ART. 25.

§ III. SUBTRACTION.

MENTAL EXERCISES.

WHEN it is required to find the difference between two numbers, the process is called Subtraction. The operation is the reverse of addition.

but 4; how many

Ex. 1. John has 7 oranges, and his sister but 4 ; more has John than his sister?

ILLUSTRATION.

We first inquire what number added to 4 will make 7. From addition we learn that 4 and 3 are 7; consequently, if 4 oranges be taken from 7 oranges, 3 will remain Hence John has 3 oranges more than his sister.

2. Thomas had five oranges, and gave two of them to John how many had he left?

3. Peter had six marbles, and gave two of them to Samuel· how many had he left?

4. Lydia had four cakes; having lost one, how many had she left?

5. Daniel, having eight cents, gives three to Mary; how many has he left?

6. Benjamin had ten nuts; he gave four to Jane, and three to Emily; how many had he left?

7. Moses gives eleven oranges to John, and eight to Enoch; how many more has John than Enoch?

8. Paid seven dollars for a pair of boots, and two dollars for shoes; how much did the boots cost more than the shoes?

26. How many are 20 less 5? 20 less 8? 20 less 9? 20 less 12? 20 less 15? 20 less 19?

27. Bought a horse for 60 dollars, and sold him for 90 dollars; how much did I gain?

ILLUSTRATION. We may divide the two prices of the horse into tens, and subtract the greater from the less. Thus 60 equals 6 tens, and 90 equals 9 tens; 6 tens from 9 tens leave 3 tens, or 30. Therefore I gained 30 dollars.

28. Sold a wagon for 70 dollars, which cost me 100 dollars; how much did I lose?

29. John travels 30 miles a day, and Samuel 90 miles; what is the difference?

30. I have 100 dollars, and after I shall have given 20 to Benjamin, and paid a debt of 30 dollars to J. Smith, how many dollars have I left?

31. John Smith, Jr., had 170 dollars; he gave his oldest daughter, Angeline, 40 dollars, his youngest daughter, Mary, 50 dollars, his oldest son, James, 30, and his youngest son, William, 20 dollars; he also paid 20 dollars for his taxes ; how many dollars had he remaining?

ART. 26. The pupil, having solved the preceding questions, will perceive, that

SUBTRACTION is the taking of one number from another to find the difference.

When the two numbers are unequal, the larger is called the Minuend, and the less number the Subtrahend. The answer, or number found by the operation, is called the Difference, or Remainder.

NOTE. The words minuend and subtrahend are derived from two Latin words; the former from minuendum, which signifies to be diminished or made less, and the latter from subtrahendum, which means to be subtracted or taken away.

ART. 27. SIGNS. Subtraction is denoted by a short horizontal line, thus, signifying minus, or less. It indicates that the number following is to be taken from the one that precedes it. The expression 6-24 is read, 6 minus, or less, 2 is equal to 4.

QUESTIONS. Art. 26. What is subtraction? What is the greater number called? What is the ess number called? What the answer? -Art. 27. What is the sign of subtraction? What does it signify and indicate ?

EXERCISES FOR THE SLATE.

ART. 28. Method of operation, when the numbers are large and each figure in the subtrahend is less than the figure above it in the minuend.

Ex. 1. Let it be required to take 245 from 468, and to find their difference. Ans. 223.

OPERATION.

Minuend 468
Subtrahend 245
Remainder 223

We place the less number under the greater, units under units, tens under tens, &c., and draw a line below them. We then begin at the right hand, and say, 5 units from 8 units leave 3 units, and write the 3 in units' place below. We then say, 4 tens from 6 tens leave 2 tens, and write the 2 in tens' place below; and proceed with the next figure, and say, 2 hundreds from 4 hundreds leave 2 hundreds, which we write in hundreds' place below. We thus find the difference to be 223.

ART. 29. First Method of Proof. Add the remainder and the subtrahend together, and their sum will be equal to the minuend, if the work is right.

This method of proof depends on the principle, That the greater of any two numbers is equal to the less added to the difference between them.

8. A farmer paid 539 dollars for a span of fine horses, and sold them for 425 dollars; how much did he lose?

Ans. 114 dollars.

9. A farmer raised 896 bushels of wheat, and sold 675 bushels of it; how much did he reserve for his own use?

QUESTIONS.

Ans. 221 bushels.

Art. 28. How are numbers arranged for subtraction? Where do you begin to subtract? Why? Where do you write the difference? Art. 29. What is the first method of proving subtraction? What is the reason of this proof, or on what principle does it depend?

10. A gentleman gave his son 3692 dollars, and his daughter 1212 dollars less than his son; how much did his daughter receive? Ans. 2480 dollars.

ART. 30. Method of operation when any figure in the subtrahend is greater than the figure above it in the minuend.

Ex. 1. If I have 624 dollars, and lose 342 of them, how many remain? Ans. 282.

OPERATION.

We first take the 2 units from the 4 units, Minuend 624 and find the difference to be 2 units, which we Subtrahend 3 4 2 write under the figure subtracted. We then proceed to take the 4 tens from the 2 tens Remainder 282 above it; but we here find a difficulty, since the 4 is greater than 2, and cannot be subtracted from it. We therefore add 10 to the 2 tens, which makes 12 tens, and then subtract the 4 from 12, and 8 tens remain, which we write below. Then, to compensate for the 10 thus added to the 2 in the minuend, we add one to the 3 hundreds in the next higher place in the subtrahend, which makes 4 hundreds, and subtract the 4 from 6 hundreds, and 2 hundreds remain. The remainder, therefore, is 282.

The reason of this operation depends upon the self-evident truth, That, if any two numbers are equally increased, their difference remains the same. In this example 10 tens, equal to 1 hundred, were added to the 2 tens in the upper number, and 1 was added to the 2 hundreds in the lower number. Now, since the 3 stands in the hundreds' place, the 1 added was in fact 1 hundred. Hence, the two numbers being equally increased, the difference is the same.

NOTE. This addition of 10 to the minuend is sometimes called borrowing 10, and the addition of 1 to the subtrahend is called carrying 1.

ART. 31. From the preceding examples and illustrations in subtraction, we deduce the following general

RULE. · Place the less number under the greater, so that units of the same order shall stand in the same column.

Commencing at the right hand, subtract each figure of the subtrahend from the figure above it.

If any figure of the subtrahend is larger than the figure above it in the minuend, add 10 to that figure of the minuend before subtracting, and then add 1 to the next figure of the subtrahend.

QUESTIONS.

Art. 30. How do you proceed when a figure of the subtrahend is larger than the one above it in the minuend? How do you compensate for the 10 which is added to the minuend? What is the reason for this addition to the minuend and subtrahend? How does it appear that the 1 added to the subtrahend equals the 10 added to the minuend? What is the addition of 10 to the minuend sometimes called? The addition of 1 to the subtrahend? Art. 31. What is the general rule for subtraction?