NOTE.-It has been usual to perform subtraction, where the figure in the subtrahend is larger than the figure above it, on another principle. If to two unequal numbers the same number be added, the difference between them will remain the same. Thus, the difference between 17 and 8 is 9, and the difference between 27 and 18, each being increased by 10, is also 9. Take the last example.

TENS. UNITS.

8 15

3

7

5

8

Adding 10 units to 5 units in the minuend, and 1 ten to 2 tens in the subtrahend, we have increased both by the same number, and the remainder is not altered, being 58.

This method, which has been erroneously called borrowing ten, may be practised by those who prefer, though the former is more simple and equally convenient.

3. From 10000 subtract 9.

OPERATION. 10000

9

SOLUTION.In this example we have 0 units from which to subtract 9 units, and going to tens of the minuend, we have 0 tens, nor hundreds, nor thousands but we have 1 ten thousand from which, borrowing 10 units, we have 9990, that is, 9 thousands, 9 hundreds and 9 tens left. Taking 9 units from 10 units, we have 1 unit, then no tens in the subtrahend from 9 tens m the minuend leave 9 tens, no hundreds from 9 hundreds leave 9 nundreds, no thousands from 9 thousands leave 9 thousands.

9991

4. A man borrowed 713 dollars and paid 475 dollars; how much did he then owe? Ans. 238 dollars.

Rem. 720599.

5. From 1402003 take 681404. 6. What is the difference between 36070324301 and 28040373315? Ans. 8029950986.

7. From 81324036521 take 2546057867.

Rem. 78777978654.

T18. TO PROVE ADDITION AND SUBTRACTION.-Addition and subtraction are the reverse of each other. Addition is putting together; subtraction is taking asunder.

1. A man bought 40 sheep and sold 18 of them; how many had he left?

40-18-22 sheep left.

Ans.

2. A man sold 18 sheep and had 22 left; how many had he at first? 18+2240 sheep at first. Ans.

Hence, subtraction may be proved by addition, and addition by subtraction.

To prove subtraction, add the remainder to the subtrahend, and, if the work is right, the amount will be equal to the minuend,

To prove addition, subtract, successively, from the amount, the several numbers which were added to produce it, and, if the work is right, there will be no remainder. Thus 7+8 +6=21; proof, 21—6—15, and 15—8—7, and 7

NOTE.-Proof by excess of nines. We may cast out the nines in the remainder and subtrahend; if the excess equals the excess found by casting out the nines from the minuend, the work is presumed to be right.

From the remarks and illustrations now given, we deduce the following

RULE.

I. Write down the numbers, the less under the greater, placing units under units, tens under tens, &c., and draw a line under them.

II. Beginning with units, take successively each figure in the lower number from the figure over it, and write the remainder directly below.

III. When a figure of the subtrahend exceeds the figure of the minuend over it, borrow 1 from the next left hand figure of the minuend; and add it to this upper figure as 10, in which case the left hand figure of the minuend must be considered one less.

NOTE.-Or when the lower figure is greater than the one above it we may add 10 to the upper figure, and 1 to the next lower figure

EXAMPLES FOR PRACTICE.

1. If a farm and the buildings on it be valued at 10000, and the buildings alone be valued at 4567 dollars, what is the value of the land? Ans. 5433 dollars.

2. The population of New York in 1830 was 1,918,608; in 1840 it was 2,428,921; what was the increase in ten years? Ans. 510,313. 3. George Washington was born in the year 1732, and died in the year 1799; to what age did he live? Ans. 67 years. 4. The Declaration of Independence was published July 4th, 1776; how many years to July 4th the present year?

Questions, T 18. Addition is the reverse of what? Subtrac tion, of what? How will you show that they are so? How do you prove subtraction? How can you prove addition by subtraction? Re peat the rule for subtraction.

5. The Rocky Mountains, in N. A., are 12,500 feet above the level of the ocean, and the Anles, in S. A., are 21,440 feet; how many feet higher are the Andes than the Rocky Mountains? Ans. 8,940 feet.

NOTE.

Let the pupil be required to prove the following examples.

6. What is the difference between 7,648,203 and 928,671 ? Ans. 6,719,532. 7. How much must you add to 358,642 to make 1,487,915? Ans. 1,129,303.

much?

8. A man bought an estate for 13,682 dollars, and sold it ugain for 15,293 dollars; did he gain or lose by it? and how Ans. 1,611 dollars 9. From 364,710,825,193 take 27,940,386,574. 10. From 831,025,403,270 take 651,308,604,782. 11. From 127,368,047,216,843 take 978,654,827,352.

T19. Review of Subtraction.

Questions. What is subtraction? What is the rule? What is understood by borrowing ten? Of what is subtraction the reverse? How is subtraction proved? How is addition proved by subtraction?

EXERCISES.

1. How long from the discovery of America by Columbus, in 1492, to this present year?

2. Supposing a man to have been born in the year 1773, how old was he in 1847?.

Ans. 74. 3. Supposing a man to have been 80 years old in the year 1846, in what year was he born? Ans. 1766. is 8764; the Ans. 6923. 3794, leaves Ans. 2929.

4. There are two numbers, whose difference greater number is 15687; I demand the less. 5. What number is that which, taken from 865? 6. What number is that to which if you add 789, it will become 6350 ? Ans. 5561.

7. A man possessing an estate of twelve thousand dollars, gave two thousand five hundred dollars to each of his two daughters, and the remainder to his son; what was his son's share? Ans. 7000 dollars. 8. From seventeen million take fifty-six thousand, and what will remain ? Ans. 16,944,000.

9. What number, together with these three, viz., 1301 2561, and 3120, will make ten thousand? Ans. 3018.

10. A man bought a horse for one hundred and fourteen dollars, and a chaise for one hundred and eighty-seven dollars; how much more did he give for the chaise than for the norse?

11. A man borrows 7 ten dollar bills and 3 one dollar bills, and pays at one time 4 ten dollar bills and 5 one dollar bills; how many ten dollar bills and one dollar bills must he afterFards pay to cancel the debt?

Ans. 2 ten doll. bills and 8 one doll. 12. The greater of two numbers is 24, and the less is 16; what is their difference?

13. The greater of two numbers is 24, and their difference 8; what is the less number?

14. The sum of two numbers is 40, the less is 16; what is the greater?

EXERCISES IN ADDITION AND SUBTRACTION.

15. A man carried his produce to market; he sold his pork for 45 dollars, his cheese for 38 dollars, and his butter for 29 dollars; he received, in pay, salt to the value of 17 dollars, 10 dollars' worth of sugar, 5 dollars' worth of molasses, and the rest in money; how much money did he receive? Ans. 80 dollars. 16. A boy bought a sled for 28 cents, and gave 14 cents to have it repaired; he sold it for 40 cents; did he gain or lose by the bargain? and how much? Ans. He lost 2 cents.

17. One man travels 67 miles in a day, another man fol lows at the rate of 42 miles a day; if they both start from the same place at the same time, how far will they be apart at the close of the first day? third ?

of the fourth?

of the second? of the Ans. To the last, 100 miles. 18. One man starts from Boston Monday morning, and travels at the rate of 40 miles a day; another starts from the same place Tuesday morning, and follows on at the rate of 70 miles a day; how far are they apart Tuesday night?

Ans. 10 miles.

19. A man, owing 379 dollars, paid at one time 47 dollars at another time 84 dollars, at another time 23 dollars, and at another time 143 dollars; how much did he then owe?

Ans. 82 dollars.

20. Four men bought a lot of land for 482 dollars; the first min paid 274 dollars, the second man 194 dollars less than

the first, and the third man 20 dollars less than the second; how much did the second, the third, and the fourth man pay

The second paid 80. Ans. The third pa.a 60.

The fourth paid 68.

21 Four men bought a horse; the first man paid 21 dollars, the second 18 dollars, the third 13 dollars, and the fourth as much as the other three, wanting 16 dollars; how much did the fourth man pay? and what did the horse cost?

Ans. Fourth man paid dolls.; horse cost 88 dolls. 22. From 1,000,000 take 1, and what remains? (See ¶ 17 Ex. 3.)

MULTIPLICATION OF SIMPLE NUMBERS.

T20. 1. If one orange costs 5 cents, how many cents must I give for 2 oranges? how many cents for 3 oranges? for 4 oranges?

2. One bushel of apples cost 20 cents; how many cents must I give for 2 bushels ?

for 3 bushels? 3. One gallon contains 4 quarts; how many quarts in 2 gallons? in 3 gallons? in 4 gallons?

4. Three men bought a horse; each man paid 23 dollars for his share; how many dollars did the horse cost them? 5. In one dollar there are one hundred cents; how many cents in 5 dollars?

6. How much will 4 pairs of shoes cost at 2 dollars a pair? 7. How much will two pounds of tea cost at 43 cents a pound?

8. There are 24 hours in one day; how many hours in 2 days? in 3 days? in 4 days? — in 7 days? 9. Six boys met a beggar, and gave him 15 cents each; how many cents did the beggar receive?

In this example we have 15 cents (the number which each boy gave the beggar) to be repeated 6 times, (as many times as there were boys.)

:

When questions occur where the same number is to be repeated several times, the operation may be shortened by a rule called Multiplication.

In multiplication the number to be repeated is called the Multiplicand.

The number which shows how many times the multiplicand to be repeated, is called the Multiplier.