16. From 43 take 17. EXPLANATION. 7 units cannot be taken from 3 units; therefore, take 7 from 10, and to the 3 left, add the 3 in the larger number, 43. This gives 6 as the unit figure of the answer. Then increase the 1 ten of the smaller number, 17, by 1, making it 2, and take the 2 from the 4 tens of the larger number. The difference is 2 tens. Then, on the right of this, place the 6 units, and the true difference is obtained, viz. 26. SIMPLE SUBTRACTION, written. § 51.-1. It is obvious from the definition of subtraction, (§ 46,) that only two numbers can be employed in the operation, one of which is to be taken from, or out of, the other. The one to be taken from the other is called the subtrahend, and that from which the subtrahend is taken is called the minuend, while the number obtained is called the remainder. 2. Figures can be subtracted one from another when they represent numbers merely. If, however, they are employed to express definite kinds of objects, they can be subtracted only when both represent the same kind of objects; that is, bushels can be taken from bushels, dollars from dollars, &c.; but bushels cannot be taken from dollars. 51. What is subtraction? (§ 46.) How many numbers employed? What is the one called? What, the other? What the number ob tained? When can figures be subtracted? When, if they express definite objects? RULE FOR SUBTRACTION. Write down the two numbers, the smaller under the larger, placing units under units, and tens under tens, and take each lower figure from that standing directly above it. Whenever the lower figure is larger than that above it, take that lower figure from 10, and to the differ ence add the upper figure, remembering always, whenever an upper figure is increased by 10, that the next lower figure must be increased by 1, before it is taken from the figure above it. After the sum is performed, the accuracy of the operation is tested by adding the remainder to the subtrahend; when, if the work has been accurately performed, the sum of the two will be the minuend. 1. From 8567 take 3425. PERFORMED. Minuend 8 567 larger number. Remainder 5142 For this example no explanation is needed, each figure of the subtrahend being taken directly from the figure of the minuend standing above it. 2. From 7328 take 4167. PERFORMED. 7 3 28 min. 3161 rem. EXPLANATION. Taking 7 units from 8 units, 1 unit will remain, which is written down at the foot of the units' column. But 6 tens cannot be taken from 2 tens; we therefore take it from 10 tens, and to the remainder, 4 tens, we add the 2 tens of the minuend, and thus obtain 6 tens to be placed at the foot of the tens' column. To compensate for the 10 we assumed in subtracting the tens, we add 1 to the 1 hundred of the subtrahend, thus making it 2, and subtract the 2 from the 3 hundred of the minuend, and ob Repeat the rule. How is the accuracy of the work tested? tain a remainder of 1 hundred for the remainder. Lastly, we subtract the 4 thousand of the subtrahend from the 7 thousand of the minuend, and obtain a remainder of 3. We thus obtain the total remainder of 3161. NOTE. The assuming of 10 from which to subtract the lower figure when larger than the figure standing above it, is called borrowing 10. 3. From 63814 subtract 45263, I here say, 4-3=1, and 10-—6—4, and 4+1=5; then 2+1=3, and 8-3-5; also 10—5=5, and 5+3=8, and finally, 4+1=5, and 6-5=1. 4. From 875643 subtract 395456. 5. From 396781 subtract 287834. 6. From 987654 take 456789. 7. From 876543 take 345678. 8. From 687957 take 397668. 9. From 596317 take 384567. 10. From 1073563 take 891563. 11. From 1364578 take 954631. 12. From 3895716 take 1934564. 13. From 8643769 take 4897654. 14. From 7183456 take 1745694. 15. From 7788996 take 6939154. 16. From 3344556 take 2617478. 17. From 5566778 take 1956437. 18. From 8811772 take 3456789. 19. From 9076432 take 7819567. 20. From 2345473 take 876921. 21. From 8136741 take 5687345. 22. From 7891234 take 1234789. 23. From 8315794 take 7186432. 24. From 9108765 take 3890787. 25. From 3076059 take 1849718. 26. From 7867564 take 2948675. What is stated in the note! Ans. 480187. Ans. 108947. Ans. 530865. Ans. 530865. Ans. 290289. Ans. 211750. Ans. 182000. Ans. 1961152. Ans. 1468552. Ans. 2449396. Ans. 6656445. Ans. 1129362. 27. From 4716359 take 1975645. 28. From 9182736 take 1928378. 29. From 7891367 take 4986745. 30. From 8867547 take 3956431. 31. From 8715643 take 7146894. 32. From 9182735 take 8197673. 33. From 1678909 take 871964. 34. From 6813957 take 2468035. 35. From 8795643 take 6987869. Ans. 2740714. Ans. 1807774. § 52. It is often desirable to have some test by which to try the accuracy of an operation. The more convenient test for subtraction is, to add the remainder, or difference between the two numbers, to the smaller number or subtrahend; their sums must equal the larger number or minuend. This is done on the obvious principle, that the difference between two numbers added to the smaller one must produce the greater. The proof of the following sums is required. 1. From 875956 take 387618. 52. What is a convenient mode of proving subtraction? What must the sum of the remainder and subtrahend equal? On what principle is this done? 12. From 3896712 take 1838976. 13. From 3781895 take 1967846. 14. From 9123456 take 6543219. Ans. 2057736. Ans. 1814049. Ans. 2580237. § 53. It often becomes necessary to subtract one number from the sum of two, three, or more numbers. To do this, first add together the several numbers, and make their sum a minuend, from which subtract the given number. 1. A man who had $350, received from his neighbor $1869. He afterward paid away $847. How much had he left? Ans. $1372. It is obvious that $350+$1869 $2219 is the amount received; and that $2219-$847-$1372. 2. A man bought two farms adjoining each other, one of which contained 793 acres, and the other 968. From the two he sold 1680 acres. How many acres did he then have left? Ans. 81 acres. 3. The following three sums of money were paid a merchant, viz.: $793, $1196, and $1749, of which he paid away $1999. How much had he left? Ans. $1739. 4. A farmer raised 173 bushels of rye, 213 bushels of wheat, 136 bushels of corn, and 93 bushels of barley. Of these several kinds of grain he sold 217 bushels. How many bushels had he left? Ans. 398 bushels. 5. A man bought a farm for $3756, he then built a house and two barns which cost him as follows: the house, $1729, one barn, $1296, and the other $738. He then sold the farm and buildings for $9287. How much did he gain? Ans. $1768. 6. A man went a journey of two days, traveling by steamboat 167 miles, and by railroad 75 miles. On the first day of his return he traveled 187 miles by steamboat and railroad, how many miles then remained for his second day's journey? Ans. 55 miles 7. A collector received on Monday, $1632; on Tuesday. $963; on Wednesday, $1897, and on Thursday, $1397 53. What is the process of subtracting one number from the sum of two or more numbers? |