bers is: Let mixed numbers be reduced to improper fractions, whole numbers to the form of fractions, and fractional expressions having different denominators to equivalent expressions having a common denominator. Then the difference of the numerators placed over the common denominator expresses the difference of the proposed fractional numbers.

88. Find the differences between

89. When whole or mixed numbers occur either in the minuend or subtrahend, the subtraction may be made by other methods more easily than by the general rule already given.

For the purpose of explaining these methods, let it be required

1st. To subtract from 1.

1 being equal to the quotient of any number divided by itself, it is equal to 3.

The remainder, in this case, may be found by taking the numerator of the subtrahend from the denominator and writing the difference over the denominator.

2nd. Let it be required to subtract from 7.

7-8=6+1-3=6+3=63.

This subtraction is effected (without reducing the minuend to the form of a fraction) by subtracting the fraction from 1 and annexing the remainder to the minuend diminished by 1.

3rd. Let it be required to subtract 54 from 13.

Since 545+, it follows that 13—5413—5—4.

But 13-5--8—4=7%, .'. 13—–54–72.

The last form of calculation is the most simple. It is performed by borrowing 1 from 13, which is thus reduced to 12. Then, 1-2 and 12 -5=7.

A mixed number is thus subtracted from a whole number by subtracting the fractional part of the mixed number from 1; and the integral part from the minuend less 1. 4th. Let it be required to subtract 5 from 17.

17=12+5.

... 17}—5=12+5—5=12.

Whence, a whole number is subtracted from a mixed number by subtracting the whole number from the integral part of the mixed number.

5th. Let it be required to subtract

from 13.

Reducing and to the same denominator, 24,—

To effect the subtraction of from 1, the latter is reduced to an improper fraction. This is readily done by adding the numerator 9 to the denominator 24 and writing the sum 33 over the denominator 24. Then, 33 14 33-14 19

difference required.

from 64.

6th. Let it be required to subtract

Reducing and to the common denominator 20,—

being greater than, 1 is borrowed from 6, which is thus reduced to 5. The subtraction of 1 from 15 is made in the same manner as that of from 12 in the last example.

7th. Let it be required to subtract 41 from 78.

and reduced to the same denominator (12) are and ;

... 79=719 and 41=4 ..73-41-375.

The fractional and integral parts of this subtrahend being both less than the corresponding parts of the minuend, the subtraction is effected without borrowing. The only reduction

required is that of the fractional parts to the same denomi

nator.

It is evident that, of two mixed numbers, that which has the greater integral part is the greater.

8th. Let it be required to subtract 17% from 255.

The fractions being reduced to the same denominator, and the subtrahend placed under the minuend, it becomes necessary to subtract from 15. This is done, as in Example 5, and the remainder found to be 7 Next, 1, for that borrowed, is taken from 25 or added to 17. The difference between 25 and 18 being 7, that of the proposed mixed numbers is 717.

From the examples 1-8 of this article, and accompanying remarks, are deduced the subjoined rules for the subtraction of fractions or mixed numbers from whole or mixed numbers, and of whole from mixed numbers, without reduction of the whole or mixed numbers to improper fractions.

I. To subtract a proper fraction from unity: Subtract the numerator of the fraction from the denominator: the remainder, placed over the denominator, expresses the difference between unity and the proposed fraction.

II. To subtract a proper fraction from a whole number: Subtract the fraction from 1, and annex the remainder to the whole number, diminished by 1. The mixed number, thus obtained, is the difference between the proposed whole number and fraction.

III. To subtract a mixed number from a whole number: Subtract the fractional part of the mixed number from 1; carry 1 to the integral part of the mixed number, and subtract the sum from the whole number. The mixed number, composed of the two remainders, is the difference between the whole and the mixed number.

IV. To subtract a whole number from a mixed number: Subtract the whole number from the integral part of the minuend, and to the remainder annex the fractional part of the minuend. The mixed number thus formed is the difference between the mixed and the whole number.

V. To subtract the greater of two proper fractions, having the same denominator, from 1 + the less: -- Add together the numerator and denominator of the less fraction. Subtract the

numerator of the greater fraction from the sum, and write the remainder over the common denominator. This fraction is the difference of the proposed numbers.

VI. To find the difference of two mixed numbers :- Reduce the fractional parts to a common denominator, and write the less mixed number under the greater.

Then, if the fractional part of the subtrahend is less than the corresponding part of the minuend, subtract the fraction in the lower line from that in the upper, and the whole number in the lower line from that in the upper. The mixed number, composed of the two remainders, is the difference of the proposed numbers. But if the fractional part of the subtrahend is greater than that of the minuend, subtract the former from 1+ the latter; the remainder is the fractional part of the difference: then carry 1. for that borrowed, to the integral part of the subtrahend, and take the sum from the integral part of the minuend; the remainder is the integral part of the difference. To this annex the fractional part of the difference. The number thus obtained is the difference of the proposed mixed numbers.

When the fractional parts of the mixed numbers are equal, the remainder is a whole number; and when the integral parts are equal, the remainder is a proper fraction.

90. Find the differences between

91. The difference of two literal fractions

α с

sented thus,

с

a

-; but since signifies a times theth part of

1, and, c times the {th part of 1, the difference between

and € is (a–c) times the }th part of 1, or ac. 90

b

α

Whence the

difference of two literal fractions, having the same denominator, is obtained by dividing the difference of the numerators of these fractions by their common denominator.

Literal fractions having different denominators may be reduced to equivalent fractions having a common denominator by the method of Art. 79. or Art. 81.

Examples. Find the differences between

с

92. A fraction,, signifies that unity is divided into d equal parts, and that c of these equal parts make the fraction. The value of this fraction is, therefore, c times the dth part of unity. Consequently, when not c but is made a multiplier, the product is not e times the multiplicand, but c times the dth multiplicand c part of the multiplicand; that is, product = Whence the product of any multiplicand by a fractional mul

d

d