SECTION III. SUBTRACTION. ART. 65. Ex. 1. A merchant bought 37 barrels of flour, and afterwards sold 12 of them: how many barrels had he left ? Solution.-12 barrels from 37 barrels leave 25 barrels. Ans. 25 barrels. Obs. It will be perceived, that the object in this example, is to find the difference between two numbers. 66. The process of finding the difference between two numbers is called SUBTRACTION. The difference, or the answer to the question, is called the Remainder. Obs. 1. The number to be subtracted is sometimes called the subtrahend, and the number from which it is subtracted, the minuend. 2. Subtraction, it will be perceived, is the reverse of addition. Addition unites two or more numbers into one single number; subtraction, on the other hand, separates a number into two parts. 3. When the given numbers are of the same denomination, the operation is called Simple Subtraction. (Art. 50. Obs.) Ex. 2. What is the difference between 5364 and 9387 ? Write the less number under the greater, Operation. units under units, tens under tens, &c. Then, 9387 beginning at the right hand, proceed thus: 5364 4 units from 7 units leave 3 units. Write 4023 Rem. the 3 in the units' place, under the figure subtracted. 6 tens from 8 tens leave 2 tens; set the 2 in tens' place. 3 hundred from 3 hundred leave 0 hundred; we therefore write a cipher in hundreds' place. 5 thousand from 9 thousand leave 4 thousand; set the 4 in the thousands' place. The answer is 4023. QUEST.-66. What is subtraction? What is the difference or answer called ? Obs. What is the number to be subtracted sometimes called? The number from which it is eubtracted? Of what is subtraction the reverse ? When the given numbers are of the same denomination, what is the operation called ? 67. It will be observed, that we subtract units from units, tens from tens, &c.; that is, we subtract figures of the same order from each other. This is done for the same reason that we add figures of the same order to each other. (Art. 51.) Obs. The less number is written under the greater, simply for convenience in subtracting; and units are placed under units, tens under tens, &c., to avoid mistakes which might occur from taking different orders from each other. 68. It often happens that a figure in the lower number is larger than that above it, and consequently cannot be taken from it. Ex. 3. What is the difference between 94 and 56 ? Analytic solution, It is manifest that we cannot take 6 94=80+14 units from 4 units, for 6 is larger than 4. 56= -50+ 6 To obviate this difficulty, we may take Rem. 38=30+:8 1 ten from the 9 tens, and uniting it with the 4 units, the upper number will become 8 tens and 14 units, or 80+-14. Separating the lower number into the parts of which it is composed, it becomes 5 tens and 6 units, or 50+6. Now, subtracting as in the last example, 6 from 14 leaves 8, 50 from 80 leaves 30. The answer is 30+8, or 38. Or, we may simply take 1 ten from the 9 tens, and adding it, mentally, to the 4 units, say 6 from 14 leaves 8; set the 8 under the figure subtracted. Then, having taken 1 from the 9 tens, we have but 8 left, and 5 from 8 leaves 3. The answer is 38. PROOF.–38+56=94; that is, the sum of the remainder and smaller number being equal to the larger, the answer is right. Hence, 69. When a figure in the lower number is larger than that above it ; take 1 from the next higher order in the upper number, and add it to the upper figure ; from the sum subtract the lower figure, and diminishing the next upper figure by 1, proceed as before. Obs. 1. The process of taking one from the next higher order and adding it to the figure from which the subtraction is to be made, is called borrowing ten. It is the reverse of carrying. QOEST.-67. What orders of figures do you subtract from each other? Why not sub tract different orders from each other? 2. This method of borrowing, it will be seen, does not affect the difference between the two given numbers; for, it is simply transposing a part of one order to another order in the same number, which, it is obvious, will neither increase nor diminish its value. 3. It may be asked, how can we take one from the figure in the next higher order, when that figure is a cipher? How can nothing lend anything, and how can nothing be diminished by one ? The explanation of this apparent contradiction is this: when the next figure is a cipher, we go to the next higher column still, and take one, which, added to the figure in the next lower order, makes ten; we then take one from the ten and add it to the upper figure, and proceed as before. 70. There is another method of borrowing, or rather of paying, which, though perhaps less philosophical than the preceding, is more convenient in practice, especially when the figures in the next higher orders are ciphers. Thus, in the last example, adding 10 to the upper figure, it becomes 14, and 6 from 14 leaves 8. Set down the 8 as before. Now, instead of diminishing the next upper figure by 1, if we add 1 to the next figure in the lower number it becomes 6 tens; and 6 from 9 leaves 3, which is the same as 5 from 8. The answer is 38, the same as before. Hence, 71. When a figure in the lower number is larger than that above it, add 10 to the upper figure, and to compensate this, add 1 to the next left hand figure in the lower number. OBs. 1. This method of borrowing depends on the self-evident principle, that if any two numbers are equally increased, their difference will not be altered. That the two given numbers are equally increased by this process, is evident from the fact that the 1 added to the lower number is of the next superior order to the 10 added to the upper number, and is therefore equal to it. (Art. 35.) 2. The reason that we borrow 10, instead of 8, or 12, or any other number, is because the radix of the system of Arabic notation, is 10. (Art. 36. Note. 1.) If the radix of the system were 8, it would be necessary to borrow 8; if 12, it would be necessary to borrow 12, &c. 3. On account of borrowing, the learner will perceive it is always necessary to begin to subtract at the right hand. Ex. 4. A man bought a house for 23006 dollars, and sold it for 21128 dollars : how much did he lose by his bargain ? Operation. Proof 21128 Less number. 1878 Remainder. Ans. 1878 dolls. 23006 Larger number. 72. From the preceding illustrations and principles we derive the following GENERAL RULE FOR SUBTRACTION. I. Write the less number under the greater, so that units may stand under units, tens under tens, &c. (Art. 67. Obs.) II. Beginning at the right hand, subtract each figure in the lower number from the figure above it, and set the remainder directly under the figure subtracted. (Art. 71. Obs. 3.) III. When a figure in the lower number is larger than that above it, add 10 to the upper figure ; then subtract as before, and add 1 to the next figure in the lower number, or consider the next upper figure 1 less than it is. (Arts. 69, 71. Obs. 1, 2.) 73. PROOF.—- Add the remainder to the smaller number ; and if the sum is equal to the larger number, the work is right. OBs. This method of proof depends upon the principle, that the difference between two numbers being added to the less, the sum must be equal to the greater. For, the difference and the less number are the two parts into which the greater is separated, and the whole of a quantity is equal to the sum of all its parts. (Ax. 11.) 174. Second Method.-Subtract the remainder from the greater of the two given numbers; and if the difference is equal to the less number, the work is right. 75. Third Method.-Cast the 9s out of the larger number, and place the excess at the right. Next, cast the 9s out of the smaller number, and also out of the remainder; then cast the 9s out of the sum of these two excesses, and if this last excess is the same as the excess of the larger number, the work may be supposed to be right. Thus, Ex. 5. From 7843 Excess of 9s in the greater number is 4 Take 5675 less Rem, 2168 remainder is 8 Now, 8+5=13, and the excess of 9s in 13 is 4, the same as that of the greater number. QUEST.-72. How do you write numbers for subtraction? Why write the less number inder the greater ? Why place units under units, &c. ? Where do you begin to subtract ? When a figure in the lower line is larger than that above it, how do you proceed ? is meant by borrowing ten? How many methods of borrowing are mentioned ? Illustrate the first method upon the black-board. How does it appear that this method of borrowing does not affect the difference between the two given numbers? Explain the second method. Upon what principle does this method depend? Why do you borrow 10, instead! of 8, or 12, or any other number? Why do you begin to subtract at the right hand ? 73. How is subtraction proved? Obs. Upon what principle does this method of proof depend? Can subtraction be proved by any other methods ? is 5 (6 What Note.—This method of proof depends on the same property of the number 9, as that in addition. (Art. 58. Note.) For, since the sum of the smaller number and remainder is equal to the larger number, it follows that the excess of 9s in the larger number must be equal to the excess of 9s in the remainder and smaller number together. EXAMPLES FOR PRACTICE. 76. Ex. 1. A merchant bought a ship for 35270 dollars, and sold it for 42365 dollars : how much did he make by his bargain ? 2. A miller bought 46235 bushels of wheat, and ground 17251 bushels of it: how many bushels had he left ? 3. A speculator laid out 50000 dollars in wild land, and afterwards sold it at a loss of 19046 dollars : how much did he get for his land ? 4. A man owning a block of buildings worth 155265 dollars, keeps it insured for 109240 dollars : how much would he lose in case the buildings should be destroyed by fire ? 5. The distance from the Earth to the Sun is 95000000 of miles ; the distance of Mercury is only 37000000 : how far is Mercury from the Earth ? 6. The imports of Massachusetts in 1840, were 16,513,858 dollars, the exports were 10,186,261 dollars : what was the excess of her imports over her exports ? 7. The imports of New York in 1840, were 60,440,750 dollars, the exports were 34,264,080 dollars : what was the excess of her imports over her exports ? 8. The imports of Pennsylvania in 1840, were 8,464,882 dollars, the exports were 6,820,145 dollars : what was the excess of her imports over her exports ? 9. The imports of South Carolina in 1840, were 2,058,870 dollars, the exports were 10,036,769 dollars : what was the excess of her exports over her imports ? 10. The imports of Alabama in 1840, were 574,651 dollars, the exports were 12,854,694 dollars : what was the excess of her exports over her imports 11. The imports of Louisiana in 1840, were 10,673,190 dollars, the exports were 34,236,936 dollars : what was the excess of her exports over her imports ? |