1.6 BOMDAS
When a statement contains more than one operation, always solve the operations in this order:
Brackets, Multiplication, Division, Addition, Subtraction.
Why do you think we call this order BODMAS?
Brackets of
Multiplication and Division
Addition and Subtraction
e.g
For example: 10 × (20 - 5) + 4
First: Work out the brackets. 20 - 5 = 15, so we get 10 × 15 + 4.
Second: Work out the multiplication. 10 × 15 = 150.
Third: Work out the addition. 150 + 4 = 154.
Exercise
Work these out without using a calculator.
a
70 - 20 × 2 =
30
b
21 - (16 - 20) ÷ 2 =
23
c
12 × (3 + 17) ÷ 2 + 1 =
121
d
(100 + 13) ÷ (6 + 4) =
10 130
1.7 Problem Solving
1- Understand the problem.
2- Devise a plan.
3- Carry out the plan.
4- Look back.
1- A farmer sold 500 boxes of oranges, 1711 boxes of apples, 1202 boxes of mangoes and 777 boxes of lime. How many boxes did the farmer sell altogether?
Answer: 4,190 boxes
(500 + 1711 + 1202 + 777 = 4190)
2- A box contains 13 rows of eggs with 14 eggs in each row. The shopkeeper rearranges the eggs in a different box, in rows of 7. How many eggs are now in each row?
Answer: 26 eggs
(Total eggs = 13 × 14 = 182; 182 ÷ 7 = 26)
3- Work these out.
a
-122 + -33 =
-155
b
36 - 24 ÷ 10 × 2 =
31.2
c
18 ÷ (10 - 12) =
-9
d
53 - (2 + (0.25 - 1)) =
51.75
1.7 Problem Solving (Continued)
4- Christmas balls arrived in a big box. The box had 48 rows, with 16 balls in each.
a) How many balls were there altogether?
768 balls
(48 × 16 = 768)
b) The shopkeeper packaged 5 balls per box. How many boxes did she make, and how many balls were left over?
153 boxes, 3 left over
(768 ÷ 5 = 153 R 3)
5- If a car moves forward, it travels a positive distance, and if it moves in reverse, it travels a negative distance. The car starts from zero (0) and moves forward 5 metres. Then, it moves 8 metres in reverse. Demonstrate the car movements on a numbers line.
Final position: -3 metres
(+5 + (-8) = -3)
6- The centigrade thermometer is marked for temperatures above the freezing point of water. Complete the marking on the thermometer below the freezing point of water.
Below 0°C markings:
-10°C
-20°C
-30°C
-40°C
-50°C
Unit 2
2.1 The Simplification of Fractions
Fractions
A fraction is described as a part of a whole. The number on the bottom of the fraction is called the denominator. The number on the top of the fraction is called the numerator. In the fraction 34, 3 is the numerator and 4 is the denominator.
1 Proper Fractions
are fractions whose numerators are smaller than their denominators.
12 ,
34 ,
25
2 Improper Fractions
are fractions whose numerators are equal to or greater than their denominators.
33 ,
32 ,
125
Equivalent Fractions: Have the same value, even though they may look different. We can find equivalent fractions by
multiplying or
dividing the numerator and the denominator by the same number.
by multiplying: 12 = 24
by dividing: 68 = 34
Exercise
Complete the equivalent fractions.
a
48 = 1224
b
621 = 1863
c
1827 = 69
d
-1232 = -38
Simplifying a Fraction
This means that there is no number, except for (1), that can be divided evenly into both the numerator and the denominator to reduce a fraction to its lowest terms. To simplify a fraction, we divide the numerator and denominator by their greatest common factor (GCF).
Method
To write 2025 in its simplest form:
• Factors of 20: 1, 2, 4, 5, 10, 20
• Factors of 25: 1, 5, 25
• GCF = 5
Divide numerator and denominator by GCF:
20 ÷ 525 ÷ 5 = 45
Exercise
Reduce the following to their simplest form.
a
618 =
13
b
812 =
23
c
912 =
34
d
32120 =
415
2.2 Mixed Numbers
Numbers such as 6 23 are called mixed numbers.
Rule 1
To Change a Mixed Number into a Fraction:
Multiply the whole number by the denominator and add the numerator to the product, keeping the denominator without changing.
6 23 =
(6 × 3) + 23 =
18 + 23 =
203
Rule 2
To Change an Improper Fraction into a Mixed Number:
Divide the numerator by the denominator. Write the remainder over the divisor.
133 = 13 ÷ 3 = 4 with remainder 1 =
4 13
Fractions on the Numbers Line
We can always find a fraction between two whole numbers by using a numbers line.
003
13
23
133
43
1 2/353
263
73
83
Exercise
1 Change the following mixed numbers into improper fractions:
a
3 27 =
237
b
11 19 =
1009
c
4 23 =
143
d
16 34 =
674
2 Change the following improper fractions into mixed numbers:
a
534 =
13 14
b
257 =
3 47
c
4810 =
4 45
d
368 =
4 12
2.3 Comparing Fractions
To compare fractions, we rewrite them as equivalent fractions with the same denominators. If the denominators are the same, the fraction with the greater numerator is the greater fraction.
e.g 1
Between 34 and 24: Denominators are equal, and 3 > 2, so:
34 > 24.
e.g 2
Between 34 and 23: Change to same denominator (12):
34 = 912 and
23 = 812.
Therefore: 34 > 23.
Exercise
Use > or < or = to make the following statements true.
a
-34
<
-210
b
25
<
712
c
1731
>
514
d
-1120
<
-12
2.4 Operations with Fractions
Add & Subtract
Be sure that you have the same denominator when adding and subtracting. Then add or subtract the numerator. You may need to simplify your answer after you combine the two fractions.
e.g
• 57 + 17 = 67
• 34 - 12 = 34 - 24 = 14
Exercise
Solve each problem. Write your answer as a mixed number (if possible).
a
47 - 132 =
-5 1314
b
-514 - (-97) =
1314
c
47 + 4 45 =
5 1335
d
-1225 - 1025 + 2 13 =
1 3475
e
3 813 - 6 57 - (-12) + 57 =
-1 2326
To Multiply & Divide Fractions
To Multiply Two Fractions:
1. Multiply the numerator by the numerator.
2. Multiply the denominator by the denominator.
Formula: ab × cd = a × cb × d
e.g
Multiply 14 × 49 =
1 × 44 × 9 =
436 =
19
To Divide Two Fractions:
To divide by a fraction, multiply by the reciprocal.
Formula: ab ÷ cd =
ab × dc =
adbc
Reciprocals
When the product of multiplying two fractions is 1, the numbers are called reciprocals of each other.
23 is the reciprocal of 32 because
23 × 32 = 66 = 1.
Exercises
1 Add.
a
13 + 23 =
1
b
12 + 510 =
1
c
1116 + 24 + 78 =
2 116
d
9 38 + 3 416 =
12 58
2 Subtract.
a
916 - 716 =
18
b
79 - 13 =
49
c
11 34 - 5 23 =
6 112
d
3 710 - 2 12 =
1 15
3 Multiply.
a
916 × 815 =
310
b
56 × 1875 × 1027 =
227
c
9 38 × 5 15 =
48 34
d
7 19 × 1 1516 × 37 =
5 1921
4 Write the reciprocal of the following.
a
78 →
87
b
6 79 →
961
c
4 18 →
833
d
154 →
415
5 Divide.
a
38 ÷ 14 =
1 12
b
9 ÷ 16 =
54
c
83 ÷ 45 =
3 13
d
58 ÷ 7 =
556