PRINCIPLE: To reduce an improper fraction to an integer or a mixed number divide the numerator by the denominator. Change the following improper fractions to integers or mixed 1. Change 4 to an improper fraction. In 1 whole there are . In 4 wholes there are 4 times as many as there are in 1 whole. 4 wholes=4 times 8 eighths, or 32. 32+3=39. Therefore, 43 = 39. Change the following mixed numbers to improper fractions: 38. Change 5 to thirds; 3 to eighths; 7 to ninths; and 9 to fifths. 39. How many persons will 3 pies serve when cut into sixths? Exercise 4 The numerator and denominator are called the terms of a fraction. 1. Multiply both the numerator and the denominator of the fraction 2. How does of the top line in the diagram compare with of the other line? 3. In multiplying the denominator of the fraction by 2 how has the size of the equal parts been changed? In multiplying the numerator by 2 how has the number of equal parts forming the fraction been changed? 4. If the parts have been made half as large but there are twice as many of them, has the value of the fraction been changed? of 5. Multiply both the numerator and the denominator of the fraction by 3. How does the top line compare with 2 of the lower line in the diagram? 9 How has the size of the equal parts been affected? How has the number of equal parts forming the fraction been changed? Since the equal parts have been made as large but there are three times as many of them, has the value of the fraction been changed? 6. Multiply both terms of by 4. Draw a figure similar to those in the preceding examples and compare the result with the original fraction. From these examples we see that: Multiplying both terms of a fraction by the same number does not change the value of the fraction. Exercise 5 1. Divide both terms of the fraction by 4 as follows: = 16÷ 2. Compare the value of of the upper line of the diagram with of the lower line. 3. In dividing the denominator by 4 how has the size of the equal parts been affected? 4. In dividing the numerator by 4 how has the number of parts composing the fraction been changed? 5. Even though there are only as many parts forming the fraction, has the value of the fraction been changed if the equal parts have been made four times as large? 6. Divide both terms of the fraction by 3. How has the size of the equal parts been affected? How has the number of parts forming the fraction been changed? Show why the value of the fraction has not been changed. Also show this by the diagram. 7. Divide both terms of the fraction 18 by 5. Draw a diagram to show that the value of the fraction has not been changed. From these examples we see that: Dividing both terms of a fraction by the same number does not change the value of the fraction. The teacher should supply other examples if necessary to enable the pupils to see clearly how the size and number of the equal parts have been changed. Changing to Lowest Terms Exercise 6. We have seen in the preceding exercise that we may divide both the numerator and the denominator of a fraction by the same number without changing the value of the fraction. of The drawing of problem 1 in Exercise 5 showed that the line was the same as 1 of the line. The terms of the fraction are smaller or lower than the terms of the fraction 12. We change 12 to its lowest terms by dividing both terms by the same number, 4. Or we may say that we cast out a common factor from both numerator and denominator. A fraction is in its lowest terms when there is no number except 1 that is an exact divisor of both terms. Which is shorter: 182-8÷2-1÷2-2 ÷ 2 = 1 = 6÷ = = = This illustration shows that we save work in changing a fraction to its lowest terms by dividing both terms by the largest number possible rather than by several smaller numbers. Change the following to lowest terms: Exercise 7. Adding and Subtracting Like Fractions 1. What is the sum of $4 and $3? 2. What is the sum of 5 books and 3 books? 3. What is the sum of 5 eighths and 2 eighths? Notice that in the short form + the denominators merely stand for the name of the equal parts into which the whole has been divided. are adding 5 eighths and 2 eighths, which equal 7 eighths. We In this example we merely subtract the numerators instead of adding --}. them. Add or subtract the following: Change all answers to lowest terms and change answers that are improper fractions to integers or mixed numbers: 5 11.16.2 = 21.12+12+12· 6 = 3 9 = 3 12.12-12-17. + 22.1+1+1= 8 8. +1 = 13. 吾+香 = 18. - § 3 = = 16 23. 号十一号 19.-24. § + 3 − § 24.号+号-哥 = 15. 11-12-20.13+1 = 25. 26+28+8 = 26. Lucy bought 2 remnants of ribbon. In one there was of a yard and in the other of a yard. How many yards were there in both? 27. John bought of a bushel of onion sets from one dealer and of a bushel from another. How many onion sets did he buy from both? 94 28. Mr. Dean bought the following amounts of seed: pound of onion, pound of carrot, pound of peas, pound of beans and pound cabbage. Find the total weight oi these seeds. |