Basic Mathematics Form One Notes – Fractions
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Proper, Improper and Mixed Numbers
A Fraction
Describe a fraction
AA fraction is a number which is expressed in the form of a/b where a – is the top number called
‘numerator and b- is the bottom number called denominator.
Consider the diagram below
The shaded part in the diagram above is 1 out of 8, hence mathematically it is written as 1/8
Example 1
(a) 3 out of 5 ( three-fifths) = 3/5
Example 2
(b) 7 Out of & (ie seven-eighths) = 7/8
Example 3
a S/12(SX 3)(12x 3) =15/36
b. 3/8=(3x 2M(8X2)= 6/16
Dividing the numerator and denominator by the same number (This method is used to simplify
the fraction)
Difference between Proper, Improper Fractions and Mixed Numbers
Distinguish proper, improper fractions and mixed numbers
Proper fraction -is a fraction in which the numerator is less than denominator
Example 4
7
4/5, 1/2, 11/13
Improper fraction -is a fraction whose numerator is greater than the denominator
Example $
12/7, 4/3, 65/56
Mixed fraction -is a fraction which consist of a whole number and a proper fraction
Example 6
12,52, 364
3° 5 293 363
(a) To convert mixed fractions into improper fractions, use the formula below
(b)To convert improper fractions into mixed fractions, divide the numerator by the denominator
‘denominator of proper fraction x whole number + numerator of proper fraction|
denominator of proper fraction
Example 7
Convert the following mixed numbers into improper fractions
2 1 z 4 7
a) 11= (b)6= (c) 3- d) 16= (e) 6—
(a) 115 )6> | 35 @) 165 (&) 65
Comparison of Fractions
In order to find which fraction is greater than the other, put them over a common denominator,
and then the greater fraction isthe one with greater numerator,
A Fraction to its Lowest Terms
‘Simplify a fraction to its lowest terms
Example 8
For the pair of fractions below, find which is greater
28
76 301
@>>5 ® 55
Solution
7_7x7_ 49 6_6x9_ 54
(jy 7=2277=-“% and £=S2°-%
9 on7 68 7 729 63
6 7
ede greater than 5
3_3x3_9 1_ixe_e
@2=222-5 aa 2-228. 2
8 8×3 24 3. 3×8 24
a 1
+ 3 isgreaterthan >
Equivalent Fractions
Idemify equivalent fractions
Equivalent Fraction
+ Are equal fractions written with different denominators
+ They are obtained by two methods
(a) Multiplying the numerator and denominator by the same number
~) S _ Sx3 _ 15
@ Dias 736
A, B_3a2 16.
Park
(a) Dividing the numerator and denominator by the same number (This method is used to
simplify the fraction
ay LF , i
@ Z5=2 oth numerator and denominator are divided by 2)
) ss
(i) a= jy (both numerator and denominator are divided by 8)
NOTE: The fraction which cannot be simplified more is said to be in its lowest form
2
Example 9
‘Simplify the following fractions to their lowest terms
25 gy 26 13 100 4s
@5O2zO,; @©7%; ©
Solution
3
(a) 3 = : (each number divided by 5)
268 13 =
(6) S5=% (each number divided by 2)
201
() $$=} (each number divided by 13)
4007 OS
(@) SS=5 (each number divided by 20)
4 3 ”
(©) S5=3 each numberdivided by 15)
Fractions in Order of Size
Arrange fractions in order of size
Example 10
Arrange in order of size, starting with the smallest, the fraction
24 35
3° 7’ 8 9
Solution
Put them over the same denominator, that is find the L.C_M of 3, 7, 8 and 9
Operations and Fractions
Addition of Fractions
Add fractions
30
Operations on fractions involves addition, subtraction, multiplication and division
+ Addition and subtraction of fractions is done by putting both fractions under the same
denominator and then add or subtract
+ Multiplication of fractions is done by multiplying the numerator of the first fraction with
the numerator of the second fraction, and the denominator of the first fraction with the
denominator the second fraction.
+ For mixed fractions, convert them first into improper fractions and then multiply
+ Division of fractions is done by taking the first fraction and then multiply with the
reciprocal of the second fraction
+ For mixed fractions, convert them first into improper fractions and then divide
Example 11
Find
a 1 oS 2 2 7
abi+ 2 @i+2 ~24¢2
@stZg Oeste Oxt zg
Solution
1_ixé_4 1_ixs_ 5
(a) t=2**=4 and +=2*8 =F
5 5×4 20 4 4×5 20
i fe 5_9
ie de pes ie See
5s 4 2 2 20
5 _ 5×3 _15 2_2x7_14
(be) 2=2%3=4 and 2=2*%72-¥4
7 ae 2k 3. 327 24
5) 2_15 | 16_29_,8
eo ee
7° 3°21” 2 2 a
2_2%x13_ 26 7 _ 7×5 _ 38
(9 2=278 = and 2=225 =2
5 5×13 65 13° 1325 65
2 7 _26 , 35_ 61
Gob Sot Sais
5° 13 6 65 6s
Subtraction of Fractions
31
Subtract fractions
Example 12
Evaluate
2.3 Boa el,
a) =- = ==> @ =-=
@s- FOG“: @ as
Solution
2_2%8_16 3_3%3_9
(a2) 2=228=% and 2237822
3. 3×8 24 a ex3 24
wh, 886, Bad
ie 7 ae 7
3_ 325 _15 1_1x4_4
)2=242=4 and 2=2%t=4
47 4×5” 20 5” 5×4 20
ff oe Sm
“45> 20” 20° 20
5 _ sxe | 40 1_1x16 _ 16
(o=S=* ond 1226 –
16 16×8 128 3” 6×16 128
a a ee 16 _ 24
16 8 128 «128 «128
Multiplication of Fractions
Multiply fractions
Example 13
i 3 1s 24 3
rat @os * Mewes wat
@ *7,Me*2 Of
Solution
2, 3_2e3_€_1
(a2 x 2=272-£-3
3 ites a 2
1S 24 _ 15×24 360_ 9
2x, Baz _ 30>
16 * 25 16×25 400 10
3 a 4 ee
(Qi x 422, 423%4- 28214
rte n° 2 Gea a as
32
Division of Fractions
Divide fractions
Example 14
é.. 2 23, 4s 5
Evaluate (a) > + = (b) ~+ — (c) = + 8
MO r+s; OR+ we OF
Solution
3, 2M SB _Ball_S
(ji + 222, 3-33-27
B° 8 8″ @ oma a
23. 45 _23 | 48 23×48 1104 69
0b) 2 + BiB , BL Bx _ uo _S
36° 48° 36 * 45 36×45 1520 95
5 5 Gs: ny Sat Ss
()2 + e=5 + a8 yt = SALE
7 7 1°7 % 8 7×8 56
Mixed Operations on Fractions
Perform mixed operations on fractions
Example 15
‘ 3 3 1 5
Find (a) 22+ 32 (b) 92- 42
5 ry 2 3
Solution
3_13_13x4_ 52 3_18_15x5_ 75
(a) 22=2=5**== ond 32255 58=7
35 5×4 20 a> 4 4×5 20
3 3_ 52, 75 _127_ 67
222 432254 Sooo
5 4” 20″ 20° 20 20
1_19_19x8 _ 152 5_37_37x2_ 74
(bot = Sa Se LS ong 42a 2a Pt?2LH
2-2 228 16 8 6 8×2 16
1 5_ 152 74_78_ 414
292+ 42eS2- BaBagS
2 8 16 ~=«i6 1616
Example 16
3
2 1 a ot
Evaluate (a) 52 + 15 (b) 42 x 72
Solution
(52+ 122% 2 22M y 22227 M_ 3?
3 a oe * gs * axe se
3 ot 23) 87_ 23457 _ 1911 4531
(bl) 42 x 72=S x Sat 32=
s*7a>s * @ see 40 40
Word Problems Involving Fractions
Example 17
1 ‘Musa is years old. His father is 3%times as old as he is. How old is his father?
2. 1%of a material are needed to make suit- How many suits can be made from
Decimals
The Concept of Decimals
Explain the concept of decimals
‘A decimal- is defined as a number which consist of two parts separated by a point.The parts are
whole number part and fractional part
x“