If a sum of money yields 177. 10s. of interest at 3 per cent., what will it yield at 5 per cent. in the same time? What is a fraction? A fraction is a quantity which represents one or more of those equal parts, into which any integer or whole is divided. What is a vulgar fraction? In vulgar fractions, the numerator is the remainder after division, and the denominator is the divisor. Why is it called a vulgar fraction? Because they are supposed to have undergone no artificial preparation, they are called vulgar or common fractions. How is a vulgar fraction expressed? A vulgar fraction is expressed by placing the number of equal parts which it represents above a line, and the number of those parts into which the integer is divided below the same line; thus, ‡ is read four-fifths; and signifies four of those equal parts of which five make an integer or whole. What is the upper number called? The upper number of a fraction is called the numerator. What is the under number called? The under number is called the denominator. Thus, 4 is the numerator, and 5 the denominator. What common name applies to both these numbers? They are called terms of the fraction. What is a proper fraction? A proper fraction has the numerator less than the denominator, as . What is an improper fraction? An improper fraction has the numerator equal to, or greater than, the denominator, as or g.. What is a simple fraction? A simple fraction is that which, unconnected with any other fraction, refers immediately to its integer, as, †. What is compound fraction? A compound fraction is a fraction of a fraction, as of of. What is a complex fraction? A complex fraction is that which has a fraction or mixed number, either in its numerator, or denominator, or in both, as or or or 4ᄒ 93 When fractions of the same denomination have the same numerators, but different denominators-which is the greatest? Of fractions of the same denomination, which have equal numerators, but different denomninators, that is the least which has the greatest denominator, thus is less than 1. How do we invert a fraction, or find its reciprocal? One fraction is said to be the reciprocal of another, when the numerator of the one is the denominator of the other, and the denominator of the one the numerator of the other: thus is the reciprocal of . Reduce 11 to a fraction whose denominator is 12. 132 11 x 12 = Ans. I here multiply 11 by 12; the 12 product is 132, the numerator, under which I write 12, 47 Reduce 5 4 to an improper fraction. (5×8)+7=17. I here multiply 5, the whole number, by 8, the denominator of the fraction, the product is 40; to which I add 7, the numerator of the fraction, the sum is 47, under which I write 8, the fraction's denominator, and we have 47, the improper fraction required. 8' 37 Reduce to a whole or mixed number. 37 ÷ 5=73. 5 I here divide 37 by 5, the integral quotient is 7, and 2 over; under 2 I write the denominator 5, and the entire quotient is 7, the mixed number required. Reduce of of 3 to a simple fraction. == 3, or by cancelling, thus, & × ÷ × before. I first reduce the mixed number 3 to the improper fraction. I then write the fractions with the sign multiplication between them. The continued product of the numerators is 42, the new numerator, and the continued product of the denominators is 30, the new denominator, and divided by 6 gives 7, the fraction required. By cancelling, I here cancel 2 and 3 in the numerators, and 2 and 3 in the denominators, which leaves as before. to which gives the complex fraction 5; I then mul 8 1 tiply 32, the numerator of the upper fraction, by 1, the denominator of the under fraction, which gives 32, and 5 the denominator of the upper fraction by 8 the nume32 4 rator of the under, which makes 40; therefore = 40 5 the simple fraction required. The second example ample is performed in the same manner. Reduce to a fraction of equal value, whose nume14 rator is 30. As 9 30 14: 46%, therefore the new 30 fraction is performed by proportion. 464 Reduce to a fraction of equal value, having its 49 denominator 56. As 8:56: 7: 49, therefore is the new fraction required. 56 Reduce 432 1086 1086 108 = ÷ = 276 45 +6= 1892, lowest terms. I here 5' divide the terms by 4, which gives .8 gives and then by 9, the quotients of this last divi ; and as no number greater than 1 will now divide its terms the fraction is in its lowest terms. I here multiply 2, the first numerator, by 5 and 6, the other denominators, which gives 60; then 3, the second numerator,by 3 and 6, the other denominators, gives 54; again, 5, the last numerator, by 5 and 3, the other denominators, gives 75; therefore 60, 54, and 75 are the new numerators; and the denominators, 3, 5, and 6, multiplied together, give 90, the common denominator; therefore 8, 3, 35 is the fraction required, having a common denominator. But we can divide the terms of all these fractions by 3, which gives 36, 18, 35, their lowest common denominator. Reduce of a £. to the fraction of a farthing, and of a farthing to the fraction of a £. 1st, In working this example I write it down ÷ of 20 because 20 shillings make a £; then of 12, because 12 Τ pence make a shilling; then +, because 4 farthings make one penny. I then cancel the 5 and the 20, and re 384 ducing to a simple fraction by case 4, gives the I' fraction required. In the second example I compare it in the same manner, beginning with the farthings; thus of 4, because is the fourth of a penny; then of, because a penny is the twelfth part of a shilling; then of, because a shilling is the twentieth part of a £; then cancelling the 2 and the 4, and reducing to a single fraction, gives oo, the fraction required. Reduce 2 qrs. 16 lbs. to the fraction of a cwt. I here reduce 2 qrs. 16 lbs. to lbs., which gives 72 for the numerator, and 1 cwt. reduced to lbs. gives 112 for D |
