Lesson Notes By Weeks and Term - Junior Secondary School 1
MULTIPLICATION AND DIVISION
MULTIPLICATION AND DIVISION
SUBJECT: MATHEMATICS
CLASS: JSS 1
DATE:
TERM: 1st TERM
WEEK EIGHT
TOPIC: MULTIPLICATION AND DIVISION
Introduction
Multiplication of fractions is simply a direct method compared to division of fractions. In multiplication, there is direct multiplication of the numerator of one fraction with the other and the denominator with the other.
Division is usually technical as there is reversal of the sign ( ÷) to multiplication sign 9x ) thereby leading to the reciprocal of the right-hand value.
A x x = A x α
B y B x y
But A ÷ x = A x y = Ay
B y B x Bx.
BODMAS
When signs are combined as a result of combination of fractons, it is therefore important to apply some rules that will enable us know where to start from. Such guide is BODMAS. It states that when there is combination of signs, they should be taken in order of their arrangement
B= bracket
O= of
D= division
M= multiplication
A= addition
S= subtraction.
Simple example
Example 1
Simplify the following;
(a) 1 3/5 x 6 (b) 4/11 of 3 2/3
(c) 3 ¾ x4/9 x 1 1/5 (d) 12/25 of (1 ¼ )2
Solution
(a) 1 3/5 x 6 (b) 4/11 of 3 2/3
= 8/5 x 6 = 4/11 x 11/3
= 8 x 6 = 4/3
5 x 1 = 1 1/3
= 48
5
= 9 3/5
15/4 x 4/9 x 6/5 12/25 x 5/4
15 x 6 = 12/25 x 5/4 x 5/4 =12/16 = 2
9 x 5
Example 2
Simplify (a) 7 1/5 ÷ 25 (b) 12/25 ÷9/ 10
(c) 7 7/8 ÷ 6 5/12
Solution.
(a) 7 1/5 ÷ 25 (c ) 7 7/8 ÷ 6 5/12
= 36/5 ÷ 25/1 = 63/8 ÷ 77/12
= 36/5 x 1/25 = 63/8 x12/77
= 36/125
12/25 x 10/9 2 x 11
4 x 2 = 27/22
5 x 3 = 1 5/22
= 8/15.
Example 3
Simplify
(a) 3/10 x 35/36 (b) 5 ¼ ÷ 2 /5
14/15 3 ¾
= 7 x 1 21/4 ÷ /5
2 x 12 15/4
7/24 ÷14/15 =21/4 x 5/14
15/14
7/24 x 15/14 = 15/8
15/4
= 15 = 15/8 ÷ 15/4
24 x 2
= 5 = 15 x 4
8 x 2 8 15
= 5 = 4
16 8 = ½
5/16.
EVALUATION
Simplify the following ;
11 2/3
9 ¾ x 2/3
READING ASSIGNMENT
1.Essential mathematics for jSSI by AJS Oluwasanmipg 52
Harder examples
Example I
Simplify the following fractions
Solution
5/8 x 1 3/5 b ¾ of 3 3/7
5/8 x 8/5 =3/4 x 24/7
5 x 8 = 3 x 24
8 x 5 4 x 7
= 1 3 x 6
7 = 18/7 = 2 4/7
c.9/16 ÷ 3 3/8
9/16 ÷ 27/8
9/16 x18/27 = 6/16 = 3/8.
Example 2.
Simplify 2 4/9 x 1 7/8 ÷ 2 1/5
Solution
2 4/9 x 1 7/8 ÷ 2 1/5
= 22/9 x 15/8 ÷ 11/5
BODMAS: application
= 22/9 x 15/8 x 15/11
5 x 5
3 x 4
= 25/12
= 2 1/12.
Example 3
Simplify 3 ¾ ÷ ( 2 1/7 of 11 2/3 – 5)
Solution
3 ¾ ÷ ( 2 1/7 of 11 2/3 – 5)
BODMAS – application ( the bracket first0
= 3 ¾ ÷ ( 15/7 of 35/5 – 5)
3 ¾ ÷ (15/7 x 35/3 – 5/1)
see BODMAS also multiplication first
3 3/3 ÷ ( 5 x 5 – 5)
15/4 ÷ ( 25 – 5)
15/4 ÷ 20/1
= 3
4 x 4
3
1 6
EVALUATION
Simplify the following
III. Word problems
Example I
What is the area of a rectangle of length 12 2/3m and breadth 7 ¼ m?
Solution
7 ¼ m
12 2/3m
Area of rectangle, A = L x B
L = 12 2/3m, B= 7 ¼
Area = 12 2/3 x 7 ¼
Area = 28/3 x 29/4
Area = 19 x 29
6
Area = 551
6
Area = 91 5/6m2
Example 2
Divide the difference between 4 1/5 and 2 2/3 by 1 2/5.
Solution
Interpreting the question
= 4 1/5 – 2 2/3
1 2/5
= ( 21/5 - 8/3 ) ÷ 1 2/5
( 21/5 x 3/3 – 8/3 x 5/5 ) ÷ 1 2/5
( 63/15 - 40/15) ÷ 7/5
( 63 – 40 ) ÷7/5
15
23/15 ÷ 7/5
23/15 x 5/7
23/21
= 1 2/21
Example 3
What is three-quarters of 3 3/7 ?
Solution
= three-quarter = ¾
= ¾ of 3 3/7
= ¾ x 24/7
= 3/1 x 24/7
3/1 x 6/7.
Example 4
In a school, 9/10 of the students play sports. 2/3of these play football. What fraction of the students play football.
Solution
Fraction who play sports = 9/10
Fraction that play football = 2/3 of 9/10.
2/3 x 9/10
1 x3
5
=3/5.
:.3/5 of the students play football.
Example 5
Three sisters share some money. The oldest gets 5/11 of the money. The next girl gets 7/12 of the remainder. What fraction of the money does the youngest girl get?
Solution
Let the total money; be a unit = 1
1st girl gets = 5/11 of 1 = 5/11
remainder = 1 -5/11 = 11 – 5 = 6/11
11
2nd girl gets = 7/12 of the remainder
=7/12 of 6/11
= 7/12x 6/11 = 7/22
3rd girl will get 6/11 – 7/22
= 6/11 x 2/2 – 7/22
= 12/22 – 7/22
= 12 – 7
22
= 522
:.the fraction of the money that the youngest girl will get = 522
EVALUATION
A prime number is a number that has only two factors, itself and 1. Some examples are 2,3,5,7,11,13,…
1 is not a prime number because it has only one factor, that is, itself unlike 2 which has itself and 1 as its factors. All prime numbers are odd numbers except 2 which is an even number.
Example 1
Write out all prime numbers between I and 30
Solution
Between 1 and 30.
Prime numbers = 2,3,5,7,119………..
1 is not a prime number because it has only one factor, that is, itself unlike 2 which has itself and 1 as its factors. All prime numbers are odd numbers except 2 which is an even number.
Example 1
Write out all prime numbers between I and 30
Solution
Between 1 and 30.
Prime numbers = 2,3,5,7,11, 13, 17,19,23,29.
A factor of a given number is a number which divides the given number without leaving any remainder. For instance, 10÷ 2 = 5 without a remainder, therefore, we say 5 is a factor of 10. Thus, we say that 3 is not a factor of 10.
Example 1
Find the factors of 32
Solution
32 = 1 x 32 =2 x 16 =4x 8
= 8 x 4 = 16 x 2 = 32 x 1
:. Factors of 32 = 1, 2,4,8, 16, and 32 or
Using table
32 1 x 32
2 x 16
4 x 8
Factors of 32 = 1, 2,4,8, 16 and 32
Note; A case were you have a particular number occurring two times, duplication is not allowed. Pick it once. See the example below;
Example 2
Find the factors of 144
Solution
144 1 x 144
144 2 x 72
144 3 x 48
144 4 x 36
144 6 x 24
144 8 x 18
144 9x 16
144 2 x 12
Factors of 144 = 1,2,3,4,6, 8,9,12,16, 18, 24, 36, 48, 72 and 144
from the example, 12 occurred twice, but only one was picked.
Example 3
Find the factors of 120
Solution
120 1 x 120
120 3 x 40
120 4 x 30
120 5 x 24
120 6x 20
120 8 x15
120 10 x 12
Factors of 120 = 1,2,3,4,5,6,8, 10,12,15,20,24,30,40,60 and 120.
Prime Factors:
From our definition of prime numbers, it will be easy joining factor to it and getting the meaning of prime factors.
Prime factors of a number are the factors of the number that are prime.
To find the prime factors of a number.
1.Start by dividing the number with the lowest number that is its factor and progress in that order.
Example 1
Express the following whole numbers as product of prime factors.
(a) 12 (b) 18 (c) 880 (d) 875.
Solution
2 2 3 3
1 1
12 is expressed as a product of 18 = 2 x 3 x 3
primes.
12 = 2 x 2 x 3
( c ) 2 880 5 875
2 440 5 175
2 220 5 35
2 110 7 7
5 55
11 11 875 =5 x 5x 5 x 7
1
= 880 = 2 x 2 x 2 x 2 x 5 x 11
example 2
Express 1512 as a product of prime factors.
Solution
Following the example above
2 1512
7 27
1512 = 2 x 2 x 2 x3 x 3 x 3 x 7
EVALUATION
Express the following as product of prime numbers
Index Form
If we have to write the following 4, 18, 16 as a product of prime factors, it will pose no challenge
4 = 2 x 2
8 = 2 x 2 x 2
16 = 2 x 2 x 2 x 2
As their products increase, the challenge of how to write 2 or whichever number is multiplying itself will arise.
A way of writing this in a shorter form is called index form.
The general form is xn
Where x = the base, that is the multiplicative value and
n=index or power or the number of times a particular number multiplies itself.
Example 1
Express the following index
Solution
(a) 3 x 3 x 3 x 3, this shows that four 3’s are to be multiplied together.
Writing index form
= X n
x = 3, n = 4
:. 3 4
(b) 5 x 5 x 5 = three 5’s in general form Xn
= x =5, n = 3
= 5 3
(c) 2 x 2 x2 x 2 x 2 x 2 x 2
= seven 2’s
= X n.
x = 2, n =7
= 2 7
As product of primes
800 …… 2 x 2 x 2 x 2 x 2 x 5 x 5
800 = 25 . 52
example 3
Express the following as a product of primes in index form.
(a) 720 (b) 1404
(a) 720 (d) 1404
2 720 2 1404
2 360 2 702
2 180 3 351
2 90 3 117
3 45 3 39
3 15 3 13
5 5 13 1
720 = 2 x 2 x2 x 2 x3 x 2 x 5 1404 = 2 x 2 x 3 x 3 x3 x 13
= 24x 32 x 5 = 22 x33 x 13
READING ASSIGNMENT
1.Essential mathematics for JSSI by AJS Oluwasanmipg29, 46-51
WEEKEND ASSIGNMENT
1.The fractions C/D ÷ a/b is same as
(a) Ca/Db (b) Cb/Da (c ) C x a (d) a x C (e) a x b
D x b b x D D x b
(a) 2/3 (b) 3/5 (c) 2/5 (d) 4/5 (e) 1 ¼
3.Find the length of a rectangle whose breadth and area are 7/20m and 8 1/5m2
(a) 23 3/7 (b) no answer ( c) 21 2/7 (d) 1 7/20 (e) 8 11/20.
(a) 5 11/4 (b) 2 ¾ ( c) 3 1/12 (d) 1 ¾ ( e) 3 - 3/2
(a) 2 x 3 x 7 (b) 2 x 4 x 7 (c ) 4 x 7 (d) 2 x 2 x 7 (e) 2 x 2 x2 x 7.
THEORY
4/5 + ½
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