der, which makes it 30; and this subtracted from 36, leaves 6 to be written under that order.

Let the following sums be explained as above.

SUBTRACTION OF VULGAR FRACTIONS.

If a boy has 6 ninths of an apple, and gives away 4 ninths, how much remains?

If he has 8 ninths, and gives away 5 ninths, what re. mains?

If he has 7 twelfths, and gives away 4 twelfths, what remains?

In doing these sums let the pupil tell first which is the minuend and which the subtrahend.

A man has 9 twentieths of a dollar and loses 5 twenti. eths, how much remains ?

If he has 11 twentieths and loses 7 twentieths, what remains?

If he has 8 sixteenths, and loses 5 sixteenths, what remains?

Subtract from Subtract from 1.

12

RULE FOR SUBTRACTING VULGAR FRACTIONS.

Subtract the numerator of the subtrahend, from the numerator of the minuend, and place the remainder over the common denominator.

Let the pupil, in doing the sums, use the signs in this

៖៖៖.

20

38.

66 12

៩.

A man owns of a pasture, and sells 3, how much re

mains his own?

A boy has of a guinea, and gives away, how much has he left?

from 25, are how many? from are how

many?

409

18 from 38 are how many?

from 1 ? from 1o ?

MULTIPLICATION.

Multiplication is repeating a number, as often as there are units in another number.

The number to be repeated is called the multiplicand. The figure expressing the number of times the multiplicand is to be repeated, is called the multiplier.

The answer is called the product, because it is the sum produced by multiplication.

The multiplier and multiplicand are called the factors, from the Latin word factum, (made,) because they are the numbers by which the product is made.

What is the rule for subtracting Vulgar Fractions?

There are four processes of multiplication.

The first is Simple Multiplication, where the factors are whole numbers, and ten units of one order make one unit of the next higher order.

The second is Decimal Multiplication, where one, or both the factors are decimals.

The third is Compound Multiplication, where the multiplicand consists of orders, in which other numbers besides ten, make units of a higher order.

The fourth is the multiplication of vulgar fractions, where one, or both the factors, are vulgar fractions.

SIMPLE MULTIPLICATION.

A boy gives 8 apples to each of 7 companions, how many does he give to them all?

A man travels 7 miles an hour, how far will he travel in 9 hours?

If one pound of raisins cost 11 cents, how much will 6 pounds cost?

One boy has 7 cents, and another twelve times as many, how many has the last?

At six cents apiece, how much will 9 lemons cost?

At 12 cents a dozen, how much will 8 dozen marbles cost?

One pound of sugar costs 6 cents, how much will 5 pounds cost? 8 pounds 8 pounds? 11 pounds? 12 pounds? Multiplication has been defined as repeating, or taking one number as often as there are units in another number. Let this process be illustrated by the coins; thus, d. cts.

2 66 4 (c 3

Let the multiplier be 2.

Now the pupil is to take 3 cents, as often as there are units in 2, and give the answer. Then he is to take 4

dimes as often as there are units in 2, and then 2 dollars in like manner.

Describe the four different kinds of multiplication.

Let the following sum be done by the coins.

When the pupil has taken 4 cents three times, he will have 12 cents. Let a dime be substituted for ten of these cents, to be carried to the next product, and there remain two cents, to be placed in the order of cents. Then let 4 dimes be taken 3 times, which make 12, and the one dime of the other product is added, making 13 dimes. Let a dollar be substituted for ten of the dimes, and carried to the next product, and three dimes will remain to be placed in the order of dimes. Two dollars taken three times, will make 6 dollars, and adding the one dollar of the other product, the amount is 7 dollars, to be placed in the order of dollars.

The pupil should practice in this way until the principle is fully understood.

RULE FOR MULTIPLYING, WHEN THE MULTIPLICAND HAS SEVERAL ORDERS, AND THE MULTIPLIER DOES NOT EXCEED TWELVE.

Place the multiplier below the multiplicand. Beginning at the right, multiply each order of the multiplicand, by the multiplier. Place the units of the product, under the order multiplied, and carry the tens to the next product. Write the whole of the last product.

Let the pupils at first be exercised thus :—

EXAMPLE.

249

8

1992

Eight times 9 units are 72 units; which is 2 units to be written under that order, and 7 tens to be carried to

What is the rule for Simple Multiplication, when the multiplier does not exceed 12?