78304 LESSON 6 ADDITION OF FRACTIONS AND MIXED NUMBERS 3 PROBLEM. Add 1, 2, and 12. SOLUTION 21 18 24 10 2 EXPLANATION Since the given fractions do not have the same denominator, we find the lowest common multiple of 8, 4, and 12, which is 24, and change all the fractions into 24ths. To change into 24ths, it is evident that the denominator 8 must be multiplied by 3 to produce the denominator 24; hence, the numerator 7 must also be multiplied by 3 to keep the value the same (Prin. 3, page 18). Therefore ; and similarly = 18, and 12-11Next, adding the new numerators, 21, 18, and 10, we have 49, which is written over the common denominator, 24, giving 41, or 224. PROBLEM. Add 182, 271, and 36§. SOLUTION EXPLANATION Adding the fractions only, we get 183. Adding the whole numbers, we get 81. Combining 183 with 81, we obtain 8233 as the final sum. Hence, we have the following: To add fractions: 1. If the fractions have the same denominator, add the numerators, write the sum over the common denominator, and reduce to lowest terms. 2. If the fractions do not have the same denominator, find the lowest common multiple of the given denominators, change the fractions into equivalent fractions having a common denominator, add the numerators of the equivalent fractions, write the sum over the common denominator, and reduce to simplest form. To add mixed numbers: First, add the fractions only; second, add the whole numbers; and, third, combine these two sums. Using the forms given above as models, find the sums of the following fractions, writing your answers in a ruled form. Check your work by again solving, compare results, correct errors, and when everything agrees, O. K. your work. EXERCISE 14 Add the following mixed numbers, solving and proving as in previ SUPPLEMENTARY WORK: EXERCISE I Add the following mixed numbers, and prove each addition: SUBTRACTION OF FRACTIONS AND MIXED NUMBERS PROBLEM. From 1 subtract §. EXPLANATION Changing the fractions to a common denominator, we have and, respectively. Subtracting 24 from 22, we have as the difference in value. 3 24 PROBLEM. From 181 subtract 53. SOLUTION 9 36 36 181 57 28 45 4 EXPLANATION 36 First subtract the fractions; then, the whole numbers. Changing the fractions to a common denominator, we have and, respectively. Since we cannot subtract 38 from 39, we add 1, or 38, to the, giving, from which we can subtract, producing. Subtracting 5 from 17, we have 12, to which we annex the fraction, giving 1237, the difference between 181 and 53. 36 Hence, we have the following: To subtract fractions: If necessary, change the fractions to equivalent fractions having the same denominator. Subtract the numerators of the equivalent fractions, write the result over the common denominator, and reduce the result to its lowest terms. To subtract mixed numbers: First subtract the fractions, borrowing 1 in fractional form if necessary; second, subtract the whole numbers, and annex the difference between the fractions. Using the form given as a model, subtract these mixed numbers. Solve, record answers; to prove, again solve on the back of your first solution, record answers, compare, correct errors, and O. K. your work. 6. 34-338-? 11. 87-293=? 1. 182-63=? 2. 75-33=? 3. 1411-8= ? 8. 819-563-? 4. 921-177=? 9. 354-12=? 5. 561-2918-? 10. 67-44% =? 14. 42-291=? 15. 158-3=? SUPPLEMENTARY WORK: EXERCISE J Subtract the following mixed numbers and prove each: 1. 597-183. 6. 831-28176. 11. 73352-1812. LESSON 7 MULTIPLICATION OF FRACTIONS AND MIXED NUMBERS PROBLEM. Multiply & by §. PROBLEM. Multiply 16 by 512. SOLUTION 7 13 16号X5 = 0x00 16X591 EXPLANATION First, change 16 into 8, and 5 into. Cancelling 5 into 65, and 12 into 84, and multiplying, we have 91 as the result. Hence: To multiply fractions and mixed numbers, change the mixed numbers into improper fractions, use cancellation if possible, and write the product of the numerators over the product of the denominators. In simple cases, we may use the following method: EXPLANATION First, multiplying the whole numbers, 15 and 6, we get 90. Second, multiplying by 6, and 15 by, we get 3 and 5, respectively. Third, multiplying the fractions, and, we get . Adding, we have 98 as the product. Hence: To multiply mixed numbers, we may multiply the whole numbers, multiply each fraction by each whole number, multiply the fractions, and add. |