4. A bicycler rode 27 miles on Monday, 33} miles on Tuesday, 37* miles on Wednesday, and 427 miles on Thursday. How far did he ride in the four days ? 5. A dry-goods merchant sold a lady 184 yards of flannel, 21% yards of silk, and as many yards of calico as of both the other goods. How many yards in all did he sell ? a SUBTRACTION OF FRACTIONS. INDUCTIVE STEPS. 2. 3 1. From 5 subtract . - what? 3. If you have $3 (of a dollar) and spend $, how much have you left ? 4. If you have $} and spend $1, how do you find the remainder? 5. What kind of fractions can be subtracted without reduction. 6. What kind require reduction ? Reduction to what? 7. Give four brief directions for such reduction. 8. What introductory step is sometimes necessary ? PRINCIPLES. 1. Only like fractions can be subtracted. 2. Unlike fractions can be reduced to like fractions and then subtracted. EXERCISES. 1. Find the difference between and : Explanation. Since and are like fractions, having a common denominator, 11, their difference is 8 elevenths 5 elevenths, or 3 elevenths. 80 2. What is the difference between 1 and ? Explanation. tions, we reduce them to eightieths, * -*= 46 — 3= 13 용 making them like fractions. 16 - } = 45 – 3%= %. Explanation. rately. cannot be taken from ; but 1, 75 75 taken from 11, equals ; + 3 1; } 39 33 10 - for -7 = 3. Uniting the two results, we have 33, the remainder. 11ģ = 1014 RULE. 1. Reduce unlike fractions to a common denominator. 2. Write the difference of the numerators over the common denominator. 3. Subtract integers and fractions separately, and unite the results. 4. From take . From take }. . 7. 1198 1493 8. 5893 – 674 3. 3 - 175 . 9. 754 – 6341 . - 1 10. 721 5814 11. 4912 361 12. 42 100 39 85 58 85 2. 237 900 4. 13 5. 77 336 29 25 12. Find the value of : 1. 51 — 204 8. 48 222 15. 76-31 - 6876 2. 66 — 364 9. 358 - 294. - 16. 2445 - 9 3. 61 - 593. 10. 443 - 273. 17. 4311 — 2838. 4. 38 - 374. 11. 483 -9. 18. 1011 - 938. 5. 59 — 3214. 12. 731 — 27. 19. 2834 – 1635. 6. 111 — 3114. 13. 22} --- 7. 20. 561 – 294 . 7. 36% -- 27. 14. 881 - 53:14. 21. 651 - 304. } 13. Answer the following inquiries : 1. 5 + - 12 = ? ģ $ ? 304 — 1712 : = - NotE.—The teacher will suggest the shortest method of answering the above inquiries. 49 19 17 200 27 20 To = ? PROBLEMS. 1. 31 yards, 45 yards, and 124 yards were cut off from a piece of silk containing 301 yards. 301 yards. How many yards remained ? 2. A man spent of his income for rent, 1 for food, and } What part of his income remained ? 8 for other expenses. 3. A farmer sold of his corn to one man, & to another, and had 50 bushels remaining. How much corn had he at first? 4. Show that the fraction is greater than and less than 4. 5. If I pay my grocer $18, my coal dealer $274, and my tailor $221, how much will I have left out of four 20-dollar bills? 6. ^ of a pole is in the mud, Í of it is in the water, and the rest of it is in the air. What part of it is in the air ? 7. Show that 13 2015 — 616 + 3 — 18 + 8} — 43 28 – 1013 442 is a correct equation. 8. Find the second members of these : 1. 450 + (12 X 5) - 8646 501 +4=? + 12 5 MULTIPLICATION OF FRACTIONS. INDUCTIVE STEPS. 1. How much is 2 times 3 dollars ? 6. Have you been multiplying numerators or denominators ? 7. Then what effect has multiplying the numerator? 8. What is 3 times f? What are the lowest terms of f? 1 9. Then 3 times ] $How could you have obtained f more directly than by multiplying the numerator? 10. What effect, then, has dividing the denominator? 11. Is that effect in agreement with principle 4, page 68 ? Why? PRINCIPLE. Multiplying the numerator or dividing the denominator multiplies the fraction. EXERCISES. 1. Multiply to by 4. Process. Explanation. To X 4 = 1 According to the principle we may multiply the numerator or divide the denominator. Since the denominator, 16, is divisible by 4, we divide and obtain the result, 7. 2. Multiply 4 by 17. Process. 4 X 1 = 2 = 111 4 X 77 28 Explanation. Since the denominator, 17, is not exactly divisible by 4, we multiply the numerator by 4 and obtain the result, íj = 117. RULE. To find the product of an integer and a fraction divide the denominator or multiply the numerator by the integer. |