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RULE FOR REDUCING FRACTIONS TO WHOLE NUMBERS.

Divide the numerator by the denominator; write the remainder, if there be any, over the denominator, and annex the fraction, thus formed, to the quotient.

EXAMPLES.

1. Reduce to a whole or mixed number.

2. Reduce 7. Ans. 93.

15. . Ans. 23.

3. Reduce 1. Ans. 523.

Ans. 2425.

Ans. 94.

5. Ans. 94. 7. Ans.

2835. Ans. 565. 21825.

4. Reduce 8131, 6276. 518432, 915873, 1325965. 5. Reduce 987654321 700070007. 600304002. 6. Reduce 71123450 7112345499. 4956250217, 3322211136.

59248321768.

REDUCTION OF WHOLE NUMBERS TO
FRACTIONS.

1. In three units, how many fourths, and how is the answer expressed in figures?

2. How many fifths is three units and two fifths, and how is the answer written?

3. Reduce 9 units to sixths.

4. Reduce 7 units and two twelfths to twelfths.

RULE FOR REDUCING WHOLE NUMBERS TO FRACTIONS.

Multiply the whole number by the denominator of the fraction to which it is to be reduced, and place the product over this denominator. If there is with the units, a fraction of the same denominator, add the numerator of this fraction to the product, before placing it over the denominator.

What is the rule for reducing fractions to whole numbers? What is the rule for reducing whole numbers to fractions?

EXAMPLES.

1. How many 4ths in 1? How many in 11 ? In 14 ? In 1f ?

2. How many 5ths in 1 ? In 5 ? In 1}? In 13? In 74 ? 3. How many 7ths in 7? In 8? In 12! In 73 ? In

? 59 ?

4. How many 12ths in 9 ? In 7 ? In 34? In 5 ? In 8 ? 5. How many 6ths in 3? In 4? In 52? In 7 ? In

흡 8? In 94? In 12 ?

6. How many 27ths in 3? In 2? In 5 27 44

7. How many 19ths in 15? In 13 is? In 17 ik? Ans. 285 . . ?

.

12

2

? Ans. .

144

19

19

250
19

Tg.

19

REDUCTION OF VULGAR TO DECIMAL

FRACTIONS.

Decimal Fractions are generally used in preference to Vulgar, because it is so easy to multiply and divide by their denominators.

Vulgar Fractions can be changed to Decimals by a process which will now be explained.

In this process, the numerator is to be considered as units divided by the denominator.

Thus į is 3 units divided by 4, for is a fourth of 3 units.

We can change these 3 units to an improper decimal thus, 3:0 (30 tenths), and then divide by 4; remembering that the quotient is of the same order as the dividend.

4)3.0(,75

2-8

,20

,20 Thus the 30 tenths are divided by 4, and the answer is 7 tenths, which is placed in the quotient, with a separatrix

prefixed. 4 times 7 tenths (or 28 tenths) are then subtracted, and the remainder is,2. This, in order to divide it by 4, must have a cipher annexed, making it 20 hundredths. The quotient of this is 5 hundredths, and no re

mainder.

(In performing this process, particular care must be taken in using the separatrix, both for proper and improper decimals.)

Let be reduced in the same way.

The two units are first changed to an improper decimal, thus:

8)2'0(,25
1.6

,40

,40

00

We proceed thus. 20 tenths divided by 8, is 2 tenths, which is placed in the quotient. 8 times,2 or 16 tenths (16) is then subtracted, and ,4 remain.

This is changed to 40 hundredths (,40) by adding a cipher, and then divided by 8. The quotient is 5 hundredths, which is put in the quotient and there is no remainder.

NOTE. After 3 or 4 figures are put in the quotient, if there still continues to be a remainder, it is not needful to continue the division, but merely to put the sign of addition in the quotient to show that more figures might be added.

EXAMPLES.

Reduce to a decimal, and explain as above.

Reduce

value.

each to a decimal of the same

Let the pupil be required to explain sums of this kind as directed above, until perfectly familiar with the principle.

When fractions of dollars and cents are expressed, their decimal value is found by the same process.

For example, change a dollar to a decimal.

Here the 1 of the numerator, is one dollar, divided by 2. By adding a cipher to this I and using the inverted separatrix, the dollar is changed to 10 dimes, and when this is divided by 2, the answer is 5; which being of the same order as the dividend, is 5 dimes.

The answer is to be written with the sign of the dollar before it, thus $0,5.

The only difference between the answer when is reduced to a decimal, and when a dollar is reduced to a decimal, is simply the use of the sign of a dollar ($) and a cipher in the dollar order.

1. Reduce to a decimal. Ans. ‚5.

2. Reduce a dollar to a decimal. Ans. $0,5. 3. Change

4. Change

of a dollar to a decimal. Ans. $0,125. of a dollar to a decimal. Ans. $0,0625. In this last sum there must be two ciphers added to the numerator, changing the 1 dollar to cents, instead of dimes; and in this case a cipher is put in the order of dimes, and the quotient (being of the same order as the dividend) is placed in the order of cents.

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Change the numerator to an improper decimal, by annexing ciphers and using an inverted separatrix. Divide by the denominator, placing each quotient figure in the same order as the lowest order of the part divided.

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3. Reduce TO to a decimal.

Ans. .028.

4. Reduce

to a decimal.

Ans. .05625.

Ans. .3333333+

Why are decimal fractions used in preference to vulgar? What is the rule for reducing vulgar to decimal fractions?

NOTE. We see here, that we may go on forever, and the decimal will continue to repeat 33, &c. therefore, the sign of addition + in such cases may be added, as soon as it is found that the same number continues to recur in the quotient.

REDUCTION OF FRACTIONS TO A COMMON DENOMINATOR.

Before explaining this process, it must be remembered that &c. or a fraction which has the numerator 88 and denominator alike, is the same as a unit. If therefore we take a fourth of it is the same as taking a fourth of If we take a sixth of it is the same as taking a sixth

one.

of one.

If we take of it is the same as taking of one.

Whenever therefore we wish to change one fraction to another, without altering its value, we suppose a unit to be changed to a fractional form, and then take such a part of it, as is expressed by the fraction to be changed.

For example, if we wish to change to twelfths, we change a unit to twelfths and then take of it, and we have of 12, which is the same as of one.

If we wish to change to eighths, we change a unit to and then take of it, for of is the same as of one. Change to twelfths, thus, a unit is 12.

12

One third of

12 is Two thirds is twice as much, or .

are 12.

Then

Change to twentieths. A unit is. One fifth of 20 is. Four fifths is four times as much, or 18.

Change the following fractions, and state the process in the same way.

Change to twenty-fourths.

Change to twelfths.

Reduce to twenty-sevenths.

Reduce to sixty-fourths.

Reduce to twenty-fifths.

Reduce to twenty-sevenths.

Reduce to thirty-sixths.

Reduce to forty-ninths.

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