SUBTRACTION. 37. SUBTRACTION is the process of taking one number from another of the same kind, to find the difference. The number which is subtracted is called the subtrahend, from the Latin subtrahendus, to be taken from under, as that is the number taken away. The number from which the subtrahend is taken is called the minuend, from the Latin minuendus, to be made smaller, as that is the number to be diminished. The result is called the difference, or remainder. A short horizontal line, —, read minus or less, is the sign of subtraction, and, placed between two numbers, signifies that the number after it is to be taken from that before it; thus, 7-34, read, seven minus three equals four, shows that, if 3 be taken from 7, the remainder is 4. ILLUSTRATIVE EXAMPLE, I. 38. From 267 take 135. OPERATION. Minuend, 267 Remainder, 132 Ans. For convenience, we write the subtrahend under the minuend, placing units under units, tens under tens, hundreds under hundreds, and draw a line beneath; 5 units from 7 units 2 units, which we write in the units' place, under the units; 3 tens from tens 3 tens, which we write in the tens' place; 1 hundred from 2 hundreds 1 hundred, which we write in the hundreds' place; and the result is 132, which is the difference between 267 and 135. 29. PROOF. If 132 is the difference between 267 and 135, it is evident that, if we add 132 to 135, the sum will equal 267. Hence, prove subtraction, add the difference to the subtrahend. If the sum thus obtained is equal to the minuend, the work may be presumed to be correct. NOTE. The pupil should prove each example, till he is sure that he makes no mistakes. 40. EXAMPLES. 1. 368-334 =? 2. 2769-2631=? Ans. 138. 3. 362785-250122=? 5. 974968-721265 =? Sum of the last four answers 37828933. 41. ILLUSTRATIVE EXAMPLE, II. 98613674 Ans. 6187 what? OPERATION. Here a difficulty presents itself. We cannot take 9861 4 units from 1 unit. In order to perform the opera3674 tion, we must reduce one of the tens in the minuend to units, which with the 1 unit we already have, equals 11 units; 4 units from 11 units=7 units, which we put in the units' place. Having reduced one of the tens to units, we have but 5 tens left, and as 7 tens cannot be taken from 5 tens, we must reduce one of the hundreds to tens, which 10 tens; 10 tens +5 tens 15 tens; 7 tens from 15 tens = 8 tens, which we write in the tens' place; 6 hundreds from 7 hundreds 1 hundred; 3 thousands from 9 thousands 6 thousands, and the answer is 6187. Hence the RULE FOR SUBTRACTION. Write the subtrahend beneath the minuend, units under units, tens under tens, etc. Begin to subtract at the units' place, taking each term * in the subtrahend from the one above it, and placing the remainder beneath. If the upper term is less than the lower, increase it, by adding to it one of the next higher denomination reduced to its own denomination, and then subtract, bearing in mind, in the next operation, that the upper term has been diminished by the one reduced. Sum of the last four remainders, 82,302. 9. A man had 375 oranges in a box; if he should sell 259 of them, how many would he have left? Ans. 116 oranges. 10. A man, having 451 acres of land, gave 349 acres to his son; what remained? Ans. 102 acres. 11. If a teacher is now 57 years old, and has taught 38 years, at what age did he begin to teach ? Ans. 19 years. 12. How old was a person in 1865 who was born in 1789? Ans. 76 years. 13. If I had $625 in a bank, and withdrew $249, what remained? Ans. $376 42. ILLUSTRATIVE EXAMPLE, III. From 20000 take 9. OPERATION. (1) (9) (9) (9) (10) 2 0 0 0 0 9 1999 1 Here we have no tens to reduce to units, no hundreds, and no thousands. We must then take one of the 2 ten-thousands (leaving 1 ten-thousand), and reduce it to thousands, making 10 thousands. Reducing one of the thousands to hundreds, one of the hundreds to tens, and one of the tens to units, we leave 9 thousands, 9 hundreds, 9 tens, and have 10 units, from which, if we take 9 units, 1 unit will remain. Having no tens to take from 9 tens, no hundreds to take from 9 hundreds, no thousands to take from 9 thousands, and no ten-thousands to take from 1 ten-thousand, we write these figures in their respective places below the line, and have for a remainder 19991. 43. EXAMPLES. 1. From 2017 years take 1028 years. Ans. 989 years. 2. A man, who had 1205 yards of cloth, sold 429 yards. How many yards were left? Ans. 776 yards. 3. There are 205 sheep in a flock; if 109 of them should be driven to market, how many would remain? Ans. 96 sheep. 4. A merchant bought goods for $1084, and sold them for $177 less than he gave; how much did he receive for them? Ans. $907. 5. 30070 men went into battle; 4564 were slain, and 1300 were taken prisoners; how many were left? 6. Take 229 oxen from 2006 oxen. 7. Subtract 25 hundred from 81 thousand. 8. How many more in 47000 than in 702? Ans. 24,206 men. 10. 9832147 less 3472108 equals how many? Ans. 78,500. Ans. 46,298. Ans. 702. 11. What number added to 9213628 will give 23475310? 12. What number subtracted from 7654321 will leave 369? 13. 86293210 minus 329876 equals how many? 14. 987621085 329875232 how many? 15. Find the sum of the last five answers. Ans. 771,984,860. 17. 3478921+ 368754- 2878796 what? - 19. From 4673214+2792 take 98264. 20. 98432231 - 32636841 - 808994 what? 21. 8087670 - 7549094- 89699 what? 22. Find the sum of the last six answers. Ans. 77,642,007. 23. What is the difference between 19360742 and 9643278? 24. How many times can I take 7642 gallons from 21002 gallons, and what will remain? 25. If the minuend is 36 quadrillion and the subtrahend 95 million 86, what is the remainder. Ans. 35,999,999,904,999,914. 26. If the minuend be 69 trillion and the difference 85 billion, what is the subtrahend? 27. Philadelphia was founded in 1682. In what ear was New York city settled, it having been settled 68 years before? 28. Victoria ascended the throne of England in 1837. How many years has she reigned? 29. Napoleon commenced his brilliant career in 1795. How many years before his final defeat in 1815? 30. The Israelites left Egypt in 1491 B. C., and 40 years after entered the land of Canaan. In what year did that event happen? 31. In the year 1851, London had 2362000 inhabitants; Pekin was estimated to have 1500000. How many more inhabitants had London than Pekin? 32. The equatorial diameter of the earth is 41843330 feet, and the polar diameter 41704788 feet; required the difference. 33. The population of St. Louis in 1850 was 77860, and in 1860, 160773; required the increase in 10 years. 34. James Nye has in his possession $172; he owes $28 to A, $36 to B, and $19 to C. After paying his debts, what will remain ? 35. I have saved from my income $362, and have $2180 in government bonds; how much more must I save that I may purchase a house worth $3500? 44. GENERAL REVIEW, No. 1. 1. Two persons, who are 200 miles apart, travel towards each other, one 46 miles, the other 51 miles a day; how far apart will they be at the end of one day? 2. If the above persons travel away from each other, how far apart will they be at the end of one day? 3. A man gave to his eldest son $3575, to his youngest son $4680, and to his daughter $2495 less than to the youngest son; his whole property was worth $20000; what sum remained? 4. A ship, which was valued at $15590, was sold at a loss of $4975; what did she bring? 5. If the subtrahend be 369 quadrillion, and the remainder 99 quadrillion 13 billion, what is the minuend? 6. The difference between two numbers is 95478. The larger number is 148769; what is the smaller? 7. How many times can 18640 be subtracted from 46806, and what will remain? 8. Which of the two numbers 15672 or 10560 is nearer to 13465, and how much? 9. From what number must 846 be taken twice to leave 15684? 10. To what number must 962 be added three times to make 8472? 11. Which is nearer to 348628, 63248 +93264, or 6000£3 59321 ? For Dictation Exercises, see Key. h |