If EXAMPLES FOR MENTAL EXERCISE. you have 14 apples, and give one seventh of them to each of four boys, how many do you give away? Ans. A seventh to 1 boy, would be 2, and four sevenths, would be four times as much, or 8. What is of 14? If you have 48 cents, and give a twelfth of them, to each of two boys, how many do you give away? What is of 48?. A man has 35 sheep, and sells four fifths of them, how many does he sell? A boy has 40 marbles, and loses of them, how many does he lose? What is 40 multiplied by ? What is 36 multiplied by? What is of 21 of 24? of 81? 4 of 49? 2 of 16?g of 40? & of 45? 3 of 60?5 of 96 ? of 30? What is of 18? of 100? of 40? of 64 ? of 24? 20 of 28? 1⁄2 of of 48? of 81? of 144? 19 of 99? of 49? 4 of 32? 5 of 81?% of of 16? 3 of 12 of 18? of 33! of 271 12 NOTE TO TEACHERS.-Let the younger pupils omit the following exercises in small type till a review. EXAMPLES FOR THE SLATE, for older puPILS, WHO If a number is to be both multiplied and divided by two figures, it makes no difference which is done first, provided the same figures are used as multiplier and divisor.` For example, let a number be multiplied by 2, and divided by 9. We can divide first by 9, and then multiply the quotient by 2; or we can multiply first by 2, and then divide the product by 9, and the answer is the same. Thus 18 multiplied by 2, is 36; and this divided by 9 is 4. If then we multiply 12 by we divide by 4, to find one fourth of 12, and multiply by 3, to obtain three fourths, and the answer is 9. But if we should multiply 12 by 3, and then divide the product by 4, the answer would be the same. Thus 12 X 3: = 36 and 364: 9. Thus 9 is the same answer as is obtained by dividing 12 by the denominator, and multiplying the answer, by the numerator. What are the two ways in which 18 can be multiplied by 4-6? What will be the answer, if it is divided by 6 first, and the quotient multiplied by 4? What will be the answer, if it is multiplied by 4 first, and then the product divided by 6? RULE FOR MULTIPLYING WHEN ONLY THE MULTIPLIER IS A FRACTION. Divide by the denominator, to obtain one part, and multiply by the numerator, to obtain the required number of parts. But in case this division should leave a remainder; Multiply by the numerator first, and then divide the product by the denominator. EXERCISES FOR THE SLATE, FOR OLDER PUPILS. In all these cases it is best to multiply by the numerator first, and then divide by the denominator. If any remains after division, place the divisor under it, for a fraction. What is the rule for multiplying when the multiplier only is a frac EXAMPLES FOR MENTAL EXERCISE. 1. If 15 is five eighths of some number, what part of 15 is one eighth of that number? 2. If 12 is four sixths of some number, what part of 12 is one sixth of that number? 3. If 18 is six ninths of some number, what part of 18 is one ninth of that number? 4. If 15 is § of a number, what is of that number? 5. If 15 is of some number, what is that number? Let such exercises be stated thus. 6. If 15 is five eighths, a fifth of 15 is one eighth. A fifth of 15 is 3. If 3 is one eighth, then the whole is 8 times as much, or 24. 7. 24 is 8. 36 is of what number? of what number? 9. 42 is of what number? 10. If a man can do of a piece of work in 12 days, how long would it take him to do of it? (Ans.) It would take him only one sixth of the time to do one seventh that it does to do. of 12 is 2. It would take him 2 days. Let the remaining sums be stated as above. 11. If a man bought of a barrel of wine for 18 dollars, how much will cost? 12. How much will the whole cost? ៖ 13. Bought of a chaldron of coal for 24 shillings, how much will cost? How much will the whole cost? 14. If 15 is of some number, what is one eighth of that number? 15. What is the whole of that number? If 23 is of one number, what is that number? 16. If a man bought of a cask of brandy for 42 dollars, what is worth? what is the whole worth? 17. If of a month's board cost 12 dollars what is it a month? 18. If of a cord of wood cost 16 shillings, what and what would the whole cost? would cost, 23. How many times is 4 contained in 5? (Ans.) Once and one over. 24. What is of 1? What is of 1? It will be seen that in fractions, as in whole numbers, it makes no difference in the product, which factor is used as multiplier. For 12 x - 9 = 36 = 9. ་ × 12 Here when the whole number is used as multiplier, the answer is an improper fraction, which, if changed to whole numbers, is 9. Multiply 18 by and by 18, and tell in what respects the answers differ. Multiply by 14, and 14 by 3. Is there any difference in the value of the answers ? MULTIPLICATION WHERE BOTH FACTORS ARE FRAC TIONS. 1. If we had an orange and should give away half of this what part of an orange should we give away? How much is of. 2. If we have of an orange, and should give away of it, what part of a whole orange should we give away? (Ans.) If the two halves of any thing be divided into 4 pieces each, the whole is divided into 8 pieces. Takingof one of these halves then, is taking whole. of is 1. of the 3. If we have of an orange, and give away half of it, what part of the whole orange do we give away ? Ans. If an orange is divided into 4 pieces, and each of these pieces are halved, the orange is divided into 8 pieces, and each piece is of the whole. ofis. 4. If you receive of an orange, and you give of it away, what part of the whole orange do you give away? Ans. The orange is divided into 3 parts; if each of these parts is divided into 4 parts, the whole orange would be divided into 12 parts, and each part is of the whole. ofis. 5. If you have an apple and it is cut into 5 equal parts, what part of the apple is each piece? If each piece is cut into 3 equal parts, what part of the whole apple is each piece? Ans. If an apple is cut into 5 equal parts, each part is one fifth of the whole, and if each of these pieces is divided into 3 parts, each part is of the whole. 6. If you have an orange, and it is divided into 3 equal |