14. Compound numbers are such as consist of several denominations, as pounds, shillings, and pence; hundred weights, quarters, and pounds, &c. Note. Decimals are only vulgar fractions of a particular description, and were introduced in order to lessen the trouble which, in many cases, attends the use of the latter. They are much used in the practical affairs of life, particularly in Mensuration and Gauging. ADDITION OF DECIMALS. RULE. Place the figures in such a manner that those of the same denomination may stand under each other; add them together as in whole numbers, and point off as many figures, for decimals, as are equal to the greatest number of decimal places in any of the given numbers. Note. It has already been observed, that decimals and whole numbers are both subject to the same rules; but in all calculations by the former, great care must be taken to point off the proper number of decimal places in the results, or the truth of the operations will be completely destroyed. EXAMPLES. 1. What is the sum of 54.646 +3.95+46.3905+968.202 +7.0264+.2064+2.0463+5.646? 54.646 3.95 46.3905 968.202 7.0264 .2064 2.0463 5.646 1088.1136 Ans. 2. Find the sum of 367.60+4678.3609+869.563+ .2003+7.5964 + 42.67. 3. Add 53.7+2943+1.2+2.0073+1.47+637. Ans. 5965.9906. Ans. 3638.3773. Ans. 286.02699. 4. Required the sum of 124.1+.3492+84.02+6.349+ .00879+71.2. SUBTRACTION. RULE. Place the figures of the same denomination under each other; then, beginning at the right-hand subtract as in whole numbers, and point off the decimals as in addition. EXAMPLES. 1. What is the difference between 52.73 and 2.676? 52.73 2. From 2.18 take .814. 3. From .794 take .0981. Ans. 1.366 Ans. .00149. 4. What is the difference between .0943 and .09281? 5. Required the difference between 374.901 and 68.14. Ans. 306.761. MULTIPLICATION. RULE Place the figures under each other, and multiply them together, as in whole numbers; and point off as many decimal places in the product as there are in the multiplier and the multiplicand together. Note 1. When there are not so many figures in the product as there are decimals in the multiplier and multiplicand together, cyphers must be annexed to the left of the product, that the decimal places may be properly represented. 2. When a decimal is to be multiplied by 10, 100, 1000, &c., it is only necessary to remove the decimal point so many places to the right, as there are cyphers in the multiplier; thus, 4.27 × 1 = 042.7; and 379 x 100 = 37.9. 3. If a decimal be multiplied by a decimal, the product will be less than either the multiplier or multiplicand; and if a whole or a mixed number be multiplied by a decimal, the product will always be less than the multiplicand. EXAMPLES. 1. What is the product of 24.73 multiplied by 7.325? Divide as in whole numbers, and point off from the right-hand of the quotient, as many figures for decimals, as the decimal places in the dividend exceed those in the divisor. Note 1. When the figures in the quotient are too few to make up the proper number of decimals required by the rule, the defect must be supplied by prefixing cyphers to the left of the quotient. 2. When the dividend does not contain as many decimals as the divisor, cyphers must be placed on the right of the dividend, until they are made equal, previously to beginning the operation; and the quotient, to that extent, will be a whole number. 3. If there be a remainder after division, you may continue the quotient to any extent that may be thought necessary, by subjoining a cypher continually to the last remainder. 4. When it is required to divide a decimal by 10, 100, 1000, &c., remove the decimal point so many places to the left as there are cyphers in the divisor; thus, 36.410 3.64; and .5864÷ 100 = .005864. = 5. If a decimal be divided by a decimal, the quotient will be greater than either the divisor or dividend; and if a whole, or a mixed number, be divided by a decimal, the quotient will be greater than the dividend; but if a decimal be divided by a whole, or a mixed number, the quotient will be less than the dividend. EXAMPLES. 1. Divide .2843701 by .147. .147).2843701(1.9344 Ans. 147 1373 1323 507 441 660 588 721 588 133 remainder. 2. What is the quotient of 741 divided by .325? 3. Divide 839 by 5.2. Ans. 2280. Ans. 161.3461. 4. What is the quotient of .074 divided by 36? REDUCTION. CASE I. To reduce a vulgar fraction to a decimal of the same value. RULE. Place cyphers, at pleasure, on the right of the numerator, as decimals; then divide by the denominator, and the quotient will be the decimal required. Note 1. A compound fraction may be reduced to a simple one, by multiplying all the numerators together for a new numerator, and all the denominators together for a new denominator; thus, of == 2. An improper fraction may be reduced to a mixed number, by dividing the numerator by the denominator; thus, * = 29} = 27}.' 3. A mixed number may be reduced to an improper fraction, by multiplying the whole number by the denominator of the fraction, adding the numerator to the product; and placing the sum over the 28×5+4 144 denominator; thus, 281= 5 = 5 4. A whole number may be expressed like a fraction, by putting 1 for its denominator. EXAMPLES. 1. Reduce to a decimal. 4)1.00 2. Reduce to a decimal. 3. Reduce to a decimal. 4. What is the decimal of ? 5. Reduce to an equivalent decimal. 6. What is the decimal of ? 7. Let 35 be reduced to a decimal. 죽음 48 8. Express by decimals. 9. What is the decimal of 382 4693 10. Let 586 be expressed in decimals. 83367 11. Reduce of of to a decimal. 12. Reduce of of to a decimal. Ans. .9533. |