REDUCTION TO LOWER AND HIGHER TERMS 41 3. Which kind of fraction has a numerator smaller than its denominator? Which kind has a numerator equal to or larger than its denominator? A mixed number is an integer and a fraction written together. 5, 173, 4511, and 107 are mixed numbers. Classify each of the following as a proper fraction, an improper fraction, an integer, or a mixed number: 3 1 7. 12, 8, 18. 1. 12, 14, 161. 4. 1%, 25, 4. 1. Multiply by 2 both the numerator and the denominator in the fraction. What is the answer? 1 In mastering this classification, the following game will be found helpful. With the help of a rubber pen or its substitute, a sharpened eraser on the end of a pencil, numbers similar to those above are written on cards about four inches square. These cards are held before the class and flashed one at a time. As each card is displayed, the teacher names the child who is to classify the number as a proper fraction, an improper fraction, an integer, or a mixed number. If a child fails in his classification, he must stand by his seat until he can give the correct answer when another child fails. When he has given a correct answer, he takes his seat. If a teacher prefers, the numbers may be written on the board one at a time instead of being shown on cards. 2. Multiply by 4 both the numerator and the denominator in the fraction. What is the answer? 3. Which is larger, or ?or? (See diagram.) 1 x 2 2× 2 Which has the greater value? 5. Compare with 1 x 4 4 X 4 6. Which has the greater value, or 7. What is the effect on the value of a fraction, of multiplying both terms by the same number? 8. Find, in the diagram above, the number of halves that equal . Find the number of eighths that equal 18. 9. Compare with or. Which has the greater value? 10. Compare 18 with 4 ÷ 4 16 ÷ 2 or §. II. What is the effect on the value of a fraction, of dividing both terms by the same number? From the facts learned above, it can be seen that the following is true: Both terms of a fraction may be multiplied or divided by the same number without changing the value of the fraction. The process of changing the terms of a fraction without changing its value is called reduction. 1. How do these fractions compare in value: and ? and ? 2. Which fraction has the smaller numbers as terms: or †? or ? REDUCTION TO LOWER AND HIGHER TERMS 43 When the terms of a fraction are changed to smaller numbers without changing the value of the fraction, the fraction is reduced to lower terms. To reduce a fraction to lower terms, divide both its numerator and its denominator by the same number. A fraction is in its lowest terms when its terms are divisible by no number greater than one. Which of the following fractions are in their lowest terms? Reduce those that are not in their lowest terms: 27. Divide 326 by 4. Reduce the fraction in the quotient to its lowest terms. In solving the following problems, reduce each fraction to its lowest terms before placing it in the quotient: 1. Which fraction has the larger numbers as terms, or ? or &? When the terms of a fraction are changed to larger numbers, the fraction is raised to higher terms. To raise a fraction to higher terms, multiply both its terms by the same number. 26. Reduction of an Improper Fraction How many inches are there in 10 inches? In 15 inches? Notice that ÷ 3 = 10 ÷ 2 and that 15 ÷ 4 = 15 ÷ 4. In both problems above, the number of inches could have been found by dividing the numerator of each fraction by its denominator. To reduce an improper fraction to an integer or a mixed number, divide the numerator by the denominator, How many quarters of a yard are there in 23 yards? In 1 yard there are 4. In 2 (2 and 2) yards there are 2 × + §. 2 × ‡ = 4. + 2 = 4. 22 yd. = 4 yd. 1. How many eighths of an inch are there in 13 inches? Explain. How many half hours are there in 3 hours? Explain. REVIEW 3. How many thirds of an apple in 43 apples? Explain. 45 It will be noticed that the answer for each of the last four problems could have been found by multiplying the integer by the denominator of the fraction, adding the numerator to their product, and placing the result over the denominator. To reduce a mixed number to an improper fraction, multiply the integer by the denominator of the fraction, add the numerator to the product, and place the result over the denominator. 1. Classify each of the following as a proper fraction, as an improper fraction, or as a mixed number: 1 In this work it will be found helpful to give daily practice, using fraction cards covering all the forms of reduction. See footnote, page 41. |