16
Chapter 1 | Review of Basic Arithmetic
1.3 |
Fractions
­­­Definition of Fractions and Types of Fractions
A fraction is a rational number written as one integer divided by another non-zero integer. It
is usually written as a pair of numbers, with the top number being called the numerator and the
bottom number the denominator. A fraction line (horizontal bar) separates the numerator and the
denominator. The use of fractions is another method of representing numbers.
numerator
3
8
division sign
denominator
3
(
) is a fraction. It is read as "three divided by eight", "three-eighths", or "three
8
over eight", which all indicate that 3 is the numerator and 8 the denominator.
For example,
A proper fraction is a fraction in which the absolute value of the numerator is less than the
absolute value of the denominator; i.e., the absolute value of the entire fraction is less than 1.
For example,
The absolute value of a number
refers to the positive sign of that
number.
For example, the absolute value
of -2 is 2.
i.e. |-2| = 2.
■■83 ( ) is a proper fraction.
■■-52 in absolute value is 25 (
) which is a proper fraction.
-2
Therefore, 5 is a proper fraction.
(Note: the larger the denominator, the smaller the size of each piece.)
An improper fraction is a fraction in which the absolute value of the numerator is greater than
the absolute value of the denominator; i.e., the absolute value of the entire fraction is more than 1.
7
-3
For example, (
) and
( ) are improper fractions.
4
2
(negative three halves)
(seven quarters)
A mixed number consists of both a whole number and a fraction, written side-by-side, which
implies that the whole number and proper fraction are added.
For example, 3 58 implies 3 + 5 8
three
A complex fraction is
denominator.
1
For example,
,
3
b l
4
five-eighths
a fraction in which one or more fractions are found in the numerator or
2
c m
35
3
, and 6 are complex fractions.
6
1
c m
8
­Reciprocal of a Fraction
When the numerator and denominator of a fraction are interchanged, the resulting fraction is
called the reciprocal of the original fraction.
4
7
For example, the reciprocal of is
7
4
Note: The reciprocal of a fraction is not its equivalent fraction.
­­­Whole Number as a Fraction
Any whole number can be written as a fraction by dividing it by 1 (i.e., 1 is the denominator).
7
For example, 7 =
1
Chapter 1 | Review of Basic Arithmetic
­­­Converting a Mixed Number into an Improper Fraction
­­­­­­
Follow these steps to convert a mixed number into an improper fraction:
■■Multiply the whole number by the denominator of the fraction and add this value to
the numerator of the fraction.
■■The resulting answer will be the numerator of the improper fraction.
■■The denominator of the improper fraction is the same as the denominator of the
original fraction in the mixed number.
29
3 ^ 8h + 5
=
For example,
3 58 =
8
8
3 # 8 = 24 pieces 5 pieces
Thus, there is a total of 29 pieces,
each piece being one-eighth in size.
­­­Converting an Improper Fraction into a Mixed Number
Follow these steps to convert an improper fraction into a mixed number:
■■Divide the numerator by the denominator.
■■The quotient becomes the whole number and the remainder becomes the numerator
of the fraction.
■■The denominator is the same as the denominator of the original improper fraction.
For example,
29 3 5
= 8 Because,
8
3
8g 29
24
5
Quotient: whole number of the fraction
Remainder: numerator of the fraction
­­­Converting a Fraction into an Equivalent Fraction
When both the numerator and denominator of a fraction are either multiplied by the same number
or divided by the same number, the result is a new fraction called an equivalent fraction. Equivalent
fractions imply that the old and new fractions have the same value.
2
:
For example, to find 2 equivalent fractions of 5
2
2#2
=
Multiplying both numerator and denominator by 2, we obtain,
5
5#2
4
=
10
2
2#3
=
Multiplying both numerator and denominator by 3, we obtain, 5
5#3
= 6
15
4
6
2
and
are equivalent fractions of .
10
15
5
36
:
Consider another example to find 2 equivalent fractions of
60
36 ' 2
36
=
Dividing both numerator and denominator by 2, we obtain,
60 ' 2
60
18
=
30
36
36 ' 4
=
Dividing both numerator and denominator by 4, we obtain,
60
60 ' 4
= 9
15
36
18
Therefore,
and 9 are equivalent fractions of
.
60
30
15
Therefore,
17
18
Chapter 1 | Review of Basic Arithmetic
­Converting a Fraction to its Decimal Form
A proper or improper fraction can be converted to its decimal form by dividing the numerator by
the denominator.
2
For example,
= 0.4, 13 = 1.625
5
8
A mixed number can be converted to its decimal form by first converting it to an improper fraction
then dividing the numerator.
For example,
3 (2) + 5
11
3 25 =
=
= 5.5
2
2
11 (7) + 3
80
11 37 =
=
= 11.428571...
7
7
■■
■■
­­­Reducing or Simplifying a Fraction
Dividing both the numerator and denominator of a fraction by the same number, which results in an
equivalent fraction, is called reducing or simplifying the fraction.
For example, we can simplify 16 as shown:
20
16 ' 2
16
=
Dividing both numerator and denominator by 2, we obtain,
20 ' 2
20
=
8
10
16 ' 4
16
=
Dividing both numerator and denominator by 4, we obtain,
20 ' 4
20
4
= 5
8
4
16
.
Therefore, and are reduced fractions of
10
5
20
­­­Fraction in Lowest (or Simplest) Terms
A fraction in which the numerator and denominator have no factors in common (other than 1) is
said to be a fraction in its lowest (or simplest) terms. Any fraction can be fully reduced to its
lowest terms by dividing both the numerator and denominator by the highest common factor (HCF).
Example 1.3(a)
Reducing Fractions to their Lowest Terms
40
Reduce the fraction
to its lowest terms.
45
Solution
The factors of 40 are 1, 2, 4, 5, 8, 10, 20, 40.
The factors of 45 are 1, 3, 5, 9, 15, 45.
The HCF is 5.
Therefore, dividing the numerator and denominator by the HCF, 5, results in the fraction in its
lowest terms: 40
40 ' 5
=
= 8
45
45 ' 5
9
Comparing Fractions
Fractions can easily be compared when they have the same denominator. If they do not have the
same denominator, find the LCD of the fractions, then convert them into equivalent fractions with
that LCD as their denominator.
Chapter 1 | Review of Basic Arithmetic
Example 1.3(b)
Comparing Fractions
Which of the fractions,
Solution
5
3
or is greater?
12
8
We first find the LCD of the fractions, which is the same as the LCM of the denominators.
The LCM of 12 and 8 is 24.
Now convert each of the fractions to its equivalent fraction with 24 as the denominator.
Multiply the denominator by 2 to obtain the LCD of 24 and multiply the numerator by 2 as well to
maintain an equivalent fraction.
5#2
5
10
=
=
12 # 2
12
24
3
Similarly, convert to an equivalent fraction with 24 as the denominator.
8
3#3
3
9
=
=
8#3
8
24
Since the denominators are the same, we can now compare the numerators of the above to
identify the greater fraction.
10
9
5
3
2
2 .
; therefore,
10 > 9, which implies that
24 24
12 8
>
Basic Arithmetic Operations with Fractions
It is best to convert a mixed number into an improper fraction before performing any basic arithmetic
operations.
Adding Fractions
Addition of fractions requires that the denominators of every fraction be the same. To make them
the same, first find the LCD and change each fraction to its equivalent fraction having the same
denominator. Now, add the numerators of each of the equivalent fractions. The resulting fraction
will have the common denominator, and its numerator will be the result of adding the numerators
of the equivalent fractions.
Example 1.3(c)
Solution
Adding Fractions
4
Add 3 56 and
9
36 + 9 =
^3 # 6h + 5 4
+
6
9
=
23 4
+ 6
9
Finding the equivalent fraction using the LCD of 18, we obtain,
=
69
8
77
+
=
18 18
18
Converting the improper fraction to a mixed number, we obtain,
5
= 4 18
5
4
Subtracting Fractions
The process for subtraction of fractions is the same as that of the addition of fractions. First, find
a common denominator, then change each fraction to its equivalent fraction with the common
denominator. The resulting fraction will have that denominator, and its numerator will be the result
of subtracting the numerators of the original fractions.
19
20
Chapter 1 | Review of Basic Arithmetic
Example 1.3(d)
Subtracting Fractions
Subtract 2 from 12 21
3
Solution
12 21 -
^12 # 2h + 1 2
2
=
3
2
3
=
25 2
- 2
3
Finding the equivalent fraction using the LCD of 6, we obtain,
=
71
75 4
- =
6
6
6
Converting the improper fraction to a mixed number, we obtain,
= 11 56
Multiplying Fractions
When multiplying two or more fractions, first simplify the fractions, if possible, then multiply the
numerators to get the new numerator and multiply the denominators to get the new denominator.
Example 1.3(e)
Multiplying Fractions
Multiply:
(i)
3
4
# 2
11
(ii) 9 # 4
13
(iii) 7 # 4 12
21
Solution
(iv) 15 # 2
5
2
6
(i) 3 # 4 = 3 # 4 = 3 # 2 =
1 11 11
2 11 1 2 11
(ii) 9 # 4 = 9 # 4 = 36
13 1
13
13
1
1
(iii) 7 # 4 = 7 # 4 = 1 # 1 = 1
9
12 21 12 21 3 3
3
3
3
(iv) 15 # 2 = 15 # 2 = 3 # 2 = 6 = 6
5
1
5
1 1
1
1
Dividing Fractions
The division of fractions is done by multiplying the numerator by the reciprocal of the denominator.
5
c m
When a fraction is inverted, the
5
7
5
9
15
3
45
resulting fraction is called the
For example, = ' = # =
=
3
9
3
7
7
7
21
‘reciprocal’ of the original fraction.
b l
9
9
7
7
Note: Dividing by is the same as multiplying by the reciprocal of , which is .
7
9
9
When a fraction is a mixed number, convert it to an improper fraction and continue with the
arithmetic operation.
9
2
1
b l
2 41
9
45
9
8
2
4
For example, =
=
'
=
#
=
5
4
8
4
45
5
45
58
c m
1
5
8
Chapter 1 | Review of Basic Arithmetic
Example 1.3(f)
Solution
Dividing Fractions
3
by 1 45
Divide
20
3
' 1 45
20
3 9
3 9
3 9
' Multiplying
'
=
by the reciprocal of' , we obtain,
20 5
20 5
20 5
=
1
1
1
3 5
# = 3 #5 =
20 9 4 20 9 3 12
Converting a Complex Fraction into Proper or Improper Fraction
­­­A complex fraction can be converted to a proper or improper fraction by dividing the numerator
by the denominator and then simplifying the expression.
For example,
■
1.3 |
7
c m
7
7 1
7
2
= '5 = # =
2
2 5
10
5
■
8
9
2
16
= 8 'b l = 8 # c m=
9
2
9
9
b l
2
Exercises Answers to the odd-numbered problems are available at the end of the textbook
1. Convert the following mixed numbers into improper fractions:
3
a. 1 83 b. 12 4 c. 7 53 d. 9 23
2. Convert the following mixed numbers into improper fractions:
a. 5 23 b.15 67 c. 6 45 d. 4 43
3. Convert the following improper fractions into mixed numbers:
12
17
31
a.
b.
c.
7
8
9
4. Convert the following improper fractions into mixed numbers:
46
a. 23 b. 35 c.
5
12
6
5. Find the missing values:
?
15 3
12
?
2
a. =
=
b.
= =
35
25 ?
12
3
?
?
22
55
c. 12 = 6 =
?
d.
14
5
d. 34
3
d.
75
25
?
=
=
45
?
18
d.
?
9
27
=
=
20
4
?
6. Find the missing values:
a.
?
5
20
=
=
24
8
?
b.
?
12
4
=
= 18
9
?
c.
?
36
30
=
=
42 14
?
7. Reduce the following to their lowest fractions:
80
a. 12 156
b. 18 8. Reduce the following to their lowest fractions:
a. 70 b. 225 15
30
9. Which of the following fractions is greater:
a. 8 or 13 ?
b. 5 or 16 ?
7
13
39
12
68
c. 10 36
d. 144
c. 124 48
d. 75
345
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22
Chapter 1 | Review of Basic Arithmetic
10. Which of the following fractions is smaller:
a. 2 or 3 ?
b. 12 or 35 ?
15
45
8
5
11. Perform the addition, reducing to the lowest fraction:
a. 5 23 + 1 125 b. 7 + 5 c. 11 + 8
20 10
9
6
12. Perform the addition, reducing to the lowest fraction:
a. 16 81 + 1 21 1
3
b. 3 + 8 c. 9 + 4
12
5
13. Perform the subtraction, reducing to the lowest fraction:
a. 18 57 - 2 25 b. 21 - 1 c. 35 – 3 18 6
13 3
14. Perform the subtraction, reducing to the lowest fraction:
a. 12 43 - 5 31 b. 9 - 1 10
2
c. 21 - 7 25
8
15. Perform the multiplication, reducing to the lowest fraction:
4 23
6 19
1 a. 11 43 # 1 74
b. 5 # 9 c. 9 # 12
16. Perform the multiplication, reducing to the lowest fraction:
a. 9 53 # 1 29
96
3
5
b. 8 # 11 c. 3 # 7
9
17. Perform the division, reducing to the lowest fraction:
1 3
8 2
a. 10 41 ' 2 27
b. 12 ' 4 c. 7 ' 5
48
18. Perform the division, reducing to the lowest fraction:
10 3
3
13 a. 23 21 ' 8 16
b. 15 ' 7 c. 8 ' 4
For Problems 19 to 40, express your answers as a proper fraction or a mixed number, where appropriate.
19. Peter spent two-thirds of his money on rent and food and one-fourth on education. Together, what fraction of the money did
he spend on rent, food, and education?
20. Tracy invested one-fifth of her savings in the stock market and two-thirds in real estate. Together, what fraction of her savings
did she invest in the stock market and real estate?
21. Lily worked 5 21 hours, 6 41 hours, and 3 43 hours over the last three days. How many hours did she work in total over the three
days?
22. A rain gauge collected 3 32 inches, 1 41 inches, and 2 21 inches of rain over the past three months. What was the total rainfall
over the three months?
23. A wooden board measured 2 21 meters in length. It was shortened by cutting 1 85 meters from it. What is the new length of the
board?
24. A tank had 4 52 litres of water. If 1 32 litres leaked from the tank, how much water was left in the tank?
25. David spent one-fourth of his money on rent and one-third of the remainder on food. What fraction of his money was spent on food?
26. Mary spent two-fifths of her money on books and one-third of the remainder on clothes. What fraction of her money was
spent on clothes?
27. After selling two-fifths of its textbooks, a bookstore had 810 books left. How many textbooks were in the bookstore initially?
28. Rose travelled two-thirds of her journey by car and the remaining 20 km by bus. How far did she travel by car?
Chapter 1 | Review of Basic Arithmetic
29. Cheng walked 5 41 km in 1 21 hours. How many kilometres did he walk in 1 hour?
30. It took 15 41 hours to complete three-fourths of a project. How long did it take to complete the entire project?
31. The product of two numbers was 9. If one number is 3 43 , what was the other number?
32. If a wire that is 43 43 cm long is cut into several 1 41 cm equal pieces, how many pieces were there?
33. A stack of plywood sheets measures 49 21 inches high. If each plywood sheet is three-fourths of an inch thick, how many
sheets of plywood are in the stack?
34. A garment factory has 40 41 metres of cotton fabric. If 1 43 metre of the fabric is required for a dress pattern, how many
dresses can be made?
35. A bottle contained 80 mg of medicine. Each dose of the medicine is 2 21 mg. How many doses were there in the bottle?
36. It took two-thirds of an hour for a machine to make one component. How many components can be made in 40 hours?
37. A company identified one-twentieth of the 320 bulbs that it received from a supplier as being defective. How many bulbs
were not defective?
38. Matthew received a bonus of $6850. He spent two-thirds of this amount on a vacation. How much did he have left?
39. Three software programmers worked 17 21 hours, 25 43 hours, and 11 41 hours each, to develop an e-commerce site. If each
of them was paid $18 per hour, how much did they receive in total?
40. It took three consultants 27 43 hours, 21 41 hours, and 18 21 hours each, to design a product. If each of them was paid $55
per hour, what was the total amount paid to them?
1.4 |
Order of Operations (BEDMAS)
When arithmetic expressions contain multiple operations with brackets, exponents, divisions,
multiplications, additions, and subtractions, the arithmetic operation is performed in the following
sequence:
1. Perform all operations within the brackets. If there is more than one bracket, start with the
innermost bracket and move outwards to complete all the brackets.
2. Perform operations with exponents.
3. Perform the necessary divisions and multiplications in the order in which they appear from
left to right.
4. Complete the operation by performing the necessary additions and subtractions in the order
in which they appear.
The order of operations: Brackets, Exponents, Divisions, Multiplications, Additions, Subtractions
can be remembered by the acronym, BEDMAS.
Example 1.4(a)
Computing Arithmetic Expressions by Following the Order of Operations.
Compute the following arithmetic expressions:
(i) 6 + 4 # 50 ' (8 - 3)2 - 1
(ii) 12 + 32 [(8 # 5) / 5] - 7 + 2
23