 less than the four times another number.53. A number is 3 more than one-half of another number.54. A Cadilac costs 1.7 times as much as a Ford.55. A number is 4 less than three times another number.56. An 80-foot tree cut into two pieces.57. Two consecutive odd integers.58. A number and the number increased by 12%.59. A number and the number decreased by 15%.60. The cost of an item and the cost increased by a 7% sales tax.
Continue….Express as an equation.
69. One number is five times another. The sum of two numbers is 18.70. Marie is 6 years older than Denise. The sum of their ages is 48.71. The sum of two consecutive integers is 47.72. The product of two consecutive even integers is 48.73. Twice a number decreased by 8 is 12.74. For two consecutive integers, the sum of the smaller and twice the larger is 29.75. One-fifth of the sum of a number and 10 is 150.76. One train travels six times as far as another. The total distance traveled by both trains
is 700 miles.77. One train travels 8 miles less than twice the other. The total distance traveled by both
the trains is 1000 miles.78. A number increased by 8% is 92.79. The cost of a car plus a 7% tax is $13,600.80. The cost of a jacket at a 25%-off sale is $65.81. The cost of a meal plus 15% tip is $18.82. The cost of a videocassette recorder reduced by 20% is $215.83. The product of a number and the number plus 5% is 120.
3.3 Solving Application problems
To solve the word problem1. Read the question carefully.2. If possible draw a sketch to help visualize the problem.3. Determine which quantity you are being asked to find. Choose a letter
to represent this unknown quantity. Write down exactly what this letter represents. If there is more than one quantity, represent all unknown quantities in terms of this variable.
4. Write the word problem as an equation.5. Solve the equation for the unknown quantity.6. Answer the question or questions asked.7. Check the solution in the original stated problem.
Group Activity/Challenging Problems
1. To find the average of a set of values, you find the sum of the values and divide the sum by the number of values.
(a) If Paul's first three test grades are 74, 88, and 76, write an equation that can be used to find the grade that Paul must get on his fourth exam to have an 80 average.
(b) Solve the equation from part (a) and determine the grade Paul must receive.
ExponentsIn the expression xn, x is called the base and n is called the exponent. xn is read as “x to the nth power.”
xnEXPONENT
BASE
What an exponent represents??
An exponent tells us how many times a number is multiplied with itself.For example 105 means105 = 10 x 10 x 10 x 10 x 10
= 100000Similarly
73 = 7 x 7 x 7 = 343
LAWS OF EXPONENTSCertain mathematical operation can be done whenever we have variables which contain exponents and equal bases. These operations are called “laws of exponent”.These laws are
Product Rule for ExponentsQuotient Rule for ExponentsZero Exponent RulePower Rule for Exponent
Each one is explained as under
Product rule for exponents
xn . xm = xn+m
Thus when same bases are multiplied, the exponents are added
xn . x
m
Bases are same
x=n+m
Exponents are Added
Examples (Exercise Set 4.1, p#
195)Simplify.1. x2 . x4 = x6
2. x5 . x4 = x9
3. y . y2 = y3
4. 42 . 4 = 43 = 645. 32 . 33 = 35 = 2436. x4 . x2 = x6
7. y3 . y2 = y5
8. x3 . x4 = x7
9. y4 . y = y5
Quotient Rule for Exponents
0, xxxx nm
n
m
Bases are same
Exponents are subtracted
Since division by “0” is not allowed
101
Power Rule for Exponents
(xm)n = xm.n
The power rule indicates that when we raise an exponential expression to a power, we keep the base and multiply the exponents.
Exponential expression raise to a power
Keeping the base & multiplying the exponents
Expanded Power Rule for Exponents
0,0,
yb
ybxa
byax
mm
mmm
This rules illustrates that every factor within the parenthesis is raised to the power outside the parenthesis.
(As division by zero is not allowed)
104
Zero Exponent Rule
0,10 xxBy the zero exponent rule, any real number, except 0, raised to the zero power equals 1.
NOTE:00 is not a real number.This is called the undeterminedform.
Negative Exponents
2535
3 xx
xx
25
3 1....
..xxxxxx
xxxxx
5
3
xx
We will develop this rule as follows:Using Quotient rule we have
Again by dividing out common factors,
1.
2.
We see that is equal to both & . Therefore
must equal
2x2
1x
2x2
1x 107
Negative Exponents Rule
0,1 x
xx m
m
It means that when a variable or a number is raised to a negative exponent, the expressionmay be rewritten as 1 divided by the variableor number to the positive exponent.
Note: When we are asked to simplify an exponentialexpression, our final answer should contain no negativeexponent.Also when a factor is moved from the denominator to thenumerator or from numerator to the denominator, the signOf the exponent changes.
System of Linear equationBy the system of linear equations we mean a group of linear equations.Sometimes in business we deal with many variables and unknown quantities. For example a company consider overhead cost, cost of material, labor cost, maximum possible production, selling price of the item, and a host of other items when seeking to maximize their profit. The business may express the relationship between the variables in equation or inequalities. These equations or inequalities form a system of linear equations or inequalities. The solution of the system of equations or inequalities gives the values of the variables for which the company can maximize profit.
Examples
631072
yxyx
12306174
zyxzyx
105541243
31
32
321
xxxxxxx
23653
074132
21
21
21
21
xxxxxxxx
(2 variables, 2 equations) (3variables, 2 equations)
(3 variables, 3 equations) (2 variables, 4 equations)In this course we will study linear system of equations in two variables111
Solution of System of Equations
By the solution of linear system we mean the order pair(s), which satisfy all the equations in the system simultaneously.e.g. Consider the system
It has the solution (1,6), since this order pair satisfy both the equations in the system. But (2,7) is not a solution to the system as it satisfy the first equation but not the second.
425
xy
xy
112
Continue…A system can be solved by the following methods:
1. The graphical method2. The substitution method3. The addition method
While solving a linear system, we face one of the following three situations:
a) The system is consistent (sol: exists)b) The system is inconsistent (no sol:)c) The system is dependent (infinite sol:)
Continue….Consistent, Inconsistent, Dependent system
Line 1 Line 2
y
xLine 1
Line 2
y
x
Line 1
Line 2
x
y
(a)
(b)
(c)
Consistent System
Inconsistent SystemDependent System
. Solution
Solution by Graphical MethodTo solve a linear system of equations graphically, we graph each equation and determine the point(s) of intersection.We will not present here the solution by graphical method, because often the solution by this method may be incorrect since we have to estimate the coordinates of the point of intersection. Thus we follow the other two methods.
Solution of System of equations by Substitution
To solve a system of equations by substitution1. Solve for a variable in either equation. (To avoid working with
fractions, prefer to solve for the variable with the numerical coefficient of 1).
2. Substitute the expression found for the variable in step 1 into the other equation.
3. Solve the equation in step 2 to find the value of one variable.4. Substitute the value found in step 3 into the equation found in step1 to
find the other variable.
Exercise Set 8.2 (p# 454)
1.
13242
yxyx
x
Solve the system of equations by substitution
Sol:
(1)
(2)Solving (1) for
42 yx (3)Putting this value of in (2)x
1178134813)24(2
yyyyyy
Putting in (3), we get1y 2xThus the solution is (2,1) 117
Solution by Addition Method• This method is also called the elimination method.• This is often an easier method to solve a linear system of
equations.• In this method we add or subtract the given equations to
get a third equation which contains only one variable.• Note that our immediate goal is to obtain one equation
containing only one unknown.
The Method1. If necessary, shift all the terms containing variables to the left side of
the equal sign and the constants to the right side of the equal sign.2. If necessary, multiply one or both equations by a constant(s) so that
when the equations are added the resulting sum will contain only one variable.
3. Add the equations. This gives a single equation which contains only one variable.
4. Solve the equation obtained in step 3.5. Put the value obtained from step 3 into either of the original
equation. Solve this equation for the remaining variable.
CHAPTER NO. 09
ROOTS AND RADICALS
• In this chapter the concentration will be on the radical equations and their solutions.
Definition• An equation that contains a variable in a radicand is called a radical equation.
n bax )( RADICAND
RADICAL SIGNINDEX
HOW TO SOLVE A RADICAL EQUATION? (square root term)• Use the appropriate properties to write the equation
with the square root term by itself on one side of the equation. We call this isolating the radical.
• Combine like terms.• Square both sides of the equation to remove the square
root.• Solve the equation for the variable.• Check the solution in the original equation for the
extraneous roots ( a number obtained when solving an equation that is not a
solution to the original equation).
Exercises
2
1. 8 2. 5
3. 5 4. 3 6
5. 4 8 6. 3 5
7. 3 4 2 8. 6 4
9. 2 8 10. 5 3
11. 2 4 2 12. 4 9
13. 8 2 14. 2 5 4
x x
x x
x x
x x x
x x
x x
x x x x
TO SOLVE A RADICAL EQUATION WITH TWO SQUARE ROOT TERM
• Rewrite the equation, when necessary, so that there is only one square root term on each side of the equation.
• Square both sides of the equation and solve to find the solution.
Continue…• An equation of the form
is called a quadratic equation.• Since this is a polynomial equation of degree 2,
therefore it has two solutions.• These solutions may or may not equal, may or
may not real.
0,02 acbxax
Continue…• Such equations can be solved using
a) Factorizationb) Completing squarec) Quadratic formulaWe will solve these equations using (b) &(c). EXAMPLES of Quadratic Equations:
079
072
0543
2
2
2
xx
x
xx
Completing Squarei. Make the numerical coefficient of the squared term equal to
1.ii. Shift the constant term of the equation to the right side of the
equal sign.iii. Take one-half of the numerical coefficient of the 1st degree
term, square it, and add this quantity to both sides of the equation.
iv. Replace the trinomial with its equivalent squared binomial.v. Use the square root property to eliminate the square.vi. Solve for the variable.vii. Check your answer in the original equation.
Solution of Quadratic Equation by Quadratic Formula
• Given the quadratic equation
The solution to this equation is given by:
known as the Quadratic formula (Muhammad Bin Musa formula).
0,02 acbxax
aacbbx
242
Continue…• The radicand is known as the
‘Discriminant’.• Three cases arise while solving a Quadratic
equation by the Quadratic formula.a) When (two distinct & real
solutions exist)b) When (repeated solutions are
there)c) When (no real solution exist)
acb 42
042 acb
042 acb
042 acb
Remember that• If the discriminant is a negative number
We can’t go further. The equation has no real number solution. For such type of equations, your answer should
be “no real number solution”. Don’t leave the answer blank. Don’t write 0 for the answer.