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denom. The fraction thus formed, whether proper or improper, is the sum of the given fractions.

The numerators, that is to say, the fractions, being each composed of units of the order signified by the com. denom., their sum is a number of units of the same order; and, for this reason, we give it the same denominator

Thus, &+2=5.

But when they have not a common denominator, (198,) we first reduce them to a com. denom., (200 or 201,) and proceed as before. Thus, to find the sum of 1, 3, and, we first reduce these fractions to a com. denom., (200,) and have for their equivalents 21, 29, and 19. Then to find their sum we say, 21+28+18

28

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42

the sum of,, and is 125.

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Wherefore,

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Examples.

189+140+216

252

Let the expressions

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Hence, the sum of 3, 5, and is 24.

in the following examples be written in the same manner:

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new numerators, s their sum,

16+20+21

24

b

с e

a+b+c+e

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5. }+&+ 1⁄2 +¿§§; also, 7+ } + { } = 1.

6. 1+3+2+1 + % +3 +8 = 5118.

7. 10+1/2+1/+18=237983.

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8. 1 + 1 + 1 + {+}+{+}+1=12088.

Addition of Mixed Numbers.

2520

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203. Mixed numbers may be added either by first changing them to improper fractions, and then proceeding as above, or by first finding the sum of the integral parts; then, that of the fractional parts; and lastly, the sum of the two sums.

Or thus: 18 and are equivalent to 198 and 198; then,

1957

from
we take

10119%

94165

and have

71 for remainder, as before.

Here, as is greater than 11, we do not add a unit to either number, as in the preceding example.

3. To subtract 9411 from 101, we place the numbers thus:

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and, as there is nothing above 11, we subtract it from a unit, and carry a unit to the figure 4, as in the first example. Now, in reducing a unit to a fraction, having 13 for denominator, (191,) we have 13 also for the numerator; but, as this is the same as the denominator, we subtract the numerator of 1} from its own denominator, and, placing the difference 2 over 13, we have for the difference between 1 and a unit; we then carry a unit to 4, and, having subtracted, we find that 62 is the difference between the given numbers.

13

13

The difference between a fraction and a unit is called the complement of that fraction: thus, 11 and are the complements of each other.

4. 153-33117; and 153-117 = 37. 5. 931-672,9%; and 931-2% = 67. 6. 283-911-183; and 283 - 183 = 911. 7. 655-1911-45,77; and 9-43-44. 8. 101-69411; and 151-141=1. 8 9. 10437-655,978,343. Prove this and the succeeding example by addition and subtraction. 10. 31719807-21907.

Multiplication of Fractions.

21

206. To find the product of several fractions, we multiply all the numerators together for a numerator, and all the denominators for a denominator. The fraction thus formed is

the product.

For example, to multiply by . If we (165) first multiply by 2, we have §. But is (145) the third part of 2; hence, the product is 3 times too great, and must be divided by 3. Now, this is done (165) in multiplying the denomina

tor 4 by 3, which gives

2, and have

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for the quotient; therefore

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Or, to multiply by 2; we divide (165) its denominator by for the product. But this, as above, is just 3 times the true product: we, therefore, (165,) divide its numertor by 3, and have for the true product, as before.

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The striking out of a numerator and denominator, as here shown, is called cancelling. We do not always write the quotient 1 above or below the figure wholly cancelled; because we know that, when all the numerators are cancelled, the result is 1; seeing that any power of 1 is 1.

The reasoning applied above to the multiplication of two fractions, will apply to the multiplication of their product and a third fraction; to that product and a fourth, &c., and, consequently, to the continued multiplication of any number of fractions. Hence, we see that, in multiplying any number of fractions, the resulting numerator is the product of all the numerators, and the denominator, the product of all the deno

minators.

207. Let us observe that, in multiplying by a fraction, we not only perform a multiplication, but also a division; and, as the number by which we divide is greater than that by which we multiply, the quantity multiplied by a fraction, instead of being increased, is diminished.

Examples.

1. To multiply,,, and together, we proceed thus:

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The lines drawn through the numbers 3, 4, 5, show that they cancel each other; that is to say, because the value of would not be altered, (165,) if both its terms were multiplied by the product 3X4X5, we omit the multiplication of these numbers, which saves the trouble, not only of this multiplication, but also of reducing the final product to its lowest terms. When two opposite terms are equal, they may be omitted:

when one is a multiple of the other, we may divide the greater by the less, and do with the quotient as we should have done with the greater: also, when they have a common measure, divide both by it, and operate with the quotients as with the numbers; because, in each case, the effect is the same as to divide both terms of the final product by the same number, which (165) does not alter its value.

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The prime factors of 9×26×36 and of 13×24×27 are the same, namely, 3+, 23, and 13: therefore, omitting these numbers, we have for the total product

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208. To multiply a fraction by a whole number, is the same as to multiply the whole number by the fraction; for, in either case, (191,) having expressed the whole number as a fraction, we multiply the whole number and the numerator of the fraction together, and divide the product by the denominator, (206.) Thus:

2136x=2136X3=6408-1602.

4

Or, by cancelling, if practicable, first divide the whole number by the denominator of the fraction, and multiply the numerator by the quotient. Or, if the whole number divide the denominator of the fraction, divide the numerator by the quotient. Thus :

2136X534×3=1602.

Also, when the numerator and denominator differ, as in the present case, by only one unit, we may, when we have divided the whole number by the denominator, subtract the quotient from the whole number: thus, having found the quotient 534, we say, 2136-534=1602, as before.

Therefore, to take . . . §. &c. of a number, is to dimiwish it by J, 4, J, J, &c. of itself.

When the whole number is the same as the denominator of the fraction, the result is the numerator. Thus,

$X4=8; }{X12=11, &c.

Examples.

5

·1. {×3={=}=15=5=23. Or, X=5=23.

6

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Ρ 9

3. 7×11=9§; 134 × 13=123; 17×204=18210.

Multiplication of Mixed Numbers.

209. Reduce the mixed numbers to improper fractions, and multiply them (206) as fractions. Also, when there are whole numbers, express them as fractions; and, in all cases, cancel as much as possible.

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2209

By the general method, 153533X8X 24-9224, as before.

4.

5.

X28X=1; 2×33×4=39; 331×104-360. X3 X 2 X 16; 61 X3 X 2 = 39. 6. 949 7713-7341151; 69 X 3927074. 210. The calculation is often facilitated by striking a factor from one number and introducing it into another. Thus, striking the factor 11 from 3311, we have 31; then introducing it into 13, we have 152, and multiplying the results, instead of the numbers, we have 4683 for the product, as follows:

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