(19.) A person left of his property to A, 20 to B, to C, to D, to E, to F, and the rest, which was 800l. to his executor; what was the value of the whole property, and of each person's share? (20.) How many deals 12 feet long and 7 inches broad will be required to floor a room 7 yards long by 5 yards wide, allowing for a vacancy 7 feet long by 5 feet broad? (21.) There is an island 120 miles in circuit; 7 footmen all start together to travel the same way round it, and continue to travel till they all come together again: A goes 5 miles a day, B 64, C 7, D 84, E 91, F 101, and G 114. In how many days will they all be together a second time? (22.) The hour, minute, and second hands of a watch are together at 12 o'clock, when will they all be together a second time? DECIMAL FRACTIONS. 25 225 Definition 1. Decimal Fractions, or Decimals, or such as have 10, 100, 1000, &c. for their denominator; thus, 100, 1000, &c. are decimal fractions, and these are expressed by writing the numerator only, with a point before it on the left hand; thus, 1, 25, ·225, &c. 2. When the numerator of a decimal fraction is written without its denominator, it must always consist of as many figures as there are ciphers in the denominator; thus, =5, T‰。='05, 10‰0='005, &c. Hence the denominator of a decimal fraction is an unit with as many ciphers as there are figures in the decimal. 3. Ciphers on the right hand of decimals make no alteration in their value; thus, 5, 500, 5000, &c. are decimals of the same value, for 5008 the nature of vulgar fractions. by 4. Ciphers on the left hand of decimals decrease their value; thus, 5, 05, 005, &c.=70, TOO, TOO, &c. Note 1. Decimals, as well as whole numbers, decreuse in a ten-fold proportion towards the right hand; therefore decimals have the same properties as whole numbers, and are subject to the same rules. 5. A mixed number is composed of a whole number and a decimal, which are separated from each other by a point, thus, 115.5 signifies 115. 2. A mixed number, as 115.5, may be expressed thus, 1155: also, 115-005 1150-05 11500.5 115005 115.005 &c. ADDITION OF DECIMAL FRACTIONS. RULE. Place all the decimal points directly under each other, so that tenths may stand under tenths, and hundredth parts under hundredth parts, &c. in the decimals; and tens under tens, hundreds under hundreds, &c. in the whole numbers. Then add them together as in whole numbers, and from the right hand of the sum point off as many figures, for decimals, as are equal to the greatest number of decimals in any of the given numbers. Examples. (1.) Add 5.74+3.75 +94.375+745+005495 toge ther. 5.74 3.75 94.375 745 ⚫005495 104.615495 sum. (2.) Add 5.714+3.456+543+17:4957 together. (3.) Add 3-754+475+00857+37.5 together. (4.) Add 54-34+375 +14 795+1.5 together. (5.) Add 71.25 +1.749+17595+3.1 together. (6). Add 375.94+5732+14.375+1.5 together. (7.) Add '005+·0057+31·008+00594 together. SUBTRACTION OF DECIMAL FRACTIONS. RULE. Place the less number under the greater, the points under the points, tenths under tenths, hundredth parts under hundredth parts, &c. in the decimals; and the whole numbers under those of the same denomination. Then subtract as in whole numbers, placing the separating point, in the remainder, directly under those above it. Examples. (1.) From 57.439 take 5·93754. 57.439 51.50146 difference. (2.) Required the difference between 57·49 and 5·768. (3.) What is the difference between 3054 and 3·075 ? (4.) Required the difference between 1745·3 and 173:45 (5.) What is the difference between seven-tenths of an unit and 54 ten thousandth parts of an unit? (6.) What is the difference between 105 and 1.00075 ? (7.) What is the difference between 150·43 and 754.355? (8.) From 1754·754 take 375·49478. (9.) Take 75.304 from 175.01. (10.) Required the difference between 17.541 and 35'49. MULTIPLICATION OF DECIMAL FRACTIONS. RULE. Multiply the decimals, as if they were whole numbers, and from the product cut off so many decimal places as there are both in the multiplier and multiplicand. If there are not so many places in the product, supply the defect by prefixing ciphers to the left hand. Note 1. When any decimal is to be multiplied by 10, 100, 1000, &c. remove the separating point so many places to the right hand as there are ciphers; thus, 543 × 10=5·43; also, 7156×1000=715.6, &c. 2. What was observed in the third note in multiplication of vulgar fractions, respecting a proper fraction, or mixed number, is equally applicable to a pure, or mixed, decimal. Contracted Multiplication of Decimal Fractions. RULE. Put the unit's place of the multiplier under that place of the multiplicand which you intend to keep in the product, and invert the order of all the other figures, that ís, write the decimals on the left hand, and the integers, if any, on the right. In multiplying, always begin with that figure of the multiplicand which stands directly over the multiplying digit, and set the first figure in every product in a right line under each other to the right hand, observing to increase the first figure of every line with what would arise, by carrying 1 from 5 to 15, 2 from 15 to 25, 3 from 25 to 35, &c. from the product of the two figures (in the multiplicand) on the right hand of the multiplying digit. (3.) Mult. 473.54 by '057. Examples under the contracted rule. (1.) Multiply 2-38645 by 8-2175, and let there be only four places of decimals retained in the product. (2.) Let 54.7494367 be multiplied by 4.724753, reserving only five places of decimals in the product. (3.) Multiply 475.710564 by 3416494, retaining three decimals in the product. (4.) Multiply 3754-4078 by 734576, retaining five decimals in the product. (5.) Let 4745-679 be multiplied by 751:4549, and reserve only the integers in the product. DIVISION OF DECIMAL FRACTIONS. RULE I. Divide as in whole numbers, and from the right hand of the quotient point off so many figures for decimals as the decimal places in the dividend exceed those in the divisor; but, if the quotient does not contain such a number of figures as is equal to the excess, the defect must be supplied with ciphers to the left hand. If the number of decimal places in the divisor should be more than those of the dividend, annex so many ciphers to the dividend as will make them equal, and the quotient will be integers till all these ciphers are used; after which, you may continue the quotient to any assigned degree of exactness, by subjoining a cipher continually to the last remainder. RULE II. Make the divisor a whole number by removing the decimal point to the right hand of it, and remove the decimal point in the dividend the same number of figures towards the right hand as the point in the divisor has been removed. If there be not a sufficient number of figures |