Hence, IF A CYPHER OR CYPHERS OCCUR WHERE THE RULE REQUIRES A FIGURE OF THE MINUEND TO BE DIMINISHED, CALL EACH CYPHER A I, AND DIMINISH THE NEXT SIGNIFCANT FIGURE BY 1. EXAMPLES FOR PRACTICE. 11. A person had property to the amount of $62,007, of which he lost by a fire, $12, 148. What had he left ? Ans. $49,859. 12. A wine merchant bought 721 pipes of wine for $90,846 and sold 543 pipes for 86,049 dollars. How many pipes has he left, and what are they worth to him? Ans. 178 pipes--worth to him $4,797. 13. From 200,000 take 99,999. Ans. 100,001. 14. From 100,000 take 55,555. Ans. 44,445. 15. From 360,418 take 293,752. Ans. 66,666. 16. From 54,026 take 9,254. Ans. 44,772. After borrowing, the rule in most Arithmetics requires to increase the next figure of the subtrahend, instead of diminishing the next figure of the minuend by 1. The effect upon the answer is the same in both cases. We have given that which is simplest in principle, and easiest in practice. We will illustrate the other method, how. ever, by an example, and leave the student to choose for himself. From 32 16 Ans. Here we borrow, and, by our rule, must diminish the 3 tens by i ten, leaving 2 tens. From this, we take the 1 ten, and the remain. der is 1 ten, The other rule does not diminish the 3 tens, but increases the 1 ten, by 1 ten; making 2 tens. This taken from 3 tens, leaves 1 ten remainder, as before. A E B C F Then A B may be called a minuend, C D, a subtrahend, and E B, the remainder. If to CD, I should add a portion (D F) equal to the remainder E B, the whole, CF, would be just as long as the remainder A B. Hence, to prove Subtraction, I. ADD THE REMAINDER AND TIE SUBTRAHEND. Their sUM OUGHT TO EQUAL THE MINUEND. Also, if from the minuend, A B, I take the remainder, E B, a portion, A E, will be left, just as long as the subtrahend, C D. Hence, to prove Subtraction, II. TAKE THE REMAINDER FROM THE MINUEND. THE DIFFERENCE OLGUT TO BE EQUAL TO THE SUBTRAHEND, EXAMPLES FOR PRACTICE. 21. Take one from one million. 22. A man having $365.90 lays out $168.99. What has he left ? 23. In Boston, in 1,810, there were 33,250 inhabitants, and in 1,820, 43,278. What was the increase in 10 years ? 24. In 1,820, in Boston there were 43,278 inhabitants, and in New York 123,706. What was the difference in numbers ? 25. A man's income is $2,963.47 and he spends $1,596.89. What does le save ? 26. A gentleman bought a house and garden for $12,963.31. The house alone was worth $10,876.69. What was the garden worth, and what was the house worth, more than the garden ? 27. 7,800 4,390 : 47,809 – 13,389 | 30. 289; 367 – 189,367 31. 573,842 - 473,841 OBSERVATIONS ON SUBTRACTION FOR ADVANCED PUPILS. $ XXIII. The pupil will perceive that Subtraction is a process exactly the reverse of Addition. For, as Addition teaches to bring several numbers together into one sum, so Subtraction teaches to separato a single number into two others. He will also see that it is not only reverse in principle, but also in all its practical operations. For, whilo, in Addition, we often carry something forward to the next higher order ; in Subtraction, we take back, or borrow, from that order. Hence, also, it will be evident, that, in performing an operation, in which it is necessary to borrow, we must always begin on the right. For, in borrowing, we take a unit from the next higher order. Therefore, we must proceed from right to left, and of course begin at the right hand. The order of writing, prescribed by the rule, is only adopted for convenience. It brings the individual figures between which the subtractions are to take place, near each other. Which number should stand uppermost, is a matter of indifference. The arrange. mont, prescribed by the rule is the one usually adopted. Besides the modes of proof given in the last section, Subtraction may be proved by casting out 9s. As the Remainder and Sub. trahend, added together, are equal to the minuend, the rule is the same, in principle, with that for proof of Addition. The excess of Is in the Minuend, ought to equal the excess, obtained from the Remainder and Subtrahend. Hence, to prove Subtraction, I. Cast OUT THE 9A FROM THE REMAINDER AND SUBTRAHEND, AND SET DOWN THE EXCESSES OPPOSITE. II. Cast OUT THE IS FROM THESE EXCESSES, AND PLACE THE RESULTING EXCESS OPPOSITE THK MINUEND. III. CAST OUT THE 98 FROM THE MINUEND, AND, IF THE OPERATION HAS BEEN CORRECTLY PERFORMED, THE EXCERS WILL AGREE WITH THAT LAST FOUND, DIVISION. MENTAL EXERCISES. ♡ XXIV. 1. A boy divided 4 apples equally between 2 of his companions. How many did he give each ? If you divide 4 into 2 cqual parts, how many will there be in each part? 2. A gentleman bought 6 little books for 2 of his chil. dren. How many did he give each? If you divide 6 into 2 equal parts, how many will there be in each part ? 3. A boy, having 12 peaches, divided them equally among 3 of his companions. How many did he give each? If you divide 12 into 3 equal parts, how many will there be in each part ? 4. A man placed 16 sacks of grain in 4 equal heaps. Ilow many sacks in each hcap? If you divide 16 into 4 equal parts, how many will there be in each part? 5. George arranged 18 marbles in 3 equal rows. How many marbles in each row? If you divide 18 into 3 equal parts, how many will there be in each part ? 6. Lucy put 24 pins on a square pin-cushion, so that there was an equal number on each side. How many pins on each side ? If you divide 24 into 4 equal parts, how many will there be in each part ? 7. George made a five cornered figure with 30 pencils, putting an equal number of pencils on each side of the figure. How many pencils on each side ? If you divide 30 into 5 equal parts, how many will there be in each 8. In a window there are 35 panes of glass in 7 rows. How many panes in each row? If you divide 35 into 7 equal parts, how many will there be in each part ? 9. A man put 36 logs in 6 equal piles. How many logs in each pile? If you divide 36 into 6 equal parts, how many will there be in each part ? 10. 5 boys found a purse containing 25 silver dollars, and agreed to share the money equally. How many dollars had each boy? If you divide 25 into 5 equal parts, how many will there be in each part? 11. 20 boys chose sides to play ball. There were 2 sides. How many on a side ? If you divide 20 into 2 equal parts, how many will there be in each part ? part ? 12. If you travel 7 miles an hour, how many hours will it take you to travel 42 miles? If you divide 42 into 7 equal parts, how many will there be in each part ? 13. If you divide 24 into 6 equal parts, how many will there be in each part? How many, if you divide 18? Ilow many, if 36? If 42 ? If 48? If 54! If 60 ? 14. If you divide 14 into 7 equal parts, how many in cach part? How many, if you divide 21 ? How many, if 28! If 35 ? If 42? If 49? If 63 ?. If 56 ? 15. If you divide 64 into 8 equal parts, how many in each part? If 32 ? If 16? If 24? If 40 ? If 48? If 72 ? If 56 ? 16. If you divide 33 into 11 equal parts, how many in each part? How many, if 66 ? How many, if 99 ? If 22? If 44 ? If 55 ? If 77 ? If 88 ? 17. If you divide 48 into 12 equal parts, how many in cach part? If 24 ? If 96 ? If 108? If 72 ? If 84? If 36 ? If 60 ? 18. A man having 8 oxen, yoked them in pairs. How many pairs had he? How many twos in 8 ? 19. A boy having 12 oranges, found he could give 3 to each of his companions. How many companions had he? How many thrces in 12? 20. A man divided 15 cents among his children, giving them 5 cents apiece. How many children had he? How many fives in 15 ? 21. A merchant had 18 tea-cups, and he told his clerk to place them in piles of 6 cups each. How many piles were there? How many sizes in 18? 22. I wish to divide 16 boys into classes of 4 boys each. How many classes can I have ? How many fours in 16 ? 23. If a hundred dollars gain 6 dollars interest in 1 in how many years will it gain 48 dollars ? How many 6s in 48 ? 24. How many times 3 in 9? In 6 ? In 12? In 36 ? In 33 ? In 27 ? In 24 ? In 15 ? In 18? 25. How many times 5 in 25 ? In 35 ? In 45 ? In 55 ? In 10? In 15 ? In 20 ? In 30 ? In 40 ? In 50? In 60 ? 26. How many times 4 in 8? In 32? In 16? In 24 ? In 10°? In 36 ? In 44 ? In 48? In 52? 27. Repeat the year, |