Write the following numbers in figures: 2. Seven hundred, and five tenths. 3. Seven, and one ten thousandth. 4. Seven hundred and five, and five millionths. 5. Two hundred and two, and two hundred thousandths. 6. Twenty-five million, and twenty-five ten millionths. 7. Three hundred and forty thousand, and three tenths. 8. Sixty-five thousand, and sixty-five ten thousandths. 9. Forty-four million, and forty-four ten millionths, 10. Five hundred thousand, and five hundred thousandths. 11. Sixty-four million, and sixty-four millionths. REDUCTION OF DECIMALS. 113. To reduce a common fraction to a decimal, RULE. Annex ciphers to the numerator, and divide by the denominator. OBS. The number of decimal places in the quotient will be equal to the number of ciphers annexed. When there are not as many, make the number equal by prefixing ciphers. Recite the rule for reducing a common fraction to a decimal. 12. Reduce & to a decimal. 8) 3.000 .375 As three ciphers were annexed to the numerator, three figures must be pointed off for decimals in the quotient. This rule depends upon the principle stated in ART. 39, that if a number be multiplied and divided by the same number, its value remains unchanged. 13. Reduce the following fractions to decimals: 114. Any common fraction can be expressed exact.y in decimals whose denominator has no other factor but 2 and 5. If the denominator contains other factors, it cannot be expressed exactly in decimals. 115. To reduce a decimal fraction to a common fraction, RULE. Write the denominator of the decimal under the numerator, and then reduce it to its lowest terms. 14. Reduce .625 to a common fraction. 흡 652 = 5/3 1000 The denominator of .625 is 1000. (ART. 102.) 166350 What is the rule for reducing a decimal to a common fraction ? reduced to its lowest terms, (ART. 85,) gives for a common fraction. EXAMPLES. 15. Reduce .125 to a common fraction. ADDITION OF DECIMALS. 116. RULE. Write the numbers of the same order under each other, so that the decimal points shall be in the same vertical column. Add as in whole numbers, and point off in the sum from the right as many places for decimals as are equal to the greatest number of decimal places in any of the given numbers. PROOF. Addition of decimals may be proved in the same manner as addition of whole numbers. The principle of this rule is the same as that in the addition of whole numbers, (ARTS. 9, 17.) 24. Add together 3.014, 3014, 30.14, .003014, and .04045. 3.014 3014. 30.14 .003014 .04045 3047.197464 Write the numbers of the same order under each What is the rule for the addition of decimals? other, units under units, tens under tens, and tenths under tenths, hundredths under hundredths, &c., and add as in whole numbers, and point off from the right for decimals as many figures as are equal to the greatest number of decimal places in any of the given numbers. The greatest number of decimals in the above example is six; six figures must therefore be pointed off for decimals. OBS. 1. The decimal point will always be exactly under the decimal points in the given numbers. OBS. 2. The learner must be very careful in placing the decimal points according to the rule. EXAMPLES. 25. Add 4.032, 64.5010, 96.081, 310.018, 1.0012. 26. Add 63.03036, 73.46030, .090345, 41.23101, 1.0109. 27. Add 340.14561, 84.960, .759112, .000012, 2.0345. 28. Add 9.03456, 1.23456, 12.34567, 123.4567, 1234.567. 29. Add 85.05376, 5.45405, 54.04345, 540.4345, 7.00034. 30. 46.13455+9.73456+.0009345+875.+34.5=? 31. 9671.03+5.05674+8.7561+750.12+87.34=? 32. 1.45610+67563.1+2.31267+91.234+.679=? 33. Add together 1 tenth, 1 hundredth, 1 thousandth, and 1 ten thousandth. 34. What is the sum of 5 hundredths, 5 ten thousandths, and 5 millionths? 35. What is the sum of 95 millionths, 1 hundred thousandth, and 1 tenth? 36. What is the sum of 784 thousandths, 347 millionths, 75 ten thousandths, and 99 hundredths? 37. What is the sum of 465, 78 hundred thousandths, 9 millionths, 99 ten thousandths, and 9 tenths? 38. What is the sum of 475, 9 ten thousandths, 83 hundred thousandths, and 9 tenths? MULTIPLICATION OF DECIMALS. 117. RULE. Multiply as in whole numbers, and point off from the right of the product as many figures as there are decimal places in both factors. PROOF. Multiplication of decimals may be proved in the same manner as multiplication of whole numbers, or by changing them to the form of common fractions. OBS. If there be not as many decimal places in the product as in both factors, make the number equal by prefixing ciphers. As there are eight decimal places in both factors, and only four in the product, four ciphers must therefore be prefixed to the product. The reason of this rule will be evident by changing the decimal multiplicand and multiplier to the form of a common fraction, and then multiplying. 118. When the multiplier is a whole number, the multiplicand is taken as many times as there are units in the multiplier; but when the multiplier is a fraction, only such parts of the multiplicand are taken as are indicated by the multiplier. When the multiplier is less than a unit, the product will be less than the multiplicand. What is the rule for the multiplication of decimals? When the multiplier is less than a unit, what is the product? |