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Eight twenty-fifths, four twenty-fifths, and seven twenty-fifths, are how many ?

RULE FOR ADDING VULGAR FRACTIONS, WHEN ALL HAVE

THE SAME OR A COMMON DENOMINATOR. Add the numerators, and place their sum over the com. mon denominator.

EXAMPLE.

Add ਨੂੰਹ ਨੂੰ

and

Add ਸੰਤ ਫ ਬੰਤ and s : :
The sum of the numerators is 15, which being placed
over the common denominator, gives the answer 11.
Add the following sums, using the signs, thus :

Ans. Â + 4 + is.
Add i and i:. Add ਨੂੰ ਫੁੱਲ ਠ and ਫੁੱਲ •
Add and 1

When fractions having a different denominator, are ad. ded, it is necessary to perform a process which will be ex. plained hereafter.

Those fractions which have the numerator larger than the denominator, are called improper fractions, thus :

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Add $

When we use the expression seven halves, we do not mean seven halves of one thing, because nothing has more than two halves. But if we have seven apples, and take a half from each one, we shall have seven halves; and they are halves of seven things, and must be written as above.

SUBTRACTION. There are four kinds of Subtraction.

The first is Simple Subtraction, in which the minuend and subtrahend are whole numbers, and ten units of one order, make one unit of the next higher order.

The second is Decimal Subtraction, in which the minu. end and subtrahend are Decimals.

What is the rule for adding vulgar fractions ? What is meant by the expression seven halves ?

The third is Compound Subtraction, in which other num. bers besides ten, make units of a higher order.

The fourth is Subtraction of Vulgar Fractions, in which the minuend and subtrahend are vulgar fractions.

SIMPLE SUBTRACTION.

If 8 cents are taken from 12 cents, what will remain ?

If 9 apples are taken from 14 apples, how many will remain ?

If 12 guineas are taken from 20 guineas, how many will remain ?

If from 18 books, 12 be taken, how many will remain ?

Let the following examples be illustrated by the coin of the U.S.

· If $2,5 d. 6 cts. be taken from $3, 6 d. 7 cts., how much will remain? Which is the subtrahend, and which the minuend ?

Place $3, 6 d. 7 cts. on a table, side by side, and let the pupil take the amount of the subtrahend from them.

Subtract $:3, 4 d. 5 cts. from $6, 7 d. 7 cts.
Subtract 3 d. 4 cts. 2 m. from 5 d. 6 cts. 8 m.
Subtract 8 d. 7 cts. 5 m. from 9 d. 9 cts.

9 m.
Let the teacher place on the table the coins, thus :

$3,4 d. 6 cts. Under this, place for the subtrahend, the following, so that the-coins shall stand under others of the same order.*

$2, 2 d. 4 cts. What is the remainder, when the value expressed by the subtrahend, is taken from the minuend?

Now if 10 cents be added to the 6 cents of the min. uend, and 1 dime be added to the 2 dimes of the subtra. hend, will there be any difference in the answer? Let the pupil try it and ascertain.

İf 10 dimes be added to the 4 dimes of the minuend,

* The pupil must understand that the subtrahend shows how many of the same kinds of coin, are to be taken from the minuend,

What are the four kinds of subtraction ? Describe them.

and 1 dollar be added to the 2 dollars of the subtrahend, will there be any difference in the answer ?

Let this process be continued until every member of the class fully understands it, and then let them commit to memory this principle.

If an equal amount be added to the Minuend and the Subtrahend the Remainder is unaltered.

Let the following coins be placed as minuend and sub. trahend.

d. cts.

1 3 Minuend.

1 4 5 Subtrahend. Which is the largest sum, taken as a whole, the minuend or subtrahend?

If each order is taken separately, in which orders is the minuend the largest, and in which the smallest ? Can you

take 5 cents from 3 cents ? If

you add 10 cents to the 3 cents, you can subtract 5 from it, but what must be done to prevent the Remainder from being altered ?

$ d. cts. From 4 3 2 4

Subtract 1 4 5 6 In which orders are the numbers of the subtrahend larger than those of the minuend?

Can 6 mills be taken from 4 mills ?
What can you

do in this case ? If 10 mills be added to the 4 mills of the minuend, why must 1 cent be added to the 5 cents of the subtrahend ?

From 6432, subtract 3256.
Can 6 units be taken from 2 units ?
What must be done in this case?

m.

What is the principle by which the process of subtraction is performed ?

RULE FOR SIMPLE SUBTRACTION. Write the subtrahend under the minuend, placing units of the same order under each other, and draw a line under. Subtract each order of the subtrahend, from the same order of the minuend, and set the remainder under. If any order of the subtrahend is greater than that of the minuend, add ten units to the minuend, and one unit to the next higher or. der of the subtrahend. Then proceed as before.

EXAMPLE
Subtract 4356
From 2187

2169 Let the pupil subtract thus :

Seven units cannot be taken from 6; therefore add 10 to the minuend, which makes 16.7 from 16 leaves 9. As 10 units have been added to the minuend, the same amount must be added to the subtrahend. 1 of the order of tens is the same amount as 10 units, we therefore add 1 to 8 tens, making it 9 tens. We cannot subtract 9 tens from 5 tens, we therefore add 10 to the minuend, which makes 15. 9. tens from 15 leaves, 6 tens. As 10 tens have been added to the minuend, the same amount must be added to the subtrahend-1 of the order of hundreds is the same amount as 10 tens; we therefore add 1 to 1 hundred, which makes 2 hundred. This subtracted from 3 hundred leaves 1 hundred. Thus through all the orders.

Mode of Proof. A sum in Subtraction is proved to be right, by adding the remainder to the subtrahend ; and if the sum is the same as the minuend, the answer may be considered as right. Let the following sums be explained as above. Subtraet 34695 from 56943 653215

956432

What is the rule for simple subtraction ? What is a mode of proof ?

66

Subtract 500032 from 867200
6291540

8732418
354965

5360025 7985430

989763 3542685

6542169 5321543

7954324 1223345

8500642 1549768

3895463 3543257

6385241 2006935

5000623 The pupil should learn to subtract by the use of the signs, thus : Subtract 5 from 7. Ans. 7-5= 2. Subtract 8 from 11. Ans. 11-8=3.

Subtract the following numbers in the same way. 8 from 17. 9 from 14. 6 from 20. 40 from 85. 800 from 950. 1000 from 2744. 85 from 760. 95 from 700. 440 from 763.

DECIMAL SUBTRACTION.

If 2 tenths, 4 hundredths of a dollar, be taken from 4 tenths, 6 hundredths, what will remain ?

If 3 hundredths, 5 thousandths of a dollar, be taken from 5 hundredths, 7 thousandths, what will remain ?

If 5 dimes, 6 mills, be taken from 7 dimes, 8 mills, how much will remain ?

If 4 dimes, 5 cents, be taken from a dimes, 9 cents, how much will remain ?

If 4 units, 6 tenths, be taken from 6 units, 8 tenths, how much will remain ?

In simple subtraction, if the number in any order of the minuend, was smaller than the one to be subtracted, what did you do?

The same is to be done in Decimal Subtraction.

Take 4 tenths, 7 hundredths of a dollar, from 6 tenths, 5 hundredths.

In which order is the number of the subtrahend the lar.

gest?

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