Subtraction of decimals.-Arrange the numbers under each other, as before, according to their value; then begin at the right hand; subtract as in whole numbers, and point off the decimals, as in addition. Example. From 65.378 take 6.45 65.378 6.45 58.928 When the number of decimals is less in the superior, than in the inferior number, add as many cyphers to the superior number, as will make the number of decimals equal; for, as before observed, the value of decimals is not altered, by placing any number of cyphers after the last figure. Example. From 22.6000 Take 4.7325 17.8675 Multiplication of decimals.-Rule. Place the factors, and multiply them as in the multiplication of integers. Then mark off just as many decimals in the product, as there are decimals in both factors; and if there should not be a sufficient number of figures in the quotient, supply the defect by-prefixing cyphers. Example. Multiply 46.25 by 34 18500 13875 1572.50 When one, or both of the factors consists solely of decimals, the operation is founded on the same principle as the preceding. Example. Multiply 4.31 by 0.5 4.31 2.155 Divide as in Division of decimals.-Rule. whole numbers; but if the divisor, or dividend, or each of them, have decimals, add to that which has none, or to that which has less than the other cyphers, until the number of decimals in both be equal. Example. Divide 81.54 by 27 8100 5400 VULGAR FRACTIONS. A vulgar fraction is expressed by two numbers placed one over the other, and separated by a line, thus, †, 4. The number under the line is called the denominator, and shows into how many equal parts the whole quantity is divided. The number above the line is called the numerator and shows how many of those parts are expressed by the fraction. Example. 1 yard and 4. Here the denominator 3 shows that the whole yard is divided into three parts; and the numerator 2, shows that the fraction contains two of those parts. Fractions are divided into proper, improper, simple, and compound, or mixed. A proper fraction is when the numerator is less than the denominator; as 4, 3, 3. 2 An improper fraction is when the numerator is equal to, or greater than, the denominator; as 4, 3, . 6 A simple fraction denotes any number of parts of the whole number, as 3, 3. A compound fraction is the fraction of a frac tion, as of 3, of 1⁄2 of z. A mixed or compound fraction consists of a whole number, and a fraction, as 5, 8%. Reduction of vulgar fractions, is, the changing them from one denomination to another, to prepare them for the operations of addition, subtraction, multiplication, and division. To reduce fractions to their lowest terms. Rule.-Divide the terms of the given fraction by any number that will divide them both without a remainder; then continue dividing the quotients in the same manner, until there is no number greater than one, which will divide them, and the fraction will then be in its lowest term. Example. Reduce 2 to the lowest terms. 19÷6=4÷7={ 210) 252 (1 42) 210 (5 Proof 42) 23. The rule for finding the greatest common measure of two numbers is this. Divide the greater by the lesser number; and then divide the divisor by the remainder; and continue dividing the last divisor by the last remainder, until nothing remain; and the last divisor will be the common measure, by which divide both terms of the fraction as above. To reduce a mixed number to its equivalent improper fraction. Rule.-Multiply the integer by the denominator of the fraction, and to the product add the numerator. The sum is the numerator of the improper fraction sought, and is placed above the given denominator.-Example. Reduce 15 to an improper fraction. 15x8=120+3=123=13 Addition of vulgar fractions.-Rule. If the fractions have a common denominator, add all the numerators together, and place their sum over the common denominator, which will be the amount required. But if the proposed fractions have different denominators, reduce them all to the least common denominator, over which place the numerators as before. 2 Example. Add together and 4. 2) 4352 then 2×2×3×5×1=60, which is the 2351 least common denominator. Then 604.3.5.2 15.20. 12. 30. which, multiplied by all the numerators, give 28+8+3+278=372=61%. Answer. Subtraction of vulgar fractions.-Rule. Prepare the fractions in the same manner as for addition, if necessary; then subtract the less numerator from the greater, and under the difference, place the common denominator. Example. From take 9. 3X7 = 21 30 5×6= 30 Then 3-31-33. Answer. 5x7 = 35 35 Multiplication of fractions.-Rule. Reduce mixed numbers, if there be such, to improper fractions; and compound fractions to simple fractions. Then multiply all the numerators together, for a new numerator; which will give the required product. Example. Multiply 4, 3, and 4 of 8, together. 30, 5, 16 ,,,x 16=13. Answer. |