7. A farmer had $153. How much had he left, after laying out $4§ for flour? 8. If I start upon a journey of 137 miles, and go 898 the first day, how far have I to go on the second? 9. A certain garden contained 1 acres, of which of an acre were occupied with fruit, and the rest with vegetables. How much was planted with vegetables? 10. If I sell of a farm to one man, and of the remainder to another, what part have I left? MULTIPLICATION OF FRACTIONS. EXAMPLE FOR THE BOARD, Multiply 8 by 35; that is, multiply by 28. as large times 17 is If any number is multiplied by 3, the product is as if it were multiplied by 1. In other words, the same as of 7. Multiplication of fractions is therefore performed in the same way as reduction of compound to simple fractions. The answer to the above sum is 1924 3. Multiply 3 by 2; by ; by 61. 83, or 3033. 4. Multiply 7 by 4; by 93; by 10. 5. If a horse eats 64 quarts of oats a day, how much will he eat in 5 days? In 7 days? 6. If a barrel of flour costs $5}, how much will 74 barrels cost? 7. If a locomotive runs 17 miles an hour, how far will it run in 5 hours? 8. How much must I pay for 96 pounds of sugar, at 11 cents a pound? 9. When cider is $0.183 a gallon, what will be the cost of 23 gallons? DIVISION OF FRACTIONS. EXAMPLES FOR THE BOARD. Divide by. Divide 24, (19) by 15, (14.) The answer to the first question, dividing numerator by numerator, and denominator by denominator, is found to be The second example does not admit of so ready a division. But if we reduce both fractions to a common denominator, the question is resolved into the division of 2 by 33, which gives 162, or 184. [See Mental Arithmetic, Sect. XXIV.] We may obtain the quotient in another manner, as follows: 18 1 13 contains 1, 13 times. It contains, 9 times as often as 1; that is, 9 times 18, or 162 times. It contains 4, as often as; that is, of 162, or 162 times. Now, if we had inverted the divisor, and multiplied 18 by, the result would have been the same. Therefore, when one fraction cannot be directly divided by another, we may either reduce them both to a common denominator, and divide their numerators, or invert the divisor, and proceed as in multiplication. 1. Divide by 2, (); by ; by . 7. What is the quotient of 14 by 24; by 15; by 21? 8. What is the quotient of 3 by ; by 1; by 6,5? 23 9. If 5 barrels of flour cost $22, what is the price per barrel? 10. If a labourer receives $9.47 for 8 days' work, what are his daily wages? 11. Divide 131 by 9; by 217; by 423; by 16. REDUCTION OF FRACTIONS. Reduce EXAMPLE FOR THE BOARD. to its lowest terms. | | Dividing any number by 1 does not alter its value. Therefore, if we can find any number that will divide both the numerator and denominator of a frac tion, without a remainder, we may perform the division, and the resulting fraction will have the same value. In this example, we find that 7 will divide both 42 and 56. As equal 1, 6 , which is the quotient of by or 1, is equivalent to . Dividing again by 2, we obtain 2, as the lowest terms of the fraction 42. The discovery of common divisors may often be facilitated, by attending to the following rules, viz.: 2 will divide any number, whose right-hand figure is either 0, 2, 4, 6, or 8. 3 will divide any number, if the sum of its figures is divisible by 3. 4 will divide any number, if its two right-hand figures are divisible by 4. 5 will divide any number, whose right-hand figure is either 0 or 5. 9 will divide any number, if the sum of its figures is divisible by 9. 10 will divide any number, whose right-hand figure is 0. 11 will divide any number, if the sum of its odd digits, (the 1st, 3d, 5th, &c.,) differs from the sum of its even digits, (the 2d, 4th, 6th, &c.,) by 0 or 11. 1. Reduce each of the following fractions to its lowest terms. ; ; 27; 7; 700; 10; } }; 1; 13. 2. Reduce each of the following fractions to its 81.. 45. 17. 180. 500. lowest terms. 36" 901 6 27 210 820 600 2160 TO REDUCE DECIMALS TO FRACTIONS. Write the decimal for a numerator, and the denomination tenth, hundredth, &c. for a denominator, and reduce this fraction to its lowest terms. Thus, .5 is or ; .25 is or 4. 5 100 1. Reduce each of the following decimals to a fraction. .3; .07; .009; .216; .00309; .0007803; .91604; .0007. TO REDUCE FRACTIONS OF A HIGHER DENOMINATION, TO WHOLE NUMBERS OF A LOWER, AND THE REVERSE. This may be done in the same way as reduction of whole numbers, by multiplying, or dividing, as the case may require. EXAMPLES FOR THE BOARD. Reduce of a mile to furlongs, &c. Reduce 24min. 31sec. to the fraction of a day. 1. Reduce of a bushel to pecks, &c. 2. Reduce 3. Reduce of a day to hours, &c. of a gallon to pints, fluidounces, &c. 4. Reduce 5s. Od. 3qr. to the fraction of a £. 5. Reduce 1qr. 2na. to the fraction of a yard. 6. Reduce 9d. 1gr. to the fraction of a shilling. 7. Reduce .934£ to s. d., &c. 8. Reduce 5cwt. 3qr. to the fraction of a ton. 9. Reduce .076 miles to furlongs, &c. 10. Reduce 1R. 30r. to the fraction of an acre. 11. Reduce .89m to the fraction of a gallon. 12. Reduce 5.7min. to the fraction of a year. 13. Reduce .9889 T. to cwt., qr., &c. To drams and the fraction of a dram. CHAPTER IX. ANALYSIS. In the following chapter, a mental question is first given, to illustrate the succeeding example for the slate. Let the pupil learn to first find the answer for one, and afterwards for many. 1. If 4 barrels of flour cost $20.00, what will be the cost of 1 barrel? Of 7 barrels? 2. If 3.5 barrels of flour cost $17.75, what will be the cost of 1 barrel? Of 9.25 barrels ? 3. If 5 bushels of wheat cost $5.00, what will 1 bushel cost? 3 bushels? 4. If 6.25 bushels of wheat cost $7.125, what will 1 bushel cost? 9.5 bushels? 5. If 9 horses eat 54qts. of oats in a day, how much will 1 horse eat? 5 horses? 6. If 19 horses eat 4bu. 2pk. of oats in a day, how much will 1 horse eat? 13 horses? 7. What part of a shilling is 1 penny? 2 pence? 7 pence ? 8. What part of a pound (240 pence) is 1 penny? 78. 6d. (90 pence.)? 9. What part of a month is 1 day? 2 days? 16 days? 10. What part of a year is 1 day? (79 days.)? 11. What part of .8 is .1 .2 .3? 2mo. 19dy., 12. What part of 3.5 (3.50), is .01 .25? 9.25? 13. What part of 19 is 1? 13? 14. What part of 11s. 4d. 1qr. (545qr.) is 4s. 3d. (204qr.)? |