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RULE FOR REDUCING FRACTIONS TO WHOLE NUMBERS.

Divide the numerator by the denominator ; write the re. mainder, 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. Ans. 94.

2. Reduce. Ans. 93. . Ans. 94. 9. Ans. 15. 7. Ans. 23.

3. Reduce 118. Ans. 523. 245. Ans. 565. 1995. Ans. 2425.

4. Reduce sy. 36, 51833. 914873. 1325965. 5. Reduce 987654321, 7000 70007. 600344002,

6. Reduce 711235499 4 9 5 6 345 02 17. 33224 1136. 592 4 8 32 1768.

REDUCTION OF WHOLE NUMBERS TO

FRACTIONS.

1. In three units, how many fourths, and how is the an. swer 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 frac. tion 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 ?

54 ?

EXAMPLES. 1. How many 4ths in 1? How many in 'lį? In 14 ? In 11 ?

2. How many 5ths in 1? In 5 ? In 11? In 13 ? In 74 ? 3. How many 7ths in 7? In 8? In 12? In 7%? In

4. How many 12ths in 9 pa ? ln 7 ? In 34 ? In 54? In 8 ?

5. How many 6ths in 3? In 4? In 5f? In 7 ? In 8? In 94 ? In 12? 6. How many 27ths in 3? In 2 ?

In 2? In 5 ? Ans. . 4. 44

7. How many 19ths in 15? In 13 ? In 17 18? Ans. 285. Bu ?

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 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)3:00,75

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,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

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prefixed. 4 times 7 tenths (or 28 tenths) are then sub. tracted, and the remainder is,2. This, in order to divide it by 4, must bave a cipher annexed, making it 20 hun. dredths. 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)20(,25

166

,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 ci. pher, and then divided by 8. The quotient is 5 hundredths, which is put in the quotient and there is no re. mainder.

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 addi. tion in the quotient to show that more figures might be added.

EXAMPLES.
Reduce ito a decimal, and explain as above.

Reduce 1 44 45 $ each to a decimal of the same value.

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

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 l 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 of a dollar to a decimal. Ans. $0,125. 4. Change is 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.

5. Reduce } of a dollar to a decimal. Ans. $0,2. 6. Reduce of a dollar to a decimal. Ans. $0,625. 7. Reduce o of a dollar to a decimal. Ans. $0,1871. 8. Reduce o to the decimal of a doilar. Ans. $0,01.

RULE FOR THE REDUCTION OF VULGAR TO DECIMAL

FRACTIONS.

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.

1. Reduce o iz to a decimal. Ans. .0016. 2. Reduce to a decimal. Ans. .028. 3. Reduce Téo to a decimal. Ans. .05625. 4. Reduce į to a decimal. Ans. .3333333+ Why are decimal fractions used in preference to vulgar? What is the rule for reducing vulgar to decimal fractions ?

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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.

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REDUCTION OF FRACTIONS TO A COMMON

DENOMINATOR. Before explaining this process, it must be remembered that it &c. or a fraction which has the numerator 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 f it is the same as taking a sixth of one.

If we take of it is the same as taking i 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 1 of it, and we have of 13, which is the same as į of one. If we wish to change to eighths, we change a unit to and then take 1 of it, for ļof is the same as į of one.

Change to twelfths, thus, a unit is 1. One third of ij is inTwo thirds is twice as much, or . Then are i

Change to twentieths. A unit is f. One fifth of 3: 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 z to twelfths.
Reduce to twenty-sevenths.
Reduce to sixty-fourths.
Reduce to twenty-fifths.
Reduce to twenty-sevenths.
Reduce 1 to thirty-sixths.
Reduce to forty-ninths.

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