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18. The following table should be committed to memory, and read, two from two, naught remains; two from three, one remains, &c.

SUBTRACTION TABLE. 2-2=0 3-3=0 4-4=0 5-5=0 3—2=1 4-3=1 5-4=1 6-5=1 4-2=2 5-3=2 6-4=2 7-5=2 5-2=3 6-3=3 7—4=3 8-5=3 6—2=4 7-3=4 8-4=4 9-5=4 7-2=5 8-3=5 9-455 10-5=5 8-2=6 9-3=6 10-4-6 11-5=6 9-2=7 10-3=7 11-4-7 12-5=7 10-2=8 11-3=8 12-4-8 13-5=8 11-2=9 12-3=9 13-439 14-5=9 12–2–10 13-3=10 14-4=10 15-5=10

6—6=0 7-7=0 8-8=0 9-9=0 7-6=1 8-7=1 9-8=1 . 10-9=1 8-6=2 9-7=2 10-8=2 11-9=2 9-6=3 10-7=3 11-8=3 12-9=3 10-6=4 11-7=4 12-8=4 13-9=4 11-6=5 12-7=5 13-8=5 14-9=5 12-6=6 13-7=6 14-8=6 15-9=6 13-6-7 14-7=7 15-8=7 16-9-7 14-6=8 15-7=8 16-8=8 17-9=8 15-6=9 16-7-9 17-8=9 18–9=9 16-6=10 17-7=10 18--8=10 19-9=10 12-2=10

17–7= how many ? 12—3= how many ?

16-8= how many ? 16–4= how many

?
19-9 =

how many ? 11-6= how many ?

20–4= how many ? 18-9= how

many
?
13-7= how

many

? 25–8= how many ?

14-2= how many ? When the numbers are small, as in the above examples, they may be subtracted in the mind, without writing them down. When the numbers are large, we write one number under the other, and then make the subtraction by parts.

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EXAMPLES.

1. James has 27 apples, and gives 14 to John: how many will he have left?

The 27 is made up of 7 units and 27 Minuend. 2 tens; and the 14, of 4 units and 1 14 Subtrahend. ten. If then we subtract the units

13 from each other, 3 units will remain, and if we subtract the tens also, one“ ten will remain. Hence, the remainder is 13.

2. What are the remainders in the following exam. ples:

(1.) (2.) (3.) (4.) Minuends, 874

972
999

8497 Subtrahends, 642 631

367

7487 Remainders, 232

1010 3. A farmer had 378 sheep, and sold 256 : how many has he left?

We first write the number 378, and then 378 the 256 under it, so that units of the same 256 order shall fall under each other. We then

122 take 6 units from 8, 5 tens from 7 tens, and 2 hundreds from 3 hundreds, leaving for the remainder 122.

4. A merchant has 578 dollars in cash, and pays 475 dollars for goods : how much has he left ?

Ans. 103 dollars. 5. What are the remainders in the following exam. ples: (1.) (2.)

(3.)
62843
278846

894862
51720
167504

170641
11123

6. What is the difference between 4 units of the first order and 1 of the second ? We write the ten, and then place the 4

10 under the units' place. Then we say, 4

4 from ten leaves 6.

6

7. What is the difference between two units of the 2d order and 6 of the first ?

20

6 Here we may say, 6 from 20 leaves 14.

14

8. What is the difference between 14 and 8?

Having written the 8 under the 14, so 14 that units shall stand under units, we see 8 that 8 cannot be subtracted from 4, but we 6 can say, 8 from 14 leaves 6.

9. What is the difference between 3 units of the 2d order and 7 of the first? What is the difference between 4 tens and 2 tens? What is the difference between 5 tens and 1 ten ?

90 10. What is the difference between 9 6 units of the second order and 6 of the first ?

84

11. What is the difference between a unit of the second order and a unit of the first order ?

10
1

9

12. What is the difference between a unit of the third order and a unit of the second order ?

100 10 90

400

18. What is the difference between four units of the third order and three tens?

30 370

14. What is the difference between four units of the 4th order and 2 of the third ?

4000

200 3800

15. What is the difference between eight units of the 6th order and 5 of the 4th ?

800000

5000 795000

EXAMPLES ILLUSTRATING PRINCIPLES.

OPERATION.

10

1

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19. We are now going to explain the process of making the subtraction when a figure of the subtrahend is greater than that of the minuend directly over it.

Beginning with the units of the low. est order, we say, 2 from 3 leaves 1, which we write down in the units' place.

843 At the next step we meet a difficulty, for 562 we cannot subtract 6 from 4. If, now,

281 we add 10 to the 4, (which is written in small figures above,) and 10 also to the 6 directly under it, it is plain that the difference will not be affected, since both the numbers will thus be equally increased. But adding 10 to 4 is the same thing as adding 1 to the figure next to the left: hence,

We may consider 10 to be added to any figure of the minuend, provided we add 1 to the next figure of the subtrahend to the left.

20. Hence, to find the difference between two num. bers, we have the following

19

RULE.

I. Set down the less number under the greater, so that units of the same order shall fall under each other, and beginning with the simple units, subtract each figure from the one directly, over it.

II. When any figure of the minuend is less than the one directly under it, suppose ten to be added and then make the subtraction ; after which, add one to the next figure of the subtrahend and subtract as before.

19. Can you subtract a greater number from a less ? When the upper figure is the least, how do you proceed? Does this change the difference between the numbers? What then may we always do?

20. How do you set down the numbers for subtraction ? Where do you begin to subtract? How do you subtract ? Give the rule. How do you prove subtraction ?

PROOF.

Add the remainder to the subtrahend. If the sum is equal to the minuend the work may be regarded as right.

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9. From 2637804 take 2376982. Ans. 260822. 10. From 3762162 take 826541. Ans. 2935621. 11. From 78213609 take 27821890. Ans.

12. From thirty thousand and ninety-seven, take one thousand six hundred and fifty-four. Ans. 28443.

13. From one hundred million two hundred and fortyseven thousand, take one million four hundred and nine.

Ans. 99246591. 14. Subtract-one from one million. Ans.

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Ans. 10 years.

APPLICATIONS. 1. Suppose John were born in eighteen hundred and fifteen, and James in eighteen hundred and twenty-five : what is the difference of their

ages

? 2. A man was born in 1785: what was his age in 1830 ?

3. Suppose I lend a man 1565 dollars, and he dies, owing me 450 dollars : how much had he paid me ?

Ans. 1115 dollars. 4. In five bags are different sums of money to the amount in all of 1000 dollars. In the first there are 100 dollars ;- in the second, 314 dollars; in the third, 143

Ans. 45 years.

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