Chapter Eight Fractions PDF
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This document contains exercises on fractions. It includes addition and subtraction of fractions with the same and different denominators, comparison of fractions, and word problems involving fractions. Suitable for a middle school math class.
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FOR ONLINE USE ONLY DO NOT DUPLICATE Chapter Eight Fractions Introduction In Standard Four, you learnt about addition and subtra...
FOR ONLINE USE ONLY DO NOT DUPLICATE Chapter Eight Fractions Introduction In Standard Four, you learnt about addition and subtraction of fractions with the same denominator. In this chapter, you will learn about types of fractions, LY comparison of fractions, addition and subtraction of fractions with different denominators. You will also learn how to solve word problems involving the concept of fraction. The competence on fractions will help you in measuring N time, budgeting, medical prescriptions, assessing progress, preparing meals O and the like. Exercise 1: Revision Answer the following questions: SE U 1 _ _ 1 _6 _5 3 _ _ 4 1. + = 2. + = 3. + = 3 3 12 12 8 8 E _5 11 _ 13 _ _ 4 _4 _1 4. + = 5. + = 6. + = N 20 20 17 17 10 10 LI 15 _ _ 8 _9 12 _ _11 _30 7. ‒ = 8. + = 9. + = 35 35 36 36 53 53 N O _5 _9 _6 _5 25 _ _ 13 10. + = 11. ‒ = 12. ‒ = 22 22 20 20 48 48 R _7 _4 30 _ _ 20 5 _ _ 2 13. ‒ = 14. ‒ = 15. ‒ = FO 13 13 60 60 7 7 117 MATH std 5.indd 117 30/07/2021 14:49 FOR ONLINE USE ONLY DO NOT DUPLICATE 15 _ _8 _9 _5 16. ‒ = 17. ‒ = 40 40 19 19 18. Half of the farm was planted with maize. What part of the farm was not planted with maize? LY _ 1 19. Pupils planted seedlings of fruit trees. If of the seedlings dried up, 4 what fraction of the seedlings survived? N _ 4 20. Zawadi walked of her journey. What fraction of her journey remained? O 5 3 _ 21. The villagers sold of their cotton to a certain cooperative union 9 _4 SE and to another cooperative union. What fraction of cotton did they sell? 9 U 1 _ 3 22. Maua ate of bread and Amani ate _ of the same bread. What 12 12 fraction of the bread was eaten by Maua and Amani? E N Types of fractions There are three types of fractions, proper fractions, improper fractions and LI mixed fractions. (i) A proper fraction is a fraction in which the numerator is smaller than the N denominator. The following are examples of proper fractions: O _ 1 2 _ 1 3 _ 10 15 , _ , , _ , , _ . 4 4 9 8 12 40 R (ii) An improper fraction is a fraction where the numerator is bigger than the FO denominator. The following are examples of improper fractions: _ 6 9 _ 10 11 _ 12 , _ , , _ , . 5 4 7 9 11 118 MATH std 5.indd 118 30/07/2021 14:49 FOR ONLINE USE ONLY DO NOT DUPLICATE (iii) A mixed fraction or mixed numbers is a fraction which consists of a whole number and a proper fraction. This type of fraction is obtained when an improper fraction is simplified. The following are examples of mixed numbers: 1 _ 1 3 1 1 _ , 5 _ , 15 _ 3 , 4 , 2 2 _ . 2 2 4 4 3 LY Example N Which of the following are (a) proper fractions? (b) improper fractions? O _6 _ 2 _ 7 _ 1 _9 , , , and . 9 3 2 3 4 Answer 6 _ (a) Proper fractions are , 9 _ 2 and _ 3 1 SE . 3 U 7 _ 9 _ (b) Improper fractions are and . 2 4 E Exercise 2 N Answer the following questions: LI 1. Which of the following are improper fractions? N _ 3 10 1 9 20 16 , _ , _ , _ , _ and _ . 4 8 4 15 18 9 O 2. Which of the following are proper fractions? _ 1 1 11 5 9 , _ , _ , _ and _ R . 6 7 6 4 10 FO 3. What is the difference between a proper fraction and an improper fraction? 119 MATH std 5.indd 119 30/07/2021 14:49 FOR ONLINE USE ONLY DO NOT DUPLICATE Equivalent fractions Any fraction whose numerator is the same as the denominator has a value of 1. Example 1 _ 2 3 4 5 6 = _ = _ = _ = _ = 1. 2 3 4 5 6 If a fraction is multiplied by 1 its value remains unchanged. LY Example 2 N Write four fractions which are equivalent to . _ 1 O 2 Solution Multiply the numerator and the denominator by the same number. 1 1×2 2 SE _ = _ = _ U 2 2×2 4 _ 1 1×3 _ 3 _ = = 2 2×3 6 E _ 1 1×4 4 = _ = _ 2 2×4 8 N _ 1 1×5 _ _5 = = . LI 2 2×5 10 _ 1 _2 _3 _4 _ 5 Therefore, = = = = . N 2 4 6 8 10 O Example 3 R _ 3 Write four fractions which are equivalent to . FO 4 Solution Multiply the numerator and the denominator by the same number. 120 MATH std 5.indd 120 30/07/2021 14:49 FOR ONLINE USE ONLY _ 3 3 2 6 DO NOT DUPLICATE = _ × _ = _ 4 4 2 8 3 _ 3 _ 3 _ _9 = × = 4 4 3 12 3 3 4 12 = × _ = _ _ _ 4 4 4 16 3 _ 3 _ 5 _ 15 _ = × = . 4 4 5 20 LY Therefore, four fractions which are equivalent to _ 3 6 9 12 15 are _ , _ , _ and _ . N 4 8 12 16 20 O Exercise 3 Answer the following questions: 1. For each of the following fractions, list any three fractions that are SE equal to the given fractions: U _ 1 1 _ 1 _ 1 _ _ 1 (a) (b) (c) (d) (e) 7 11 9 15 4 E 2. Fill in the blanks with the missing fractions: N 1 _ 2 4 (a) , _ , ____, _ . LI 4 8 16 _1 2 5 (b) , _ , ____, ____, _ . N 8 16 40 _3 9 (c) , ____, _ , ____. O 5 15 _3 _9 (d) , ____, . 16 48 R 3. Which among the following fractions are equivalent? FO _ 1 1 1 25 2 3 , _ , _ , _ , _ , _ . 4 6 8 100 16 18 121 MATH std 5.indd 121 30/07/2021 14:49 FOR ONLINE USE ONLY DO NOT DUPLICATE Comparing fractions Fractions with different values may be illustrated using a table showing different fractions. Study carefully the following table: Table of fractions 1 LY 1 1 2 2 1 1 1 3 3 3 N 1 1 1 1 4 4 4 4 O 1 1 1 1 1 5 5 5 5 5 1 1 1 1 1 1 6 6 6 6 6 6 1 7 1 7 1 7 1 7 SE 1 7 1 7 1 7 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 U 1 1 1 1 1 1 1 1 1 9 9 9 9 9 9 9 9 9 1 1 1 1 1 1 1 1 1 1 E 10 10 10 10 10 10 10 10 10 10 1 1 1 1 1 1 1 1 1 1 1 N 11 11 11 11 11 11 11 11 11 11 11 1 1 1 1 1 1 1 1 1 1 1 1 LI 12 12 12 12 12 12 12 12 12 12 12 12 In the table of fractions, the length of a shaded part represents the size or N value of the particular fraction. O Example 1 R 1 _ _ 1 Of the two fractions, and , which one is greater than the other? FO 2 4 1 _ The table of fractions shows that the shaded part representing is longer 2 1 _ 1 _ 1 _ than the shaded part representing . Therefore, is greater than . 4 2 4 122 MATH std 5.indd 122 30/07/2021 14:49 FOR ONLINE USE ONLY DO NOT DUPLICATE Example 2 _2 2 _ Of the two fractions, and , which one is smaller than the other? 6 5 _2 The table shows that the shaded part representing is shorter than the 2 2 6 2 _ _ _ shaded part representing . Therefore, is smaller than . 5 6 5 Exercise 4 LY Use the table of fractions to answer the following questions: N 1. For each of the following pair of fractions, identify which one is greater than the other. O 2 _ 5 _ _1 1 _ _ 4 _2 (i) and (ii) and (iii) and 3 6 8 9 5 10 SE 2. For each of the following pairs of fractions identify which one is smaller than the other. _ 5 _2 _ 4 _ 6 6 _ 3 _ (i) and (ii) and (iii) and U 11 3 12 10 7 9 3. Write the following fractions in ascending order: E _1 _ 5 _ 1 _3 _ 7 _ 2 7 _ _ 5 _ 3 (i) , , (ii) , , (iii) , , 3 6 2 4 12 3 8 6 4 N 4. Write the following fractions in descending order: LI _ 5 _ 3 _ 2 _7 _ 5 _ 3 _ 7 _ 1 3 (i) , , (ii) , , (iii) , , _ 6 4 3 12 8 7 10 2 5 N O Adding fractions with different denominators R Example 1 FO _ 1 1 + _ = 2 4 123 MATH std 5.indd 123 30/07/2021 14:49 FOR ONLINE USE ONLY DO NOT DUPLICATE Solution 1 _ _ 1 1 _ Write and as fractions with the same denominator by multiplying 2 2 4 2 _ by . 2 1 _ _1×2 2 That is, = = _ . 2 2×2 4 LY + = N O 2 _ 1 3 + _ = _ . 4 4 4 _1 1 _ 2 _ _ 1 2+1 _ 3 _ Therefore, + = + = = . 2 4 4 4 4 4 SE + = + + = U Example 2 2 2 + _ _ E = 3 3 N Solution LI N + = + O _ 2 _2 3 _ _ 1 _ 4 + = + = . R 3 3 3 3 3 FO 2 _ _ 2 4 1 Therefore, + = _ = 1 _ . 3 3 3 3 124 MATH std 5.indd 124 30/07/2021 14:49 FOR ONLINE USE ONLY DO NOT DUPLICATE Example 3 _ 2 3 + _ = 5 9 Solution _45 Since 5 × 9 = 45, multiply each fraction by to obtain a common 45 denominator. LY _ 2 45 3 45 2×9 _ 3×5 × _ + _ × _ = _ + 5 45 9 45 45 45 18 _ _ 15 N = + 45 45 O 33 or _ _ 11 = . 45 15 2 _ _ 3 33 _ _ 11 Therefore, + = or . 5 9 45 15 SE Example 4 U _ 4 3 1 + _ + _ = 15 5 30 E Solution N Since 15 and 5 are factors of 30, we proceed as follows: LI _ 4 4 2 8 3 3 6 18 = _ × _ = _ ; _ = _ × _ = _ 15 15 2 30 5 5 6 30 N It follows that; O _ 4 3 1 8 18 1 + _ + _ = _ + _ + _ 15 5 30 30 30 30 R 27 9 _ or _ = . 30 10 FO _ 4 3 _ _ 1 27 _ _ 9 Therefore, + + = or . 15 5 30 30 10 125 MATH std 5.indd 125 30/07/2021 14:49 FOR ONLINE USE ONLY DO NOT DUPLICATE Exercise 5 Answer the following questions: _ 3 1 3 1 3 4 1. + _ = 2. _ + _ = 3. _ + _ = 5 6 4 5 7 21 _ 1 _ _ 1 5 1 1 1 + _ = 6. _ + _ + _ = 30 4. + = 5. 36 18 5 7 2 4 5 LY _ 5 7 2 5 1 2 7. + _ = 8. _ + _ = 9. _ + _ = 20 60 3 9 7 5 _8 24 _ _ 5 1 _ _ 1 1 1 + _ + _ = N 10. + = 11. + = 12. 20 40 65 13 2 3 8 O 3 _ _ 2 _ 2 _ 1 _ 1 _ 2 9 _ 13. + = 14. + + = 15. + = 8 6 3 9 18 15 10 _ _ 3 _ 3 1 1 7 10 + _ + _ = _ _ 20 16. + = 50 25 12 _ _ 13 17. 15 4 7 4 _ _ 45 SE 18. + 9 27 = 19. + = 20. + = 25 100 30 90 U Subtraction of fractions with different denominators E N Example 1 LI _ 3 1 − _ = 4 2 N Solution O 1 _ _2 Multiply by so as to obtain fractions with the same denominator. R 2 2 _ 1 1 _ _ 2 2 _ That is, = × = FO 2 2 2 4 3 _ _ 1 3 _ _ 2 3−2 _ _ 1 − = − = = . 4 2 4 4 4 4 126 MATH std 5.indd 126 30/07/2021 14:49 FOR ONLINE USE ONLY DO NOT DUPLICATE Alternatively, using diagram; 3 4 this equals to 1 1 2 4 LY Or N _ 3 _ 1 _ 3 _ 2 − = − O 4 2 4 4 3−2 _ = 4 1 = _ . 4 SE 3 _ 1 _ 1 _ U Therefore, − = . 4 2 4 E Example 2 N _ 3 1 − _ = LI 5 4 Solution N _ 3 1 3 4 1 5 − _ = _ × _ − _ × _ O 5 4 5 4 4 5 ( 3 × 4) − (1 × 5) ___________ = R 20 12 − 5 _ 7 _ FO = = . 20 20 3 _ _ 1 7 _ Therefore, − = . 5 4 20 127 MATH std 5.indd 127 30/07/2021 14:49 FOR ONLINE USE ONLY DO NOT DUPLICATE Example 3 _ 5 4 − _ = 6 7 Solution _ 5 _ 4 5 _ _ 7 _ 4 _ 6 − = × − × 6 7 6 7 7 6 LY ( 5 × 7) − ( 4 × 6) = ___________ 42 _35 − 24 _ 11 N = = . 42 42 O _ 5 _ 4 11 _ Therefore, − = . 6 7 42 Exercise 6 SE U Answer the following questions: _ 8 _ 2 _ 4 2 1 _ 1 1. − = 2. − _ = _ 3. − = 9 6 8 10 2 3 E _ 15 14 25 13 3 _ 2 4. − _ = 5. _ − _ = _ 6. − = N 20 40 30 90 7 14 LI _ 3 4 6 _ 2 54 _ 9 7. − _ = _ 8. − = 9. _ − = 4 9 8 5 60 20 N 5 _ _ 2 8 10 6 _ 4 10. − = 11. _ − _ = _ 12. − = 9 4 15 30 7 14 O 9 _ 3 8 1 3 _ 3 13. − _ = 14. _ − _ = _ 15. − = 20 10 12 5 5 8 R 20 _ _ 8 _ 24 19 10 5 16. − = 17. − _ = 18. _ − _ = FO 22 11 28 56 13 39 46 _ _ 40 _ 18 7 19. − = 20. − _ = 50 100 30 60 128 MATH std 5.indd 128 30/07/2021 14:49 FOR ONLINE USE ONLY DO NOT DUPLICATE Multiplication of fractions with different denominators Example 1 1 1 3 2 (a) _ × _ = (b) _ × _ = 3 4 4 3 Solution The solution is the fraction whose numerator is the product of the two LY numerators and denominator is the product of the two denominators. 1 1 1×1 1 × _ = _ (a) _ _ N = . 3 4 3×4 12 O _ 1 1 _ _ 1 Therefore, × = . 3 4 12 Altenatively, _ 1 1 SE × _ = 3 4 U Steps: 1. Construct a rectangle and subdivide it into 3 horizontal rectangles and E 4 vertical rectangles. _ 1 N 2. Shade the first vertical rectangle representing and the first horizontal 1 4 _ rectangle representing . LI 3 3. Count the number of small rectangles which have been shaded twice. N There is only one such rectangle. This is the numerator. 4. The total number of all small rectangles is 12. This is the denominator. O R FO _ 1 _ 1 _1 Therefore, × = . 3 4 12 129 MATH std 5.indd 129 30/07/2021 14:49 FOR ONLINE USE ONLY DO NOT DUPLICATE _ 3 _ 2 (b) × = 4 3 Steps: 1. Subdivide a rectangle into 4 horizontal and 3 vertical rectangles using equally spaced lines. 3 _ 2. Shade 3 out of 4 horizontal rectangles which represent . Also, shade LY _ 2 4 2 out of the 3 vertical rectangles which represent . 3 3. Count the number of small rectangles that have been shaded twice. N There are 6 such small rectangles. This is the numerator. O The total number of small rectangles is 12. This is the denominator. SE U E _ 3 _ 2 _6 _ 1 Therefore, × = or . 4 3 12 2 N Example 2 LI _ 5 3 × _ = N 8 4 O Solution Steps: R 1. Subdivide a rectangle into 4 horizontal and 8 vertical rectangles using equally spaced lines. FO 5 _ 2. Shade 5 of the 8 vertical rectangles. This represents . 8 3 _ 3. Shade 3 of the 4 horizontal rectangles. This represents . 4 130 MATH std 5.indd 130 30/07/2021 14:49 FOR ONLINE USE ONLY DO NOT DUPLICATE 4. Count the number of small rectangles that have been shaded twice, once horizontally and once vertically. There are 15 such small rectangles. This is the numerator. The total number of small rectangles is 32. This is the denominator. LY N O 5 _ _ 3 15 _ Therefore, × = . 8 4 32 Exercise 7 SE U Answer the following questions: _ 2 _ 3 _ 6 1 2 _ 3 _ 1. × = 2. × _ = 3. × = E 5 7 9 7 4 5 3 _ _ 5 _ 13 _ 2 _ 7 1 _ N 4. × = 5. × = 6. × = 8 10 15 4 12 6 LI 4 _ _ 6 1 _ 2 _ 2 _ _ 7 7. × = 8. × = 9. × = 5 11 4 7 3 10 N 8 _ 1 _ _ 2 _ 3 _ 11 _ 1 10. × = 11. × = 12. × = 9 3 4 4 13 5 O _ 3 1 _ 5 _ 4 _ 7 _ 1 _ 13. × = 14. × = 15. × = 14 2 9 6 8 5 R _ 2 5 _ 3 _ 5 _ 3 _ _ 6 16. × = 17. × = 18. × = 17 7 5 9 4 10 FO 1 _ 4 _ _ 17 _ 1 19. × = 20. × = 4 5 19 2 131 MATH std 5.indd 131 30/07/2021 14:49 FOR ONLINE USE ONLY DO NOT DUPLICATE Multiplication of fractions by a whole number Example 1 1 4 × _ = 5 Solution When a fraction is multiplied by a whole number, write the whole number LY over one. 4 _ In this example, 4 = . 1 N Thereafter, multiply numerator by numerator and denominator by denominator: O 1 4 1 4 × _ = _ × _ 5 1 5 4 _ . = 5 1 _ 4 _ SE Therefore, 4 × = . U 5 5 Example 2 E _ 2 × 12 = N 3 Solution LI _ 2 2 12 × 12 = _ × _ N 3 3 1 O 24 _ = 3 R = 8. FO _ 2 Therefore, × 12 = 8. 3 132 MATH std 5.indd 132 30/07/2021 14:49 FOR ONLINE USE ONLY DO NOT DUPLICATE Exercise 8 Answer the following questions: 3 _ 2 _ 5 _ 1. 6 × = 2. × 9 = 3. × 30 = 4 3 6 1 _ 1 _ 2 _ 4. × 24 = 5. 2 7 × = 6. 1 5 × = LY 2 9 5 1 _ _2 5 _ 7. 1 2 × = 8. × 4 = 9. × 18 = 6 10 9 N 1 _ _ 1 1 _ 10. 21 × = 11. × 36 = 12. × 32 = O 7 3 8 1 _ 1 _ _1 13. × = 5 14. 2 0 × = 15. × 50 = 4 10 16. 3 6 _3 × = 11 17. 1 SE _1 3 × = 13 18. 4 1 _ 0 × = 20 _7 2 _ U 19. × 10 = 20. × 25 = 10 5 Word problems involving fractions E N Example 1 LI 5 _ A household had one bag of sugar. If Anna used of the sugar in the bag 1 6 N _ and Magoma used of the sugar in the bag, what fraction of the sugar 8 O was used by both Anna and Magoma? Solution R _ 5 _ 1 Anna used of the sugar and Magoma used of the sugar. FO 6 8 Add the two fractions: 133 MATH std 5.indd 133 30/07/2021 14:49 FOR ONLINE USE ONLY DO NOT DUPLICATE _ 5 1 20 + 3 23 + _ = _ = _ 6 8 24 24 23 _ . = 24 23 _ Therefore, Anna and Magoma used of the sugar. 24 Example 2 6 _ LY Ali harvested of the onions grown in a certain farm, and Zaituni 7 1 _ harvested of the onions from the same farm. What is the difference 9 N between the fraction of onions harvested by Ali and the fraction harvested by Zaituni? O Solution _ 6 1 _ Ali harvested and Zaituni harvested of onions. 7 6 _ _ 1 _ Subtract the two fractions: − = 9 54 − 7 SE 47 = _ . 7 9 63 63 U 47 _ Therefore, the difference between the two fractions is . 63 Example 3 E _3 Grandmother drinks of a litre of milk per day. How many litres of milk 4 N will she drink in 6 days? LI Solution 3 3 6 N _ × 6 = _ × _ 4 4 1 O _3×6 18 _ = = 4×1 4 R 9 _ = 2 FO 1 4 _ . = 2 1 _ Therefore, the grandmother will drink 4 litres of milk in 6 days. 2 134 MATH std 5.indd 134 30/07/2021 14:49 FOR ONLINE USE ONLY DO NOT DUPLICATE Exercise 9 Answer the following questions: 1 _ 1. A teacher spent of her salary to pay school fees for her daughter 2 3 _ and for domestic expenses. What fraction of her salary was used 5 for both expenses? LY 3 _ 2. Half of the garden was planted with okra and was planted with 8 green vegetables. What fraction of the garden was planted with okra N and green vegetables? O _3 2 _ 3. John eats of bread and Elizabeth eats of the same bread. What 5 7 fraction of bread is eaten by both? _ 1 SE 4. On the first day, a mother cooked kilogram of rice. The second day 1 2 _ U she cooked kilogram of rice. What fraction of a kilogram of rice was 3 cooked in both days? E 1 _ 1 _ 5. During the holidays, of the students travelled by bus and of the 4 3 N students travelled by train. What fraction of the students travelled by both bus and train? LI 3 _ _ 1 6. Baraka took of the bags of beans and Asha took of the same N 5 10 bags of beans. What fraction of the bags of beans was not taken? O _ 2 7. Muungano Primary School has 80 cattle. The school decided to sell 5 R of the cattle. How many cattle remained at the school? FO _ 2 _ 1 8. Amina took of a piece of cloth to make a gown. Anna took of 5 10 the same piece of cloth. What fraction of the piece of cloth was taken by both? 135 MATH std 5.indd 135 30/07/2021 14:49 FOR ONLINE USE ONLY DO NOT DUPLICATE 9. Juma drew a line segment of length 9 centimetres. Then, he drew 1 _ another line segment whose length is of the length of the first line 3 segment. What is the length of the second line segment? 10. One day, my brother, sister and uncle helped my mother to cultivate her _ 1 _ 3 garden. My brother cultivated of the garden, my sister cultivated 5 2 10 _ of the garden and my uncle cultivated of the garden. What fraction LY 5 of the garden was cultivated by all three? _ N 1 11. Eliudi had 20 bags of sorghum. He gave of the bags to his brother. 2 How many bags of sorghum did the brother get? O _ 1 12. A hospital has 48 beds. If of the beds are made of iron, how many 12 beds at the hospital are made of iron? SE _ 1 13. A Kangaroo ran of a distance of 15 kilometres. What distance did U 3 the Kangaroo run? 1 _ E 14. Mwembechai primary school enrolled 30 pupils in kindergarten. If 6 N of the enrolled pupils did not turn up, how many pupils did not turn up? 3 _ LI 15. A technical school has 80 teachers. If are female teachers, what is 4 the number of male teachers? N O Exercise 10 Answer the following questions: R _ 7 _ 8 FO 1. Which of the fractions and is greater than the other? 8 9 3 _ 1 _ 2. What is the difference between and ? 5 6 136 MATH std 5.indd 136 30/07/2021 14:49 FOR ONLINE USE ONLY DO NOT DUPLICATE _ 6 7 3 3. + _ = _ 0 × = 4. 1 10 9 4 _ 54 3 3 × _ = _ _ 11 5. 6. Subtract from 60 4 7 12 _ 13 9 5 7. × _ = 8. _ × 28 = 15 10 7 _2 1 _ 9. Find of 200 people. 10. Multiply by 8 LY 5 8 3 _ 17 _ _ 19 11. Multiply 24 by 12. − = 12 34 68 N _1 _ 2 _15 _ 38 52 13. + + = 14. − _ = O 10 20 40 40 80 _ 2 _ 2 _ 2 _ 2 1 1 15. + + = 16. + _ + _ = 4 3 6 5 3 5 4 _ _ 5 17. − = 6 9 SE 8 9 U _ 18. Find the difference between and . _ 9 27 1 _ 19. Masalu had 88 mangoes. If he sold of the mangoes, how many E 8 mangoes were sold? N _ 5 _1 20. Furaha cultivated of his farm and his friend cultivated of the same LI 7 8 farm. What fraction of the farm was not cultivated? N O Summary 1. In multiplying a fraction by another fraction, multiply numerator by R numerator, and denominator by denominator. FO 2. In multiplying a whole number by a fraction, the whole number is first expressed as a fraction. 3. When an improper fraction is simpl
