EXAMPLES. 1. Suppose a king's crown to weigh 4 lb. 9 oz. 5 pwt. 6 gr. of gold; how many grains are there in the crown? (Multiply by 12 oz. 20 pwt. and 24 gr., 2. A farmer sold a yoke of oxen, which weighed 22 cwt. 3 qrs. and 8 lbs., for 7 cts. a pound; how much did the oxen weigh, in pounds, and how much did he receive, in mills? 3. In 15 lb. 8 oz. 1 dr. 2 sc. 5 gr. of medicines, how many grains? 4. In 136 bushels, how many pecks, quarts, pints, and gills? 5. Bought of Cyprian Nichols 2 hhd. 3 gal. 2 qt. 1 pt. 3 gi. of Molasses. How much did it cost, at two cents a gill? 6. If, on June 10th, 1844, Charles Waterhouse was 26 years, 5 mo. 27 days old, what is his age at present, or then, in seconds? REDUCTION ASCENDING: Because the numbers are constantly decreasing by division, making the quotient of a higher denomination. RULE.-Divide the given denomination by the number of itself that it takes to make one of the next higher order, and the quotient thus again, if necessary, until the quotient is the denomination required; that is, ounces must be divided, first by 16, to make the quotient pounds, for 16 oz. make a pound, &c. Place the remainders, if any, at the right of the quotient, according to their several orders. Questions. Why is it called Reduction Ascending?-Do you divide or multiply?-Divide what?-By what?-Why?-Explain it. Lastly, what? EXAMPLES. 1. If a king's crown should weigh 27486 grains, how many lbs. would it weigh? (Divide by 24 gr. 20 pwt. and 12 oz.) 2. If a yoke of oxen weigh 38896 oz., it is how many pounds, quarters, and cwt.? -3. In 90345 grains, how many of the fourth higher order? 4. In 34816 gills, how many pints, quarts, pecks, and bushels? 5. In 4151 gills, how many hogsheads? 6. In 83602819,293 seconds, how many years? 7. In a pile of wood 96 feet long, 5 feet high, and 4 feet wide, how many cords? NOTE-By Table of Solid Measure, the propriety of multiplying the length, depth, and width together, will readily be seen. Lastly, divide their continued product by 128. (See Table.) 8. In 82 tons of round timber, how many inches? 9. What is the content of a pile of wood 703 feet long, 6 feet high, and 74 feet wide? NOTE.-Change the fractions to those having a common denominator and corresponding numerators. Or, change the mixed numbers to equivalent improper fractions, and to a common denominator, having corresponding numerators; then proceed by the foregoing note, and then find the inches. 1 10. What is the content of a load of wood 73 feet long, 311 feet high, and 511 feet wide, in inches? 11. How many shingles will it take to cover the roof of a house 40 feet in length, and of 18 feet rafters, allowing each shingle to be 4 inches wide, and each course to be laid out 6 inches? 12. How many boards will cover a barn that is 50 feet long, and 30 feet wide; the height of the gable ends to be 13 feet, and the rafters 20 feet each; and the posts of the frame, 15 feet in height? (4790 ft.) 13. How many bricks, 8 inches long, 4 wide, 2 thick, will build a house 44 feet long, 40 feet wide, and 20 feet high, with walls 12 inches thick? (88560.) 14. The Sun's mean distance from the earth is 95000000 of miles; a cannon ball at the first discharge, flies about a mile in 8 seconds. How long would a cannon ball be, at that rate, in flying from the earth to the Sun? 15. The forward wheels of a waggon are 144 feet in circumference, and the hind wheels, 15 feet 9 inches; how many more times will the forward wheels turn round than the hind ones, in running from Hallowell to Boston, it being 174 miles? (50284.) 16. The planet Venus, at its greatest elongation, appears about 166800 seconds from the Sun. Required the number of degrees. (46° 20′.) DECIMAL FRACTIONS. Enumeration shows that these are parts of a whole number, having the same ratio; therefore, we need only to notice, for adding or subtracting such, that there must be as many decimals * Circumference, (L. circumferentia,) the line that bounds a circle. in the result, of either case, as there are decimals in the number, having the most places of decimals. Write down the given questions as in simple numbers, of either case. Explanation-Those places at the left of the point are deci mals. MULTIPLICATION OF DECIMALS. RULE.-Proceed as in simple numbers-pointing off for decimals, from the result as there are decimals in the factors. If at any time there are not so many of these places in the product, supply the defect by prefixing ciphers at the left of the product. Demonstration.-It is obvious from what has been given on common fractions, that multiplying whole numbers by any fraction, is taking a certain part of the multiplicand for the product, consequently, multiplying one fraction by another, must produce a frac tion smaller than either of the factors. And it is evident, by Enumeration, that .25 is 35 or 4, and .5 is or; and it is obvious, that, of of, is of 125 X,=.125, because Decimals read the same as whole numbers, being the same as equivalent common fractions; therefore, .25X.5.125. Hence the RULE. 25 Questions. What are Decimal Fractions?-What should be noticed in their addition or subtraction?-What do you call the places at the left of the point?-How do you point off, in Multiplication of Decimals?-You sometimes supply what defect?-How?-Why?-(See Dem.) Does it affect decimal places to have ciphers placed at their left?-How? Explanation. Hence by the operations, we not only see the propriety of the rule, but infer that if a cipher is placed at the right of Decimals it does not alter their value, but does very much, if placed at their left. DIVISION OF Decimals. RULE.-Proceed as in simple numbers-pointing off as many figures from the right hand of the quotient, as the decimal places in the dividend exceed those in the divisor. If there be not so many figures in the dividend as in the divisor, annex ciphers to the dividend, until the places, in each, are alike, then calling the quotient whole numbers, and annexing ciphers after, if there be a remainder. If the number of places in the quotient be not sufficient to make out the requisite number of decimals, place as many ciphers at the left hand of the quotient, as will make the number of places required. Demonstration.-The multiplicand and multiplier, in proving Multiplication, become the divisor and quotient in Division. Therefore, the number of decimals in the quotient, must be equal to the difference between the number of those in the dividend, and the like places in the divisor; for, it is obvious, that dividing one fraction by another, is increasing the dividend. Hence the RULE. Questions-How do you point off for decimals, after their division? Why? (See Dem.) How do you point off if the places in the quotient are not sufficient, &c.?-Why?-(See Dem.) When do you annex ciphers to the dividend?-What then do you call the quotient?-If there be a remainder, how? What do you understand by the demonstration? 1. 12.1)34.21 (2.8 242 1001 968 Rem. 33 EXAMPLES. 2. .101) 1302.000 (12891.089 EXPLANATIONS. 1. There being one more decimal place in the dividend than in the divisor, one figure is pointed off from the right of the quotient for decimals. Ciphers might have been annexed to the remainder, and the quotient carried to a greater degree of exactness. 2. Here the dividend is whole numbers, and the divisor has three figures in decimals; therefore three ciphers are annexed to the right of the dividend before dividing. 3. By reason of the rule, two ciphers are prefixed at the left of the quotient, and all pointed off, for ciphers, are to be prefixed, when? 4. By the rule, the quotient is pointed off; because the decimal places in the dividend exceed those in the divisor, by three places. REDUCTION OF DECIMALS. 1. To reduce a vulgar fraction to its equivalent decimal. RULE-Annex ciphers to the numerator, which divide by the denominator; and the quotient is the decimal sought. Demonstration.-Since every decimal fraction has 1, with as many ciphers annexed as there are places in the decimal, for a denominator, it is obvious, proceeding thus takes but a part of 1. EXAMPLES. 1. Reduce to a decimal of the same value. Explanation-The first cipher at the right of the figure in the dividend, was annexed, that being needed first; the others being annexed to the remainders, as needed, and then placed in the dividend. NOTE. By reducing the quotient, with 1 and Os annexed for a denominator, to its lowest terms, we find the numbers used in the operation. 2. Reduce,, 4, and §, severally, to decimals. 3. Reduce,,, and, severally, to decimals. 2. To reduce numbers of different denominations to their equivalent decimal values. RULE. Write down the several denominations under each other, the least name first, and ending with the highest. At the left of this column, draw a line from top to bottom. At the left of the line, and opposite each denomination, place such a number for a divisor, as it takes of that denomination to make one of the next higher. Annex ciphers to the top row, and divide by the number standing against it, writing the quotient, decimally, at the right of the next higher denomination, which is written below, · dividing it and the quotient annexed, by the number which is opposite, annexing the quotient to the right of the next number below. Proceed thus, and the last quotient is the decimal sought. Demonstration. The preceding is only expanding or contracting the ratios in question, as the denominations are larger or less |