Denominator, we generally speak of reducing Fractions so that they may have the same Denominator.

would be-?" by 4, you would

"If, therefore, I wish to add to, what must I do to the ?" Turn it into eighths. "What must I do to the in order to turn it into eighths?" Multiply it by 4. "No, for 4 times Two. "If, therefore, you multiplied the fraction alter its value, and not reduce it to the same Denominator as; you wish to leave its value unaltered, and yet to turn the Denominator into 8."

If the pupil cannot, upon consideration, tell you what is to be done, you must remind him of Rule 2, that a fraction is not altered by multiplying the Numerator and Denominator by the same number. Then you ask, "By what number must I multiply the Denominator of in order } to make the new Denominator 8?" By 4. "And by what number must I multiply the Numerator in order not to alter the value of the fraction?" By 4. "And what does then become?" .. "And we obtained this result by multiplying the smaller Denominator by such a number as to make it equal to the larger Denominator, and by multiplying the Numerator by the same number."

After several instances of this kind, in which one fraction is added to another by reducing the former to a fraction with the same Denominator as the latter, we proceed to instances where both fractions are altered by being reduced to fractions with the same Denominator. "Add a pound to a florin." Twenty-two shillings. "In order to get this answer, what did you do to the pound and the florin?" I turned them into shillings. "Yes, in order to add money of different denominations you turned them both into money of the-?" Same denomination. "And so, in order to add two fractions of different denominations, you must often turn them both into fractions with the same denominator."

Required, to add together and

"Take your stick, which is divided into twelve equal parts, or twelfths; suppose I have to add and of the stick. I see that of the stick contains a certain number of these twelfths. How many twelfths?" 4 twelfths. "Write down in arithmetical signs, that one third is equal to four twelfths."

"Here you have multiplied the Numerator and Denominator of the first fraction by 4, have you not?" Yes. "Well, note that 4 is the Denominator of the second fraction. And now look at the stick, and tell me how many twelfths there are in . Write down the result in

arithmetical signs." = ·

"Here you have multiplied the Numerator and Denominator of the second fraction by-?" 3. "Yes, and note that 3 is the Denominator of the first fraction.

"Now, therefore, knowing that one third is equal to four twelfths, and that one fourth is equal to three twelfths, we know that one

fourth added to one third is equal to how many twelfths?" Seven twelfths. "Write down this result in arithmetical signs, viz., that a third added to a quarter is equal to four twelfths added to three twelfths, and that this is equal to seven twelfths."

"Now, in order to add any two fractions in this way, we want a rule to guide us. Let us see what we have been doing. In order to alter the shapes of the two fractions above, so that, without having their values altered, they should have a Denominator common to both, we multiplied the Numerator and Denominator of the first by 4, which was the Denominator of the second. "And we multiplied the Numerator and Denominator of the second by-?" 3. "Which is the-?" Denominator of the first.

Try the same method with

and . "What does the first become when its Numerator and Denominator are multiplied by the Denominator of the first?" . "Add the results." "Verify these results on the stick. Are they true?" Yes. "Then now repeat the

=

4. Rule.-In order to add two fractions, multiply the Numerator and Denominator of the first by the Denominator of the second, and the Numerator and Denominator of the second by the Denominator of the first. Then add the two Numerators, retaining the Common Denominator.1

38. THE VALUE OF A FRACTION.

"What is the meaning of of an orange?" That one orange is divided into 4 parts, and 3 of these are taken together. "True; but I shall now show you that has another meaning. Suppose I take 3 oranges at once and divide them among 4 people, what will each receive? You cannot at once answer. How many quarters will there be in 3 oranges?" 12. "And twelve quarters divided amongst 4 people give to each -?" 3 quarters. "Then you see that of an orange is the same as 3 oranges divided by 4." Yes. "And, similarly, of a hundred is the same as 3 hundreds divided by 4." Yes. "And therefore of a unit (i.e.) is the same as 3 units divided by 4 (i.e. 34)?" Yes.

Now let us see whether this rule holds true in other cases, viz., that a Fraction is the same as the Numerator divided by the Denominator. "According to this rule, what would be the value of 48?" 12. "How many quarters are there in 12 things?" 48. "Then is it true that 48 = 12?" Yes. Again are-?" 4. " are-?" 4. Then from all these cases wee that

66

1 As for the Rule of Least Common Multiple, it can be advantageously deferred. When the pupil has to add three fractions, let him (at first) add two together, and add the result to the third. He will thus all the more appreciate the rule of the L. C. M. when he reaches it, as shortening a lengthy process.

5. Rule. The value of a Fraction is the same as that of the Numerator divided by the Denominator.

Hence when we speak of five sixths we mean either (1) that one thing is divided into 6 parts, five of which are taken together, or (2) that five things are divided by 6.

39.

MULTIPLICATION AND DIVISION OF FRACTIONS.

(i) To Multiply a Fraction by a Whole Number.1 What is 7 times 5 oranges?" 35 oranges. 35 ounces.

"7 times 5 millions?" 35 millions.

7 times 5 ounces?" "7 times 5 quarters

of an orange? 35 quarters of an orange. 7 times 5 halves? 35

halves. 7 times 5 sixths?" 35 sixths.

"Write down in arithmet

ical signs that 7 times 5 sixths (i. e. 7 multiplied by 5 sixths) is 35 sixths."

7 × 8 = 35.
x

"What is 5 times 7 eighths?" 35 eighths. "Write this down."

5 × 1 = 35.

'Hence, in order to multiply a fraction by a number, what must we do to the Numerator?" Multiply it by the number.

the Denominator?" Nothing.

Then write down the

"And what to

6. Rule.-In order to multiply a Fraction by a Whole Number multiply the Numerator by it, and leave the Denominator unchanged.

(ii) To Divide a Fraction by a Whole Number.

Suppose I have three separate quarters of an orange, and I wish to give half of my three quarters to a companion, I can cut each quarter into two eighths, can I not, and keep three of the eighths, while I give him the other three?" Yes. "What, therefore, is when divided into two equal parts, or, in other words, when divided by 2?" Three eighths. "Write down in arithmetical signs that three quarters divided by 2 is equal to three eighths."

"In the same way, suppose there is a stick of chocolate twelve inches long, cut into separate twelfth parts (or inches), of which I have received five; and suppose I wish to share my five twelfths (or inches) equally with a companion, or, in other words, to divide it by 2. I can divide each of my inches into half, can I not, and give him five half inches, while I retain five half inches myself?' ' other words, five inches, when divided by 2, is five half inches?” Yes. "Now an inch is a twelfth part of a foot; what part of a foot is half an inch? If you cannot tell at once, count how many half-inches

"" Yes. "In

Here it may be explained that a number that is not a fraction is sometimes called a whole number, in order to distinguish it from a fraction, or broken number.

there are on the foot rule." 24. "Then a half-inch is what part of a foot?" A twenty-fourth. "Therefore, in saying that five inches, when divided by 2, are equal to five half-inches, we really say that five twelfths divided by 2 are equal to-?" Five twenty-fourths. down in arithmetical signs that five twelfths divided by 2 are equal to five twenty-fourths.”

"Write

"Now here we have been dividing first by 2, and then by 2. Have we in either case altered the Numerator?" No. "Have we altered the Denominator?" Yes. "What have we done to it?" Multiplied it by 2. "But 2 is the Whole Number by which we are to divide, is it not?" Yes. "Then, in order to divide a Fraction by any Whole Number, what must you do?" Multiply the Denominator by the Whole Number. "Write that down."

7. Rule.'-In order to divide a Fraction by a Whole Number, multiply the Denominator by it, and leave the Numerator unchanged.

(iii.) To multiply one Fraction by another.

Required to Multiply

by

"If I multiplied by 3, the result (by Rule 6) would be §.

"But this would be too much; for I have multiplied by 3 instead of by, i.e. (Rule 5) 3 ÷ 4. The multiplier has therefore been 4 times too great; what must I do to diminish the result?" Divide by 4. "And 4 is (by Rule 7) what?" .

Hence

86

"How have we obtained our new Numerator?" By multiplying the two old Numerators together. "And how the new Denominator?" By multiplying the two old Denominators together. "Then now you can write down the following:

8. Rule. In order to multiply two Fractions together, multiply the two Numerators to obtain the new Numerator, and the two Denominators to obtain the new Denominator.

(iv.) To divide by a Fraction.

-:" 2. "How many quarters are there

"Then 1 divided by
in 1?" 4. "Then 1

"How many halves are there in 1?" 2. is-: divided by is-?" 4. "What is 1÷?" You cannot tell at once. How many quarters are there in 1?" 4.

"Then 1

is the same as "And this is ? "

4 quarters divided by 3 quarters, is it not?" Yes. Yes. "Write down your results."

1 Rules 6 and 7 may afterwards be amplified by showing that multiplying the Denominator produces the same result as dividing the Numerator, and that dividing the Numerator produces the same result as multiplying the Denominator.

Not 4-3 quarters (see Rule, Par. 34), but 4-3 times, or units.

9. Rule.-In order to divide by a Fraction, invert the Fraction and

multiply.

Another Method.

The following method is not experimental; but it is brief, and has the advantage of applying to the division of a Fraction, as well as of a Whole Number, by a Fraction.

(1) Required to divide 12 by .

If I divide by 3, instead of by, the answer would be ; but as I have divided by a divisor 4 times too large, the result is 4 times too small, and must be multiplied by 4; it is therefore

4 x 12
3

Here we have inverted the Fraction and multiplied. The pupil will readily see that (by Rule 5) the result is 16, and can verify the result on a foot rule by ascertaining that there are 16 three quarters of an inch in 12 inches.

(2) Required to divide by §.

3

be divided by 5 instead of by 4, the result is (Rule 7) 4 x 5

; but, as the divisor is 7 times too large, the quotient is

times

of

les

4.

be

20

too small, and requires to be multiplied by 7; this is

3 × 7 21

20

From both these instances we obtain the Rule given above.

or

20

40. THE MODERN RULE OF THREE, OR METHOD Of unity. The "Rule of Three," as it used to be taught with the old-fashioned method of "stating," affords little, if any, opportunity of appealing to the reason; but when it is taught according to what is called the Method of Unity, presupposing a knowledge of Fractions, it is a most valuable mental exercise.

Thus, suppose the question to be, "If 5 apples cost 24d., what will 21 apples cost?"

1. In time, but not at first, the pupil may substitute ... for "therefore," for "is," and x for "of " and for "times." He may also be allowed, instead of repeating the same words three times, to indicate them by

1 The process of canceling factors common to the Numerator and Denominator follows at once from Rule 3, Page 56.