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(c.) In order that numbers may be subtracted, they must be of the same name or denomination, i. e. they must represent units of the same name or kind.
ILLUSTRATIONS. - -3 pens from 7 pens leave 4 pens, and 3 pencils from 7 pencils leave 4 pencils; but 3 pens cannot be subtracted from 7 pencils, because they are not of the same denomination.
(d.) For convenience in subtracting, numbers are so written that figures of the same denomination shall come under each other, the minuend being usually written above the subtrahend, and the remainder beneath. The denominations are subtracted separately, beginning at the right.
1. What is the value of 8.795 — 3.281?
SOLUTION.-Writing the numbers as opposite, we have 1 thousandth from 5 thousandths leaves 4 thousandths; 8 hundredths from 9 hundredths leave 1 hundredth; 2 tenths from 7
tenths leave 5 tenths; 3 units from 8 units leave 5 units. Therefore, the remainder is 5 units, 5 tenths, 1 hundredth, and 4 thousandths, or, 5.514.
(e.) Methods of Proof. From the nature of subtraction, it is evident that, if the work be correct
1st. The sum of the subtrahend and remainder will equal the minuend.
2d. The difference between the minuend and remainder will equal the subtrahend.
(g.) Perform the following subtractions, and prove the work.
2. 5786 2351?
7. 8397 2064?
10. 98762345 ?
40. Reductions sometimes Necessary.
(a.) When, as is frequently the case, a figure of the subtrahend is larger than the corresponding figure of the minuend, we reduce a unit of the next higher denomination of the minuend to the
denomination of the figures we are considering, and add its value to the value of the given minuend figure. From the sum thus obtained, we subtract the subtrahend figure.
(b.) Allowance should be made for the reduced unit, either by calling the next minuend figure one less, or the next subtrahend figure one greater than it is.
1. What is the value of 5283. 1468?
Writing the numbers as opposite, and beginning at the right hand, we proceed thus: since 8 units cannot be subtracted from 3 units, we reduce 1 of the 8 tens of the minuend to units; 1 ten 10 units, which, added to the 3 units, gives 13 units. 8 units from 13 units leave 5 units, which we write. 6 tens from the 7 tens left in the minuend leave 1 ten, which we write. Since 4 hundreds cannot be subtracted from 2 hundreds, we reduce one of the 5 thousands to hundreds; 1 thousand 10 hundreds, which, added to the 2 hundreds, gives 12 hundreds. 4 hundreds from 12 hundreds leave 8 hundreds, which we write. 1 thousand from the 4 thousands left in the minuend leaves 3 thousands, which we write.
The answer, then, is 3 thousands, 8 hundreds, 1 ten, and 5 units; or, 3815.
(c.) The following form of writing the work requires more figures, but shows the reduction more clearly than did the preceding.
Minuend, changed in form.
2. What is the value of 4003 — 2715?
SOLUTION.-Writing the numbers as opposite, and beginning at the right hand, we subtract thus: since 5 units cannot be subtracted from 3 units, and there are no tens or hundreds in the minuend, we reduce 1 of the 4 thousands to hundreds, giving 10 hundreds; then reduce 1 of the 10 hundreds to tens, giving 10 tens; and then reduce 1 of the 10 tens to units, giving 10 units. 10 units plus 3 units equal 13 units, and 5 units from 13 units leave 8 units; 1 ten from the 9 tens left the minuend leaves 8 tens; 7 hundreds from the 9 hundreds left in the minuend leave 2 hundreds; 2 thousands from the 3 thousands left in the minuend leave 1 thousand. The answer, then, is 1 thousand, 2 hundreds, 8 tens, and 8 units; or, 1288.
The following form of writing the changed minuend illustrates clearly the nature of the reductions made:
2. 6984 3. 10000 4. 21057 5. 70502
(d.) In the preceding explanation, we subtracted 1 from the minuend figure from which a unit was taken for reduction, before subtracting the subtrahend figure; but we might as well have subtracted the 1 with the subtrahend figure.
ILLUSTRATION. In the first example, instead of subtracting 1 ten, on account of the reduction, from the 8 tens, and then subtracting the 5 tens, we might have added the 1 ten to the 5 tens, and subtracted both together.
(e.) Many always subtract by the latter method. One method is as convenient as the other, but the one to which we are accustomed will seem the easiest.
41. Examples and Practical Problems.
Minuend, changed in form.
31. A man bought a farm for $4728, and sold it for $6253. How many dollars did he gain?
32. Boston contained 93383 inhabitants in 1840, and 136881 in 1850. How many more were there in 1850 than in 1840?
33. Mount Washington, in New Hampshire, is 6226 feet high; Mount Monadnock, in the same State, is 3718 feet high. How many feet higher is the former than the latter?
34. How many more Troy grains are there in a pound Avoirdupois than in a pound Troy? See 21.
35. A man who owned 2578 acres of Western land, sold 1389 How many had he left?
36. When a yard of broadcloth costs $5.375, and a yard of cassimere costs $1.25, how much more does a yard of broadcloth cost than a yard of cassimere?
37. If I have in my possession $3206.36, and owe $140.489, how many dollars shall I have left after paying my debts?
38. By the Tribune Almanac for 1857 it appears that, in the presidential election of 1856, Buchanan received 1,834,337 votes, Fremont 1,341,812, and Fillmore 873,055. How many more votes did Buchanan receive than Fremont?
39. How many more did Buchanan receive than Fillmore ? 40. How many more did Fremont receive than Fillmore?
41. How many more votes did Buchanan and Fremont together receive than Fillmore?
42. How many more votes did Buchanan and Fillmore together receive than Fremont?
43. How many more votes did Fremont and Fillmore together receive than Buchanan?
44. By selling a lot of land for $975.36, I gained $197.84. How much did the land cost me?
45. If one man travels 271 miles, and another travels 320 miles, how much farther does the second travel than the first?
46. I bought goods to the amount of $71.58, giving in payment a hundred-dollar bill. How much change ought I to receive back?
47. Edward has $675, George has $813, and William has as many dollars less than Edward, as Edward has less than George. How many dollars has William ?
48. Joseph says that he has 837 cents, and that if he should spend 198 cents, he would still have 237 cents more than Otis. How many cents has Otis?
42. Subtraction of Several Numbers.
(a.) The best method of subtracting several numbers from a single number, is to write them under the minuend, and then to subtract the sum of each column of the subtrahends from the appropriate part of the minuend, reducing from the higher denominations, as before explained.
1. What is the value of 651
Now uniting 4 tens (on account of the reduction) with the tens of the subtrahends, gives 28 tens, which cannot be subtracted from the 5 tens of the minuend. To obtain tens enough to perform the subtraction, we must reduce 3 hundreds to tens. 3 hundreds 30 tens, and adding the 5 tens, gives 35 tens, from which subtracting the 28 tens leaves 7 tens.
EXPLANATION.-The sum of the units of the subtrahends is 33, which cannot be taken from 1 unit. To obtain units enough to perform the subtraction, we must reduce 4 tens to units. 4 tens = 40 units, and adding the 1 unit gives 41 units, from which subtracting 33 units leaves 8 units.
Subtracting 3 hundreds (on account of the last reduction) from the 6 hundreds of the minuend leaves 3 hundreds.
The answer, then, is 3 hundreds, 7 tens, and 8 units, or 378.
(b.) Proof-To test the correctness of the work, see if the remainder added to the subtrahends equals the minuend.
(c.) Practically, the subtraction may be performed thus:
9, 17, 26, 33, from 41, leaves 8.
4, 11, 16, 20, 28, from 35, leaves 7.
3 from 6 leaves 3.
Giving for an answer, 378, as before.
(d.) What is the value of —
2. 9748 879 643 3. 5246472-631