REDUCTION OF DECIMALS. Reduction of decimals consists in changing their form without altering their value. CASE I.-TO REDUCE A DECIMAL TO A COMMON FRACTION. RULE.-Write the decimal as a decimal fraction and reduce this fraction to its lowest terms. PROBLEMS. Reduce .25 to a common fraction. Solution: .25==, Ans. PROBLEMS. Reduce to common fractions 1. .25625. Ans. . 2. .003125. Ans. 10. 3. 2.125. Ans. 21. 4. 19.01750. Ans. 1947o. 5. 3.33. Ans. 33. 6. 11.05. Ans. 11. CASE II.-TO REDUCE COMMON FRACTIONS TO DECIMALS. RULE.-Annex ciphers to the numerator and divide it by the denominator. Then point off from the right of the quotient as many decimal places as there are ciphers annexed. Any fraction in its lowest terms having in its denominator any factor other than 2 or 5 can not be reduced to a pure decimal. Thus: .0833, or .0833+; and = .1666, or = .1667-. The sign + is used at the end of a decimal to indicate that the last figure is too small; the sign -, to indicate that it is too great. By the rule, a mixed number may be converted into a mixed, or a complex, decimal; and a complex decimal having no other factor than 2 or 5 in the denominator of the common fraction into a pure decimal. Thus: 99.25, since =.25; 33.33, since =.33, and .26.2604, since =.04. RULE.-Write the numbers to be added so that the decimal points shall be in the same column; then add as in whole numbers and place the decimal point in the sum directly under the decimal points above. Complex decimals, if there be any, must be made pure as far as the decimal places extend in the other numbers. 1. PROBLEMS. 14.034+25+.0000625+.0034-what? Ans. 39.0374625. 2. 216.86301+48.1057+.029+1.3=what? Ans. 266.29771. 3. 163+.37+3.43+.0003-what? Ans. 3.9804. 4. .11.66663+.222222 what? Ans. 1. = 5. 35.+3.5+.35+.035-what? Ans. 38.885. 6. .144+.0188+920+.01394 what? Ans. 920.1754. = SUBTRACTION OF DECIMALS. RULE. Write the less number under the greater so that the decimal points shall be in the same column; then subtract as in whole numbers and place the decimal point in the remainder directly under those above. If either or both of the decimals be complex, extend them to the same decimal place before subtracting. If the greater number has not as many decimal places as the smaller, annex O's until it has the same. 1. = PROBLEMS. = 19.54-8.00717 what? Ans. 11.53283. 5. = 4.9-.01=what? Ans. 4.9225. 6. .01.00§=what? Ans. 0. MULTIPLICATION OF DECIMALS. RULE.-Multiply as in whole numbers and point off in the product, from the right hand, as many decimal places as there are in both numbers multiplied together; if there be not so many in the product, supply the deficiency by prefixing 0's. DIVISION OF DECIMALS. RULE.-Divide as in whole numbers and point off in the quotient, from the right hand, as many decimal places as those in the dividend exceed those in the divisor; if there be not so many in the quotient, supply the deficiency by prefixing O's. When there are more decimal places in the divisor than in the dividend, annex O's to the latter until the number in both is the same. The quotient will then be a whole number. When it is necessary to continue the division farther than the figures of the dividend will allow, annex O's and consider them as decimal places of the dividend. CHAPTER IV TABLES OF MEASURE In pure mathematics, as before stated, there are but eight different kinds of quantity. For measuring each kind of quantity, there are, however, one or more systems of measurement, and each system has, for convenience, a number of subdivisions. Quantities expressed in terms of the same subdivision are said to be of the same denomination. A table of measure is a series of numbers showing the relation between the different subdivisions of a system of measurement for a particular kind of quantity. A measure is a unit for measuring quantities of the same denomination. A standard unit is a measure made a standard, by law or custom, for comparison of all measures of the same system. The following are some of the tables of measure in ordinary use. MEASURES OF LENGTH. LONG MEASURE-ENGLISH SYSTEM (established in United States by act of Congress in 1834). TABLE. 12 inches, marked in., make 1 foot, marked ft. 3 feet, 5 yards, 320 rods, make 1 yard, marked yd. make 1 rod, marked rd. make 1 mile, marked mi. |