Here the lowest denomination in the given quantity is rods, and therefore I reduce the quantity to rods, for a numerator. Then, because the quantity is to be reduced to the fraction of a mile, 1 mile is the integer; which I reduce to rods, for a denominator. Then I place the numerator over the denominator, and the fraction is ; which I reduce to its lowest terms, by Prob. I., and have 13 for the answer. 320 2. Reduce 13 s. 6 d. 2q. to the fraction of a pound. Ans. 658-657. 96096 3. Reduce 4d. 2q. to the fraction of a shilling. Ans. s. 4. Reduce 8 oz. 12 dr. to the fraction of a pound, Avoirdupois. Ans. 35 lb. 5. Reduce 2 ft. 9 in. 1 b. c. to the fraction of a yard. 64 Ans. yd. 6. What part of a yard of cloth is 2 qr. 2 na.? Ans. yd. 7. What part of an acre is 3 roods, 8 sq. rods? Ans. . 8. What part of a cord of wood is 85 cub. ft. 576 cub.in.? Ans.. 9. What part of a week is 1 day, 18 hours? 10. What part of a degree is 20 min. 40 sec.? Ans. 31. QUESTIONS ON THE FOREGOING. 1. What is a fraction? 2. How are fractions produced? 3. How many kinds of fractions are there? 4. How is a vulgar fraction represented? 5. What are the upper and lower parts of any vulgar fraction called? 6. What does the denominator denote? 7. What does the numerator denote? 8. What is a mixed number? 9. How do we reduce a fraction to its lowest terms? 10. How do we find the value of a fraction of a higher denomination in whole num bers in a lower? 11. How do we reduce any quantity to the fraction of a higher denomination? DECIMAL FRACTIONS. A Decimal Fraction, is a fraction whose denominator is a unit (1), with a cipher or ciphers annexed; as 10, 150, 65 &c. 10009 45 100 As the denominator of a decimal fraction is always 10, or 100, or 1000, &c. it need not be expressed; for the numerator only may be made to express the value of the fraction: For this purpose it is only required to write the numerator with a point before it at the left hand, to distinguish it from a whole number, when it consists of so many figures as the denominator has ciphers: So, is written thus .5, and thus .45. But if the numerator has not as many places as the denominator has ciphers, then ciphers must be prefixed to make up that number of places : So,, must be written thus .05, and 1, thus .007, &c. Thus do these fractions receive the form of whole numbers. Any decimal may be expressed in the form of a common fraction by writing under it its proper denominator, (viz. a unit with as many ciphers annexed as there are figures in the given decimal,) rejecting from the numerator the decimal point, and also the ciphers, if any, to the left hand of the significant figures. Thus, .75, expressed in the form of a common fraction, is 75%; and .027, is 1800. 100 When a whole number and decimal parts are expressed together, in the same number, it is called a mixed number. Thus, 25.48 is a mixed uumber, 25., or all the figures on the left hand of the decimal point, being whole numbers, and .48, or all the figures on the right hand of the decimal point, being decimals. The point prefixed to decimals is called the separatrix, or the decimal point. This point must never be omitted; because, without it, decimals and mixed numbers cannot be distinguished from whole numbers. Decimals are numerated from left to right, (which is contrary to the way of numerating whole numbers,) and each figure takes its value by its distance from the unit's place: If it be in the first place after units, (viz. in the first place to the right hand of the decimal point,) it signifies tenths; if in the second place, hundredths; and so on. For in decimals, as well as in whole numbers, the values of the places increase towards the left hand, and decrease towards the right, both in the same tenfold proportion; as in the following .0 5 .0 25 4.0 0 0 9 8 4 8.1 2 6 8 49 5 9 1.0 0 00008 Integers. Decimal parts. read 5 Tenths. Ciphers placed at the right hand of a decimal de not alter its value, since every significant figure continues to possess the same place: So .5, .50, and .500, are all of the same value, each being equal to f, or 2. Therefore, when there are ciphers at the right hand of any decimal fraction, they may be omitted. 5 09 But ciphers placed at the left hand of decimals, decrease their value in a tenfold proportion, by removing them farther from the decimal point: Thus .5, .05, .005, &c. are 10, 100, 1000, &c. respectively. It is therefore evident that the value of a decimal fraction, compared with another, does not depend upon the number of its figures, but chiefly upon the value of its first left hand figure: for instance, a fraction beginning with any figure less than .9, such as .89978, &c, if extended to an infinite number of figures, will not equal .9 or 1. Decimals are read in the same manner as whole numbers, giving the name of the lowest denomination, or right hand figure, to the whole. Thus, .7854, (the lowest denomination, or right hand figure, being ten-thousandths,) is read, 7854 ten-thousandths. Place the numbers, (whether pure decimals or mixed numbers,) according to the values of their places, so that the decimal points shall stand exactly under one another, and then proceed as in addition of whole numbers, only taking care to put the decimal point in the sum exactly under those in the numbers added. Note. The methods of proving Addition, Subtraction, &c., of Decimals, are the same as in whole numbers. * As decimals increase from right to left, and decrease from left to right, in the same tenfold ratio as whole numbers, it is evident that Addition and Subtraction of Decimals may be performed in the same manner as in whole numbers, if care be taken to write down the decimals in such a manner that tenths shall stand under tenths, hundredths under bundredths, &c. Thus, the sum of 5 and 4 is evidently 9; and the difference between 8 and 3 is .5. 6. What is the sum of 429+21.37+355.1+1.07+1.7? Ans. 808.24 7. What is the sum of 972+20+1.75+.7164+65.4? Ans. 1059.8664 8. To .9 add one-tenth part of a unit.. SUBTRACTION OF DECIMALS. RULE. Ans. 1. Place the subtrahend under the minuend, so that the decimal points shall be one under the other, and then proceed as in subtraction of whole numbers, only putting the decimal point in the remainder under those of the other numbers. 8. From a unit, or 1, take the hundredth part of itself. Ans. .99 MULTIPLICATION OF DECIMALS. RULE.* Place the factors, and multiply them together, as in *The reason of this rule will be easily understood after the student has become acquainted with the method of multiplying vulgar fractions together. In Multiplication of Vulgar Fractions, we multiply together the numerators, for a numerator, and the denominators, for a denomin |