met in 4 days, A traveling at the average rate of 3 miles per hour and 63 hours per day, and B at the average speed of 24 miles per hour. How many hours per day did B travel? 127. Two cisterns at first contained equal quantities of water; but after 75 gallons were taken from one and 58 gallons added to the other, both together contained 518 gallons. How many gallons did each contain at first? 128. If a barrel contains 35 gallons of cider, and of its contents are 용 taken at one drawing, 5 gallons at a second drawing, and 3 of the remainder at a third drawing, what part of its original contents will remain ? 129. A dealer sold 468 barrels of a consignment of potatoes and had 312 barrels remaining. What part of the consignment did he sell? 130. The average distance traveled by each of five men in 8 days was 190 miles. If 4 of the men averaged respectively 251, 243, 185, and 16 miles per day, what was the average distance per day traveled by the fifth man? 131. If 8 men can do as much work as 12 boys, and 13 men can complete a task in 9 days, how long will it take 3 men and 2 boys to complete it? 132. At 75 cents per bushel, a man can buy 18 more bushels of oats with of his money than he can with of it. How much money has he? 133. 58 bushels are 9 bushels less than the quantity of wheat in two bins; and the greater bin contains 13 bushels more than the less. How many bushels in each bin? 134. A grocer sold a quantity of sugar, and afterwards bought 625 pounds which is 34 of what he then had, and of what he had at first. How many pounds did he sell? 135. A dealer bought 575 barrels of flour and sold out so as to gain of the sum paid therefor. What did he obtain for all the flour if his average gain was 50¢ per barrel? 136. A and B have 3 having as much as B. bushels of corn, and A lacks of a bushel of How much corn has each? 137. If goods 20 inches wide are worth 40 cents per yard, what is the equivalent value of the same grade of goods and 28 inches wide? 138. If a man sells of of his farm, how many thirds of that will he have remaining? How many thirds of of his farm will he have remaining? INDUCTIVE EXERCISE. 1. If an orange be divided into 10 equal parts, what is each part called? What are 3 parts called? 7 parts? 9 parts? What is the fractional unit of each of the preceding fractions? 2. If one-tenth of an orange be divided into 10 equal parts, what is each part called? What are 5 parts called? 7 parts? 11 parts? 19 parts? What is the fractional unit of each of the preceding fractions? 3. If a whole orange be divided into 100 equal parts, and one-tenth of an orange be divided into 10 equal parts, will there be any difference in the value of the equal parts of the whole orange and of the equal parts of the tenth of an orange? Why not? How many hundredths of an orange are therefore equal to one-tenth of an orange? To 2 tenths of an orange? To 3 tenths of an orange? To 7 tenths of an orange? To 9 tenths of an orange? 4. If a whole orange be divided into 1000 equal parts, or one-hundredth of an orange into 10 equal parts, what in either case will each part be called? 3 parts? 13 parts? What is the fractional unit of each of the preceding fractions? Would the fractional unit be the same if one-tenth of an orange be divided into 100 equal parts; and why? Numeration and notation of decimals. [a] The numerator of a decimal is the number of fractional units [number to the right of the decimal point] which it expresses. [b]The denominator of a decimal is 1 followed by as many ciphers as it has places to the right of the decimal point. [c] The numerator of a decimal is expressed as any whole number; and its denominator, by pointing off from the right of the numerator as many figures as the denominator has 0's. Express, orally, the value of the following decimals: Express by figures, decimally, the following quantities: 21. Five, and three hundred and sixteen thousandths dollars. 22. Seventy-four, and four thousand and nine ten-thousandths pounds. 23. Eight hundred and sixty-five, and three hundred and nine hundred-thousandths gallons. 24. Nine thousand eight hundred and seventy-six, and two thousand five hundred and nine millionths miles. 25. Seventy-five thousand three hundred and forty ten-millionths of a circle. 26. Ninety-one, and one million five hundred and three thousand seven hundred and ninety-eight hundred-millionths miles. 27. Five thousand and thirty-eight trillionths of a distance. 28. Four hundred and seventy-nine, and three thousand four hundred and forty-nine and three-fourths millionths of a year. 29. Thirty-seven, and nine and five-sixths thousandths of a rod. 30. Two-thirds of a ten-thousandth of a century. 1. What is the value of every fraction? In $, what is the dividend? The divisor? If the numerator of $ be multiplied by 10, what will be the product? What is the value of $10, expressed as an integer? How much greater is this value than the value of $? If the value of $, or $5, be divided by 10, will the quotient express the value of $; and why? How may $5 be divided by 10? What kind of fraction is $.5? Hence, how can $ be reduced to a decimal ? 2. Annexing 1 cipher to the numerator of a fraction is equivalent to multiplying the fraction by what number? Annexing 2 ciphers? Annexing 3 ciphers? Annexing 4 ciphers? If 2 ciphers are annexed to the numerator of $, what is the value of the result, expressed as an integer? How much greater is this value than thẹ value of $? If the value of $200, or $75, be divided by 100, will the quotient express the value of $3; and why? How may $75 be divided by 100? What kind of fraction is $.75? Hence, how can $ be reduced to a decimal ? 3. If 3 ciphers be annexed to the numerator of a fraction and the result divided by the denominator, how can the quotient be corrected to express the quotient or value of the original fraction? If 4 ciphers be annexed to the numerator? Reduction of decimals to common fractions. INDUCTIVE EXERCISE. 1. What is the numerator of .4 of a pound? The denominator? If the denominator of .4 of a pound be written as in common fractions, what will be the result when reduced to lowest terms? Similarly reduce to common fractions in lowest terms, .8 of a gallon, .15 of a mile, .25 of a dollar, .35 of a bushel, .75 of a hogshead, .005 of a ton, .025 of a dollar. Hence, how may simple decimals be reduced to equivalent common fractions? 2. What is the numerator of .21 of a mile? The denominator? If the denominator be written, the result will have the form of what kind of common fraction? How may this result be reduced to a simple common fraction? Hence, to what simple common fraction is .2} of a mile equivalent? Similarly reduce to simple common fractions in lowest terms, .21 of a dollar, .31⁄2 of a mile, .07} of a day, .08 of a year. EXAMPLES. Reduce the following decimals to common fractions: 84. .75 of a mile. | 90. 519.0875 tons. 96. 1.0831 years. 97. 75.98 miles. 98. 685.0023 tons. 99. 315.001 feet. 100. .000% of a barrel. 101. .001 of a bushel. Reduction of decimals to higher terms. INDUCTIVE EXERCISE. 1. How many tenths of a dollar in $1? How many hundredths? How many thousandths? How many hundredths of a dollar in 1 tenth of a dollar? In 2 tenths? In 3 tenths? In 5 tenths? How many thousandths of a dollar in 5 dollars? In 6 tenths of a dollar? In 7 hundredths of a dollar? In 18 hundredths of a dollar? 2. Does annexing ciphers to a decimal change its value; and why not? How many thousandths of a pound in .9 of a pound? In .35 of a pound? Hence, how may .9 of a pound and .35 of a pound be reduced to the common denominator thousandths of a pound? What, therefore, should you infer to be necessary in reducing two or more simple decimals to a common denominator? 3. How many cents in $1? What part of a dollar, therefore, is 1 cent? Are 5 cents? 8 cents? 15 cents? 75 cents? How many dimes in $1? What part of a dollar, therefore, is 1 dime? Are 3 dimes? Are 7 dimes? Are 9 dimes? 4. What decimal of a dollar are 5 dimes and 3 cents? How expressed, and why? What decimal of a dollar are 9 dimes and 1 cent? Are 6 dimes and 8 cents? How should 8 dollars, 2 dimes, and 5 cents be decimally expressed? Ans. Symbol, eight, point, two, five. Similarly express 15 dollars, 6 dimes, and 2 cents; 7 dollars, 1 dime, and 8 cents; 19 dollars, and 5 cents. |