15 Examples. 1. To add 5.1, 62, and 78, we may proceed thus : 51=1 6= 4; 7 = 6; (193) therefore, 5.1 + 6 + 7 = 44+54 +61 꽃 = +*+=+* + 1 = 159=19% 8 Or thus: 5+6+7= 18; the sum of the integral parts: then, 4+6+5 1 +&+=+&+= = =13; the sum 8 of the fractional parts; and lastly, 18 +13=19, as before. The student may pursue the method which seems most convenient, according to circumstances. The one also proves the other. 2. 24 + 11 +34 + 16 = 8. Subtraction of Fractions. 204. When the given fractions have a com. denom., we subtract the numerator of the less from that of the greater, and place the remainder over the com. denom.; thus, 4 — =*. The numerators being each composed of units of the order signified by the com. denom., their difference is evidently a number of units of the same order; for which reason we give it the same denominator. When the fractions have different denominators, we reduce them to a com. denom., as in addition ; because we cannot subtract units of different orders; after which we proceed as before. and 23 17 399 1 36 3 99-136 5 6 263 Subtraction of llicel Numbers. 205. We may reduce the mixed numbers to improper fractions; and, when the denominators are not alike, bring them to a common denominator, and subtract as above. For example : if, from 75, we would subtract 23, we say 73 = 5, ; therefore, 73-2=57 436 But, when the integral part of one or both numbers is very great, the following method will be found more convenient: If the fractional parts have different denominators, we first reduce them to a com. denom. We then write the less mixed number under the greater, and, having drawn a line underneath, we subtract the numérator of the lower fraction from that of the upper one, and place the remainder over the com. denom. for the difference of the fractions. Lastly, we find the difference of the whole numbers, as usual. When the numerator of the lower fraction is the greater of the two, we subtract it from the sum of the terms of the upper fraction; and, having placed the remainder over the com. denom., we carry one to the unit figure of the lower number, and proceed as usual. As the denominator is a unit, and the numerator the fraction, when we add the denominator to the numerator, we add a unit to the fraction ; therefore, having added a unit to the upper number, we add a unit to the lower, by which means both numbers are increased alike, which cannot affect the difference. Examples. 1. To subtract 27 from 75, we first find that and are equivalent to be and 3d; we then place the numbers thus : 7786 and, as 24 is greater than 7, we say 56 - 24=32; and 32 + 7=39, which we place over 56. Then, having added a unit to the upper number, we add a unit to the lower number 2, saying, 3 from 7, four. Wherefore, 7-2=438. b -6 2. 10113-9411 = 1528 d d d d = = 715 a a с 12 33 15 10193 195 Or thus : 1 and 1 are equivalent to 168 and 195; then, from and have 7165 for remainder, as before. Here, as 13 is greater than 13, we do not add a unit to either number, as in the preceding example. 3. To subtract 941} from 101, we place the numbers thus : 101 6 iz and, as there is nothing above 1], 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 11 from its own denominator, and, placing the difference 2 over 13, we have is for the difference between it and a unit; we then carry a unit to 4, and, having subtracted, we find that 6 is the difference between the given numbers. The difference between a fraction and a unit is called the complement of that fraction : thus, iš and are the complements of each other. 4. 153-33=113; and 154-113= 33. 9. 1043 10 — 6569 =978. Prove this and the succeeding example by addition and subtraction. 10. 31711=98031 = 219031 Multiplication of Fractions. 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 t. 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 21 = tor 4 by 3, which gives for the quotient; therefore, 6 1 4 X 3 12 Or, to multiply hy 2; we divide (165) its denominator by 2, and have 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. Operation. 8 R 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 denominators. 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 1, i, f, and together, we proceed thus : 1 X 8 XA X $ XXAX $ x 6 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 3 X4 X5, 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 : 3 69 3584 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. 2. XX34X36=5= : The prime factors of 9X26X36 and of 13 X 24X27 are the same, namely, 34, 23, and 13: therefore, omitting these numbers, we have =l for the total product. 3. X X=31, and *xx16=8. 7. XXX%X1X41 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: 2136X1=2136 X 1=6408 = 1602. 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 : 2136 X=534X3=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 5, 4, , $, &c. of a number, is to diminish it by d, i, j, ġ, &c. of itself. When the whole number is the same as the denominator of the fraction, the result is the numerator. Thus, 4X4=3; 11X12=11, &c. |