equal to 42 84. 1 1, is goi is2, and are is is, and are. The above numbers are therefore equivalent to 42 210 3210, and 2118, which, when added, give 10 13. Add 6, 7, and 83. 14. Add 9, 104, and 114. 15. What is the sum of 5 and 6목? 16. What is the sum of 3 and 4? 35 17. What is the sum of 140 and 26? 18. What is the sum of 윽,, and ? 19. What is the sum of 18, 14, 57, and 26? 20. What is the sum of 윽, 3, §, and 14? SUBTRACTION OF FRACTIONS. EXAMPLE FOR THE BOARD. From 14 subtract 28. 18. are equivalent to, , and to. 63 We cannot subtract from ; we therefore add 1 or to the 63 18 to the of the minuend, and carry 1 118 2 units of the subtrahend. 6. Reduce the following fractions to decimals, and find their difference. 144 and 21; 30% and 4, 17 and. 45 7. A farmer had $154. How much had he left, after laying out $4 for flour? 8. If I start upon a journey of 1374 miles, and go 89 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, multiplyby. If any number is multiplied by, the product is as large as if it were multiplied by 1. In other words, times 17 is 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 4, or 30 83 1. Multiply by 3; by; by 21. 34 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 $54, how much will 74 barrels cost? 3 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 $0.184 a gallon, what will be the cost of 237 gallons? DIVISION OF FRACTIONS. EXAMPLES FOR THE BOARD. Divide by . Divide 24, (무) by 1, (.) 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 by, which gives 162 , or 10. [See Mental Arithmetic, Sect. XXIV.] We may obtain the quotient in another manner, as follows: 1 contains 1, 4 times. It contains 3, 9 times as often as 1; that is, 9 times, , or 12 times. It contains as often as; that is, of 12, or times. Now, if we had inverted the divisor, and multiplied 162 9 by 4, 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 . 2. Divide by 4; by; by 1, (). 3. Divide by ; by; by. 4. Divide by; by; by 21. 5. Divide 1 6. Divide 7 by 2; by 3; by 4. by; by 3; by 64. 7. What is the quotient of 14 by 24; by 15; by 211⁄2? 8. What is the quotient of 3 by; by 1s; by 6? 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 13 by 91s; by 217; by 42; by 16. REDUCTION OF FRACTIONS. EXAMPLE FOR THE BOARD. Reduce to its lowest terms. 6 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 by or 1, is equivalent to . Dividing again by, we obtain 2, as the lowest terms of the fraction 42 of 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. ; ; ; ; ; ; ; ; 15 12 이이 8 2. Reduce each of the following fractions to its lowest terms. 27 500 820 256 63 1024; ووو ;;;;; 아이이 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 for 4. 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. 1. Reduce of a bushel to pecks, &c. 2. Reduce of a day to hours, &c. 3. Reduce 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. 1qr. 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 .9889T. to cwt., qr., &c. To drams and the fraction of a dram. |