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FRACTIONS

A fraction is a number that can be written in the form of C:thlbcrtzFRACTIONSF1_filesimage001.gif
Where a and b are intergers and b C:thlbcrtzFRACTIONSF1_filesimage002.gif 0
C:thlbcrtz__i__images__i__over.jpg
The number which is written on top of fraction is called Numerator and the bottom is called denominator e.g
C:thlbcrtzFRACTIONSF1_filesimage003.gif
Type of fraction

(i) Proper fractions
– A proper fraction is the one in which the numerator is less than denominator
e.g. C:thlbcrtzFRACTIONSF1_filesimage004.gif etc
(ii) Improper fractions
– These are the ones which the numerators are greater than the denominator
C:thlbcrtzFRACTIONSF1_filesimage005.gif etc
(iii) Mixed fractions or mixed numbers
– These are the ones formed after improper fractions are divided complete
e.g. C:thlbcrtzFRACTIONSF1_filesimage006.gif = 2C:thlbcrtzFRACTIONSF1_filesimage007.gif , 6C:thlbcrtzFRACTIONSF1_filesimage008.gif , 9C:thlbcrtzFRACTIONSF1_filesimage009.gif
(iv) Equivalent fractions
These are two or more fractions that can be simplified to equal lowest fraction.
C:thlbcrtz__i__images__i__equivalent.jpg
Example: Change the following into mixed numbers
1. C:thlbcrtzFRACTIONSF1_filesimage019.gif = 7C:thlbcrtzFRACTIONSF1_filesimage007.gif
2. C:thlbcrtzFRACTIONSF1_filesimage020.gif = 3
3. C:thlbcrtzFRACTIONSF1_filesimage021.gif = 6C:thlbcrtzFRACTIONSF1_filesimage022.gif
Fractions can be represented on number lines
e.g. represent C:thlbcrtzFRACTIONSF1_filesimage023.gif on a number line.
C:thlbcrtz__i__images__i__number_line.jpg
Exercise 1
1. (i)Which of the following are: –
(a) Proper fractions
(b) Mixed fractions
(c) Improper fractions
(ii) List four equivalent fractions
(i) C:thlbcrtzFRACTIONSF1_filesimage033.gif
(ii) C:thlbcrtzFRACTIONSF1_filesimage034.gif
(iii) C:thlbcrtzFRACTIONSF1_filesimage035.gif
(iv) C:thlbcrtzFRACTIONSF1_filesimage036.gif
(v) C:thlbcrtzFRACTIONSF1_filesimage037.gif
(vi) C:thlbcrtzFRACTIONSF1_filesimage038.gif
(vii) C:thlbcrtzFRACTIONSF1_filesimage039.gif
(viii) C:thlbcrtzFRACTIONSF1_filesimage040.gif
(ix) C:thlbcrtzFRACTIONSF1_filesimage041.gif
(x) C:thlbcrtzFRACTIONSF1_filesimage042.gif
(xi) C:thlbcrtzFRACTIONSF1_filesimage043.gif
(xii) C:thlbcrtzFRACTIONSF1_filesimage044.gif
(xiii) C:thlbcrtzFRACTIONSF1_filesimage045.gif
(xiv) C:thlbcrtzFRACTIONSF1_filesimage046.gif
(xv) C:thlbcrtzFRACTIONSF1_filesimage047.gif
(xvi) 3C:thlbcrtzFRACTIONSF1_filesimage048.gif

Solution
(a) Proper fraction
C:thlbcrtzFRACTIONSF1_filesimage033.gif , C:thlbcrtzFRACTIONSF1_filesimage034.gif , C:thlbcrtzFRACTIONSF1_filesimage035.gif , C:thlbcrtzFRACTIONSF1_filesimage038.gif , C:thlbcrtzFRACTIONSF1_filesimage039.gif , C:thlbcrtzFRACTIONSF1_filesimage040.gif , C:thlbcrtzFRACTIONSF1_filesimage041.gif , C:thlbcrtzFRACTIONSF1_filesimage042.gif , C:thlbcrtzFRACTIONSF1_filesimage043.gif , C:thlbcrtzFRACTIONSF1_filesimage047.gif
(b) Improper fraction
C:thlbcrtzFRACTIONSF1_filesimage049.gif , C:thlbcrtzFRACTIONSF1_filesimage037.gif , C:thlbcrtzFRACTIONSF1_filesimage044.gif , C:thlbcrtzFRACTIONSF1_filesimage045.gif , C:thlbcrtzFRACTIONSF1_filesimage046.gif
(c) Mixed fraction
1C:thlbcrtzFRACTIONSF1_filesimage050.gif , 3C:thlbcrtzFRACTIONSF1_filesimage048.gif
Four equivalent fractions are:-
1. C:thlbcrtzFRACTIONSF1_filesimage051.gif C:thlbcrtzFRACTIONSF1_filesimage052.gif C:thlbcrtzFRACTIONSF1_filesimage053.gif C:thlbcrtzFRACTIONSF1_filesimage054.gif , C:thlbcrtzFRACTIONSF1_filesimage052.gif C:thlbcrtzFRACTIONSF1_filesimage053.gif C:thlbcrtzFRACTIONSF1_filesimage055.gif
2. C:thlbcrtzFRACTIONSF1_filesimage051.gif C:thlbcrtzFRACTIONSF1_filesimage056.gif C:thlbcrtzFRACTIONSF1_filesimage053.gif C:thlbcrtzFRACTIONSF1_filesimage057.gif , C:thlbcrtzFRACTIONSF1_filesimage056.gif C:thlbcrtzFRACTIONSF1_filesimage053.gif C:thlbcrtzFRACTIONSF1_filesimage058.gif
3. C:thlbcrtzFRACTIONSF1_filesimage051.gif C:thlbcrtzFRACTIONSF1_filesimage059.gif C:thlbcrtzFRACTIONSF1_filesimage053.gif C:thlbcrtzFRACTIONSF1_filesimage060.gif , C:thlbcrtzFRACTIONSF1_filesimage059.gif C:thlbcrtzFRACTIONSF1_filesimage053.gif C:thlbcrtzFRACTIONSF1_filesimage061.gif
4. C:thlbcrtzFRACTIONSF1_filesimage051.gif C:thlbcrtzFRACTIONSF1_filesimage062.gif C:thlbcrtzFRACTIONSF1_filesimage063.gif C:thlbcrtzFRACTIONSF1_filesimage064.gif , C:thlbcrtzFRACTIONSF1_filesimage062.gif C:thlbcrtzFRACTIONSF1_filesimage063.gif C:thlbcrtzFRACTIONSF1_filesimage065.gif
2. Write the following fractions in words
(a) C:thlbcrtzFRACTIONSF1_filesimage040.gif
C:thlbcrtzFRACTIONSF1_filesimage066.gif three quarters
(b) C:thlbcrtzFRACTIONSF1_filesimage048.gif
C:thlbcrtzFRACTIONSF1_filesimage066.gif A half
(c) C:thlbcrtzFRACTIONSF1_filesimage052.gif
C:thlbcrtzFRACTIONSF1_filesimage066.gif A third
(d) C:thlbcrtzFRACTIONSF1_filesimage050.gif
C:thlbcrtzFRACTIONSF1_filesimage066.gif Five over six
(e) C:thlbcrtzFRACTIONSF1_filesimage067.gif
C:thlbcrtzFRACTIONSF1_filesimage066.gif Nine over Ten
(f) C:thlbcrtzFRACTIONSF1_filesimage068.gif
C:thlbcrtzFRACTIONSF1_filesimage066.gif A quarter
2. Write the name of the fraction of the shaded part in figures ABCD and EFGH
C:thlbcrtz__i__images__i__abcd1.jpg

C:thlbcrtzFRACTIONSF1_filesimage076.gif C:thlbcrtzFRACTIONSF1_filesimage052.gif = one over three (A third)
Which is a proper fraction
E H
C:thlbcrtzFRACTIONSF1_filesimage077.gif
F G
C:thlbcrtzFRACTIONSF1_filesimage076.gif C:thlbcrtzFRACTIONSF1_filesimage078.gif = one over four (A quarter)
Which is a proper fraction
Comparison of fraction
Fraction can be compared by using two methods
(i) Number line
(ii) L.C.M of the denominators

(I) Number line
Example 1. show C:thlbcrtzFRACTIONSF1_filesimage052.gif and C:thlbcrtzFRACTIONSF1_filesimage048.gif on a number line and then find which is greater than the other
C:thlbcrtz__i__images__i__line11.jpg
C:thlbcrtzFRACTIONSF1_filesimage080.gif C:thlbcrtzFRACTIONSF1_filesimage081.gif C:thlbcrtzFRACTIONSF1_filesimage082.gif C:thlbcrtzFRACTIONSF1_filesimage083.gif
Example 2. Which is greater between C:thlbcrtzFRACTIONSF1_filesimage056.gif and C:thlbcrtzFRACTIONSF1_filesimage084.gif?
Solution:
C:thlbcrtz__i__images__i__nominator.jpg
C:thlbcrtzFRACTIONSF1_filesimage080.gif C:thlbcrtzFRACTIONSF1_filesimage086.gif C:thlbcrtzFRACTIONSF1_filesimage082.gif C:thlbcrtzFRACTIONSF1_filesimage087.gif
(II) L.C.M of the denominators
Determine which fraction is greater between C:thlbcrtzFRACTIONSF1_filesimage056.gif and C:thlbcrtzFRACTIONSF1_filesimage084.gif
Solution
1st find the L.C.M of 5 and 7
C:thlbcrtzFRACTIONSF1_filesimage088.gif L. C. M of 5 and 7 = 35
2nd multiply by that L. C. M each fraction
C:thlbcrtzFRACTIONSF1_filesimage088.gif C:thlbcrtzFRACTIONSF1_filesimage089.gif C:thlbcrtzFRACTIONSF1_filesimage090.gif 35 = 14
C:thlbcrtzFRACTIONSF1_filesimage088.gif C:thlbcrtzFRACTIONSF1_filesimage091.gif C:thlbcrtzFRACTIONSF1_filesimage090.gif 35 = 20
Conclusion, check the one which has given us bigger number after multiplication with L.C.M
C:thlbcrtzFRACTIONSF1_filesimage091.gif C:thlbcrtzFRACTIONSF1_filesimage092.gif C:thlbcrtzFRACTIONSF1_filesimage089.gif
Example: which is greater C:thlbcrtzFRACTIONSF1_filesimage093.gif or C:thlbcrtzFRACTIONSF1_filesimage094.gif ?
Solution:
L.C.M = 44
C:thlbcrtzFRACTIONSF1_filesimage095.gif C:thlbcrtzFRACTIONSF1_filesimage093.gif C:thlbcrtzFRACTIONSF1_filesimage096.gif 44
=99
C:thlbcrtzFRACTIONSF1_filesimage095.gif C:thlbcrtzFRACTIONSF1_filesimage094.gif C:thlbcrtzFRACTIONSF1_filesimage096.gif 44
=12
:- C:thlbcrtzFRACTIONSF1_filesimage093.gif C:thlbcrtzFRACTIONSF1_filesimage097.gif C:thlbcrtzFRACTIONSF1_filesimage094.gif

Operations on Fractions

Addition and Subtraction of Fraction

NOTE:
1 .Add the numerator together if each fraction has the same denominator.

2. If the fraction has different denominator, you must find the smallest number that each denominator divides into exactly. (LCM)

3. When adding fractions, do not add the denominator.

Example:

Evaluate the following fractions

C:thlbcrtz__i__images__i__Numerator.jpg

C:thlbcrtz__i__images__i__seven_eight.jpg

Multiplication

NOTE: 1. Before multiplying a number convert mixed number into improper fractions
2. Multiply the numerators and multiply the denominator.
Examples:

C:thlbcrtz__i__images__i__multiplication.jpg

C:thlbcrtz__i__images__i__multiple.jpg
Dividing Fractions

1. When dividing fractions invert the second fraction then multiply the first fraction by the inverted fraction.

C:thlbcrtz__i__images__i__evaluate.jpg
2. Before dividing number convert mixed numbers into improper fraction.
C:thlbcrtz__i__images__i__dividing.jpg

DECIMALS AND PERCENTAGES
Are fractions of tenth, they are written using a point which is a result of division of a normal fraction
E.g. 0.34, 0.5, 0.333——–
In the fraction 0.2546 the place values are
Ones
Tenth
Hundredths
Thousandths
Ten Thousandths
0
2
5
4
6
Decimals can be converted into fractions and vice versa
E.g. Change C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage001.gif in to decimals
Solution:
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage002.gif = 0.75
This fraction which ends after dividing is called terminating fraction. Other fractions do not end, these ones are called recurring or repeating decimals.
E.g. C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage003.gif
C:thlbcrtz__i__images__i__fracti.jpg
Conversion of Repeating decimal into fractions
C:thlbcrtz__i__images__i__fr.jpg
Solution:
0.3 = 0.333…….
C:thlbcrtz__i__images__i__fro.jpg
Subtract (i) from (ii)
C:thlbcrtz__i__images__i__fra.jpg
9t = 3.0
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage004.gif = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage005.gif
t = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage003.gif
Exercise 1
Insert C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage006.gif or C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage007.gif between each pair of fractions questions 4 to 12
1. C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage008.gif , C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage009.gif
Solution
L.C.M = 3
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage010.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage011.gif 3 = 2
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage003.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage011.gif 3 = 1
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage012.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage013.gif
2. C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage014.gif , C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage015.gif
Solution
L.C.M = 63
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage014.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage016.gif 63 = 7
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage015.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage016.gif 63 = 9
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage017.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage015.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage018.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage014.gif
3. C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage019.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage020.gif
Solution
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage019.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage016.gif 12 = 10
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage020.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage016.gif 12 = 9
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage017.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage019.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage018.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage020.gif
4. C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage022.gif , C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage020.gif
Solution
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage022.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage016.gif 20 = 16
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage020.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage016.gif 20 = 15
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage017.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage022.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage018.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage020.gif
5. C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage023.gif , C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage024.gif
Solution
L.C.M of 20 and 4 = 80
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage023.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage016.gif 80 = 60
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage024.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage016.gif 80 = 140
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage017.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage024.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage018.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage023.gif
6. C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage025.gif , C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage020.gif
Solution
L. C. M of 4 and 4 = 4
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage025.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage016.gif 4 = 1
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage020.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage016.gif 4 = 3
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage017.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage020.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage018.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage025.gif
7. C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage026.gif , C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage027.gif
Solution
L. C. M of 5 and 6 = 30
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage026.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage016.gif 30 = 12
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage027.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage016.gif 30 = 5
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage017.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage026.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage018.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage027.gif
8. C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage028.gif , C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage019.gif
Solution
L. C. M of 9 and 6 = 18
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage028.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage016.gif 18 = 16
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage019.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage016.gif 18 = 15
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage017.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage028.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage018.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage019.gif
9. Which numbers are denominators in each of the following fractions?
(a) C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage029.gif 16 is the denominator.
(b) C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage030.gif 93 is the denominator
(c) 3C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage022.gif 5 is the denominator
10. Which numbers are numerators in each of the following fractions?
(a) C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage031.gif Numerators is 3
(b) 3C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage032.gif Numerators is 4
(c) C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage033.gif Numerators is 12
C:thlbcrtz__i__images__i__line_12.jpg
12. Which is greater
(a) C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage031.gif or C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage020.gif
Solution
Find the L.C.M of 5 and 4 = 20
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage031.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage016.gif 20 = 12
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage020.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage016.gif 20 = 15
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage017.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage020.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage018.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage031.gif
(b) C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage008.gif or C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage043.gif
Solution
Find the L.C.M of 3 and 2 = 6
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage044.gif x 6 = 4
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage045.gif x 6 = 3
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage017.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage044.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage046.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage045.gif
13. What is the condition for a fraction to be called improper?
The numerator is bigger than the denominator.
14. Change the following improper fractions into mixed numbers
(a) C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage008.gif = 1C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage034.gif
(b) C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage051.gif = 4C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage052.gif
(c) C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage053.gif = 3C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage054.gif
(d) C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage055.gif = 1C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage056.gif
16 15. Change the following mixed numbers into improper fractions
(a) 3 C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage022.gif
Solution
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage057.gif
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage058.gif
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage017.gif = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage059.gif
(b) 15 C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage060.gif
Solution
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage061.gif
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage062.gif
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage017.gif = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage063.gif
(c) 24 C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage064.gif
Solution
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage065.gif
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage066.gif
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage017.gif = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage067.gif
3.3 PERCENTAGES
Percentages are fractions expressed out of 100. That is – are the ones whose denominator is one hundred, they are denoted by (%) called percent
Example: 12% means 12 C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage011.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage068.gif = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage069.gif
70% = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage070.gif etc.


Examples: 1. convert the following percentage into fraction
(i) 65%
(ii)75%
(iii) 12 C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage043.gif %

Solution
(i) 65%
65 C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage016.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage071.gif = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage072.gif = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage073.gif
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage074.gif = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage073.gif
(ii) 75%
75 C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage016.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage071.gif = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage075.gif = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage020.gif
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage074.gif = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage020.gif
(iii) 12 C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage076.gif %
12C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage077.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage071.gif = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage078.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage016.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage071.gif = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage079.gif
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage074.gif 12 C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage080.gif
% = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage081.gif
2. Change
(i) 40% into decimal
(ii) 35% into fractions
(iii) 0.125 into percentage
Solution
(i) 40% = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage082.gif = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage083.gif
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage074.gif = 0.4
(ii) 35%
35 C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage016.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage071.gif = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage084.gif
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage074.gif = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage084.gif
(iii) 0.125
Solution
0.125 = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage085.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage016.gif 100%
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage074.gif = 12.5%
3. Change the recurring decimals into fractions
(i) 0C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage086.gif
Solution
Let x = 0.C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage087.gif ……………………….. (i)
100x = 21.C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage087.gif ……………………. (ii)
Take away equation (i) from (ii)
100x = 21.C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage087.gif
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage088.gif = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage089.gif
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage090.gif = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage091.gif
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage092.gif x = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage091.gif
(ii) C:thlbcrtz__i__images__i__franli111.jpg
Solution
Let x = 0.9C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage093.gif ……………………….. (i)
10x = 9.C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage093.gif …………………………. (ii)

100x = 93.3 ………………………(iii)
Take equation (ii) away from equation (iii)
100x = 93.3
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage094.gif = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage095.gif
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage096.gif = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage097.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage098.gif
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage092.gif x = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage098.gif
(iii) 0.C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage099.gif6C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage100.gif
Solution
Let x = 0.C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage099.gif6C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage100.gif ………………………….. (i)
1000x = 567.567 ……………………. (ii)
Take away equation (i) from (ii)
1000x = 567.567
X = 0.567
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage101.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage102.gif
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage092.gif x = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage103.gif
(iv) 0.C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage104.gif35C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage105.gif
Solution
Let x = 0.C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage104.gif35C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage106.gif……………………….. (i)
10000x = 1352.1352 ……………………. (ii)
Take (ii) – (i)
10000x = 1352.1352
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage107.gif = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage108.gif
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage109.gif = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage110.gif
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage092.gif x = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage110.gif
(v) 0.C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage105.gif1C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage111.gif
Solution
Let x = 0.C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage105.gif1C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage111.gif ………………………….. (i)
1000x = 219.219 ……………………. (ii)
Take away equation (i) from (ii)
1000x = 219.219
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage112.gif = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage113.gif
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage114.gif = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage115.gif
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage116.gif x = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage115.gif
(vi) 0.C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage104.gif8C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage117.gif
Solution
Let x = 0.C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage104.gif8C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage118.gif………………………….. (i)
1000x = 186.186 ……………………. (ii)
Take away equation (i) from (ii)
1000x = 186.186
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage112.gif = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage119.gif
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage101.gif = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage120.gif
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage092.gif x = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage121.gif
(vii) 0.C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage122.gif63C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage123.gif
Solution
Let n = 0.C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage122.gif63C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage123.gif ………………………….. (i)
10000n = 8634.8634 ……………………. (ii)
Take away equation (i) from (ii)
10000n = 8634.8634
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage124.gif = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage125.gif
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage126.gif = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage127.gif
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage092.gif n = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage127.gif
(viii) 0.7C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage129.gif
Solution
Let x = 0.7C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage129.gif ……………………….. (i)
10x = 0.7C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage129.gif …………………………. (ii)
1000x = 792.C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage129.gif ………………………(iii)
Take away equation (ii) from equation (iii)
100x = 792.C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage129.gif
1000x – 10x = 792.C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage129.gif – 7.C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage129.gif
990x = 785

C:thlbcrtz__i__images__i__996.jpg
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage012.gif x = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage134.gif
(ix) 0.C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage117.gif4C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage099.gif
Solution
Let y = 0.C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage117.gif4C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage099.gif ………………………….. (i)
1000y = 645.C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage117.gif4C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage099.gif ……………….. (ii)
Take away equation (i) from (ii)
1000y – y = 645-C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage117.gif4C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage099.gif
999y=645
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage137.gif = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage138.gif
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage092.gif y = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage139.gif
(x) 0.C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage140.gif
Solution
Let b = 0.C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage140.gif ………………………….. (i)
100b = 64.C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage140.gif ……………………. (ii)
Take away equation (i) from (ii)
100b – b = 64.C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage140.gif-0.C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage140.gif
99b = 64
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage143.gif = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage144.gif
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage092.gif b = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage144.gif
(xi)0.C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage117.gif2C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage100.gif
Solution
Let m = 0.C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage117.gif2C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage145.gif………………………….. (i)
1000m = 627.C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage117.gif2C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage145.gif……………………. (ii)
Take away equation (i
) from (ii)
1000m – m = 627.C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage117.gif2C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage145.gif– 0.C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage117.gif2C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage145.gif
999m = 627
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage148.gif = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage149.gif
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage092.gif m = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage151.gif


4. In question (i) to (v) change the fractions into decimals.

C:thlbcrtz__i__images__i__franeee2.jpg
Solution
1 ÷ 3 =
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage092.gif = 0.33
ii. C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage019.gif
Solution
5 ÷ 6 =
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage092.gif = 0.833
iii. C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage152.gif
Solution
4 ÷ 11 =
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage092.gif = 0.3636
iv. C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage014.gif
Solution
1 ÷ 9 =
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage092.gif = 0.111
v. C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage153.gif
Solution
7 ÷ 13 =
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage092.gif = 0.538461

C:thlbcrtz__i__images__i__frano1.jpg
Solution
Let b = 0.C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage104.gif2C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage093.gif ………………………….. (i)
1000b = 123.C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage104.gif2C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage093.gif……………….. (ii)
Take equation (i) away from equation (ii)
1000b – b = 123.C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage104.gif2C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage093.gif – 0.C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage104.gif2C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage093.gif
999b = 123
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage222.gifC:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage223.gif
b = C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage225.gif

Operations on Decimals
Operations with decimals are similar to operations with whole numbers:
Addition
Note: The decimal points must be in line, put zeros at the end to give the same number of decimal places in each number.
C:thlbcrtz__i__images__i__decimal.jpg
C:thlbcrtz__i__images__i__hesabuuu.jpg
Multiplication
Note:
  1. When multiplying decimals the answer must have the same number of decimal places as the total number of decimal places in the number being multiplied.
  2. First carry out the multiplication in the usual way, without any decimal points, then put the point to the total decimal places.
C:thlbcrtz__i__images__i__hesabubu.jpg
Division
Note:
It is not easy to divide by a decimal, so you multiply each number by a power of 10 in order that you are dividing by a whole number.
Example:- (i) Find (a) 68.32 ÷ 1.4
(b) 9.66 ÷ 0.23
Solution
(a) 68.32 ÷ 1.4 = 68.32 x 10 ÷ 1.4 x 10
682.2÷14
By long division
C:thlbcrtz__i__images__i__A70.PNG
Therefore 68.32 ÷ 1.4 = 48.8
(b)
C:thlbcrtz__i__images__i__B34.PNG
Therefore 9.66 ÷ 0.23 = 42
C:thlbcrtz__i__images__i__bbbbbbbbbbbbbbbbb.jpg
(c) 7.32 1.2 = 7.32 x 10 1.2 x 10
73.2

C:thlbcrtz__i__images__i__C50.PNG
t
Therefore 7.32 ÷ 1.2 = 6.1
C:thlbcrtz__i__images__i__hehesabu.jpg
Mariam was given 20,000 shillings by her father, she spent 48% of it to buy shoes. How much money remained.
Solution
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage226.gif C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage016.gif 20,000

=9,600
20,000
– 9,600
11,600
C:thlbcrtzDECIMALSANDPERCENTAGESF1_filesimage116.gif The remained money was 11,600/=


C:thlbcrtz__i__images__i__fruct.jpg
C:thlbcrtz__i__images__i__fre.jpg

PERCENTAGES APPLIED TO REAL LIFE PROBLEMS
The examples below show the wide range of application
Examples:-
1. In one week, Flora earned 48,000/=, she spent 4,000/= on travel to and from work. What percentage of her money was left?
Solution:
C:thlbcrtz__i__images__i__percentage.jpg
Percentage of a quantity
When finding a percentage of a quantity, it is often helps to change the percentage to a decimal and multiply it by the quantity.
Example:- Find (a) 20% of 840,000
C:thlbcrtz__i__images__i__ishirini1.jpg
C:thlbcrtz__i__images__i__perc.jpg
Percentage increase and Decrease
There are two steps to calculate percentage increase (or decrease)
Example: In 1975 the population of a village was 90. It increased by 30% the following year. What was the population in the year 1976?
C:thlbcrtz__i__images__i__method.jpg




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EcoleBooks | MATHEMATICS O LEVEL(FORM ONE) NOTES - FRACTIONS

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