3. Subtraction.

ANALYSIS.

94. 1. A boy having 18 cents, lost 6 of them; how many had he left? Here is a collection of 18 cents, and we wish to know how many there will be after 6 cents are taken out. The most natural way of doing this, would be to begin with 18, and take out one cent at a time till we have taken 6 cents; thus 1 from 18 leaves 17, 1 from 17 leaves 16, 1 from 16 leaves 15, 1 from 15 leaves 14, 1 from 14 leaves 13, 1 from 13 leaves 12. We have now taken away 6 ones, or 6 cents, from 18, and have arrived, in the descending series of numbers, at 12; thus discovering that if 6 be taken from 18, there will remain 12, or that 12 is the difference between 6 and 18. Hence Subtraction is the reverse of Addition. When the numbers are small, as in the preceding example, the operation may be performed wholly in the mind;[102] but if they are large, the work is facilitated by writing them down.

95. 2. A person owed 75 dollars, of which he paid 43 dollars; how much remains to be paid?

Operation.

From 75 minuend.
Take 43 subtrahend

32 remainder.
75 proof.

Now to find the difference between 75 and 43, we write down the 75, calling it the minuend, or number to be diminished, and write under it the 43, calling it the subtrahend, with the units under units and the tens under tens, and draw a line below, as at the left hand. As 75 is made up of 7 tens and 5 units, and 43 of 4 tens and 3 units, we take the 3 units of the lower from the 5 units of the upper line, and find the remainder to be 2, which we write below the line in the place of units. We then take the 4 tens of the lower from the 7 tens of the upper line, and find the remainder to be 3, which we write below the line in the ten's place, and thus we find 32 to be the difference between 75 and 43. From an inspection of these examples, it will be seen that Subtraction is. in effect, the separating of the minuend into two parts, one of which is the subtrahend, and the other the remainder. Hence, to show the correctness of the operation, we have only to recompose the minuend by adding together the subtrahend and remainder.

96. 3. A person owed 727 dollars, of which he paid 542 dollars; how much remains unpaid? Here we take 2 from 7, and write the difference, 5, 727 dolls. below the line in the place of units. We now proceed 542 dolls. to the tens, but find we cannot take 4 tens from 2 tenɛ. We may, however, separate 7 hundreds into two parts, one of which shall be 6 hundred, and the other 1 hundred, or 10 tens, and this 10 we can join with the 2, making 12 tens. From the 12 we now subtract the 4, and write the remainder, 8, at the left hand of the 5, in the ten's place. Proceeding to the hundreds,

Ans. 185 dolls.

What would be the most natural way of taking one number from another?

In what cases can Subtraction be

performed mentally?

When is it necessary to write down the numbers?

we must remember that 1 unit of the upper figure of this order, has already been borrowed and disposed of; we must therefore call the 7 a 6, and then taking 5 from 6, there will remain 1, which being written down in the place of hundreds, we find that 185 dollars remain unpaid.

4. A boy having 12 chesnuts, gave away 7 of them; how many had he left?

12 7

5 Ans..

Here we cannot take 7 units from 2 units; we must therefore take the 1 ten 10 units, with the 2, making 12 units;` then 7 from 12 leaves 5 for the answer.

97. 5. A man has debts due him to the amount of 406 dollars, and he Qwes 178 dollars: what is the balance in his favour?

406 178

228

Here we cannot take 8 units from 6 units; we must there fore borrow 10 units from the 400, denoted by the figure 4, which leaves 390. Now joining the ten we borrowed with 6, we have the minuend, 406, divided into two parts, which are 390 and 16. Taking 8 from 16, the remainder is 8; and then we have 390, or 39 tens in the upper line, from which to take 170, Thus the place of the cipher is occupied by a 9, and the significant figure 4 is diminished by 1, making it 3. We then say, 7 from 9 there remains 2, which we write in the place of tens, and proceeding to the next place, say 1 from 3 there remains 2. Thus we find the balance to be 228 dollars.

or 17 tens.

SIMPLE SUBTRACTION.

98. Simple Subtraction is the taking of one simple number from another, so as to find the difference between them. The greater of the given numbers is called the minuend, the less the subtrahend, and the difference between them the remainder.

RULE.

99. Write the least number under the greater, with units under units, and tens under tens, and so on, and draw a line below. Beginning at the right hand, take each figure of the subtrahend from the figure standing over it in the minùend, and write the remainders in their order below. If the figure in the lower line be greater than the figure standing over it,suppose ten to be added to the upper figure, and the next significant figure in the upper line to be diminished by 1, (96) regarding ciphers, if any come between, as 9s, (97); or, which gives the same result, suppose 10 to be added to the upper figure, and the next figure in the lower line to be increased by

What is Simple Subtraction?

What are the given numbers called?
What is the difference called?
How are the numbers to be written
down?

Where do you begin to subtract?

When the figure in the upper line

is less than the one under it, what is to be done?

Explain it by an example.

What is the sign of Subtraction?

1, with which proceed as before, and so on till the whole is fioished.

PROOF.

100. Add together the remainder and the subtrahend, and if the work be right, their sum will equal the minuend.

QUESTIONS FOR PRACTICE.

13. If you lend a neighbored in the year 1330; then how 765 dollars, and he pay you at one time, 86 dollars, and at another 125 dollars, how much is still due? Ans. 554 dolls.

26. 6485-4293-2192. 28. 48+64+93-139-66. 27.900000-1-899999.

29. 2777+11-1898-890.

4. Division.

ANALYSIS.

101. 1. Divide 24 apples equally among 6 boys, how many will each teceive?

The most simple way of doing this would be, first to give each boy 1 apple, then each boy 1 apple more, and so on, till the whole were distributed, and the number of ones, which each received, would denote his share of the apples, which would in this case be 4. Or as it would take 6 apples to give each boy one, each boy's share will evidently contain as many apples as there are sixes in 24. Now this may be ascertained by subtracting 6 from 24, as many times as it can be done, and the number of subtractions will be the number of times 6 is contained in 24; thus, 246=18, 13-6=12, 12—6—6, and 6-6=0. Here we find that by performing 4 subtractions of 6, the 24 is completely exhausted, which shows that 24 contains 6 just 4 times. Now as Subtraction is the reverse of Addition, (94) it is evident that the addition of 4 sixes, (6+6+6+6=24) must recompose the number, which we have separated by the subtraction of 4 sixes. But when the numbers to be added are all equal, Addition becomes Multiplication, (83) and 24 is therefore the product of 4 and 6, (4×6=24). A number to be divided, and which is called a dividend, is then to be regarded as the product of two factors, one of which, called the divisor, is given to find the other, called the quotient; and the inquiry how many times one number is contained in another, as 6 in 24, is the same as how many times the one will make the other, as how many times 6 will make 24, and both must receive the same answer, viz. 4. Hence to prove Division, we multiply the divisor and quotient together, and if the work be right, the product will equal the dividend.

2. How many yards of cloth will 63 dollars buy, at 9 dollars a yard?

102. When the dividend does not exceed 100, nor the divisor exceed 10, the whole operation may be performed at once in the mind: but when either of them is greater than this, it will be found most convenient to write down the numbers before performing the operation.

3. Divide 552 dollars equally between 2 men, how many dollars will each have?

2)552

400-200 140

--70 12 --6

Here we cannot say at once how many times 2 is contained in 552, we therefore write down the lividend, 552, and place the divisor, 2, at the left hand. We then proceed to separate the dividend into such parts as may readily be divided by 2. These parts wo find to be 400, 140, and 12. Now 2 18 contained in 4, 2 times, and therefore 552 -76 in 400, 200 times; 2 in 14, 7 times, and in 140, 70 times, and 2 in 12, 6 times; and since these partial dividends. 400+140+12552, the whole dividend, the partial quotients, 200+70+6=276, the whole quotient, or whole number of times 2 is contained in 552. But in

Divis. Divid. Quot.

2)552 (276

15

14

12

12

2

552 proof.

practice we separate the dividend into parts no faster than we proceed in the division. Having written down the dividend and divisor as before, we first seek how many times 2 in 5, and find it to be completely contained in it only 2 times. We therefore write 2 for the highest figure of the quotient, which, since the 5 is 500, is evidently 200; but we leave the place of tens and units blank to receive those parts of the quotient which shall be found by dividing the remaining part of the dividend. We now multiply the divisor 2, by the 2 in the quotient, and write the product, 4. (400) under the 5 hundred in the dividend. We have thus found that 400 con. tains 2, 200 times, and by subtracting 4 from 5, we find that there are 1 hundred, 5 tens, and 2 units, remaining to be divided. We next bring down the 5 tens of the dividend, by the side of the 1 hundred, making 15 tens, and find 2 in 15, 7 times. But as 15 are so many tens, the 7 must be tens also, and must occupy the place next below hundreds in the quotient. We now multiply the divisor by 7, and write the product, 14, under the 15. Thus we find that 2 is contained in 15 tens 70 times, and subtracting 14 from 15, find that 1 ten remains, to which we bring down the 2 units of the dividend, making 12, which contains 2, 6 times; which 6 we write in the unit's place of the quotient, and multiplying the divisor by it, find the product to be 12. Thus have we completely exhausted the dividend, and obtained 276 for the quotient as before.

103. 4. A prize of 3349 dollars was shared equally among 16 men, how many dollars did each man receive?

32

149
144

-

5

16.

We write down the numbers as before, and find 16) 3349 (209 5 Ans. 16 in 33, 2 times,-we write 2 in the quotient, multiply the divisor by it, and place the product, 32. under 33, the part of the dividend used, and subtracting, find the remainder to be 1, which is 1 hundred. To the 1 we bring down the 4 tens, making 14 tens; but as this is less than the divisor, there can be no tens in the quotient. We therefore put a cipher in the ten's place in the quotient, and bring down the 9 units of the dividend to the 14 tens, making 149 units, which contain 16 somewhat more than 9 times. Placing 9 in the unit's place of the quotient, and multiplying the divisor by it, the product is 144, which, subtracted from 149, leaves a remainder of 5. The division of these 5 dollars may be denoted by writing the 5 over 16, with a line between, as in the example. Each man's share then will be 209 dollars and 5 sixteenths of a dollar. (21) The division of any number by another may be denoted by writing the dividend over the divisor, with a line between, and an expression of that kind is called a Vulgar Fraction.