and ? Ans. and? 156 3. What is the difference between fa 4. What is the difference between Ans. 39 5. What is the difference between and? Ans. 19 6. What is the diff. between 53 and 4 of 41 7. What is the difference between of a pound, and of Ana. s or 10s 7d 19. 191 of a shilling? of a shilling? Ans. 4 108 of a ponad and Ans. 8 or 11 8s 11 d. MULTIPLICATION OF VULGAR FRACTIONS, * REDUCE mixed numbers, if there be any, to equivalent fractions; then multiply all the numerators together for a numerator, and all the denominators together for a denominator, which will give the product required. EXAMPLES. 1. Required the product of 2 and 3, Here X30 , the Answer. Or X = X 2. Required the continual product of 3, 3, 5, and of z. 21358 3 13X3 39 47 Ans, 3. Required the product of and . and 4. Required the product of * Multiplication of any thing by a fraction, implies the taking some part of parts of the thing: it may therefore be truly expressed by a compound fraction; which is resolved by multiplying together the numerators and denominators. Note, A Fraction is best multiplied by an integer, by dividing the denominator by it; but if it will not exactly' divide, then multiply the numerator by it, 6. Required 6. Required the product of, 4, and 3. 7. Required the product of 3, 3, and 4. 8. Required the product of, and of $. 9. Required the product of 6, and 2 of 5. 10. Required the product of 3 of 3, and § of 32. 11. Required the product of 3 and 411. 12. Required the product of 5, 3, 4 of 3, and 41. Ans, 2 30 13. 84 Ans. 23 Ans. 14114. Ans. 21 231 DIVISION OF VULGAR FRACTIONS. * PREPARE the fractions as before in multiplication; then divide the numerator by the numerator, and the denominator by the denominator, if they will exactly divide but if not, then invert the terms of the divisor, and multiply the dividend by it, as in multiplication. * Division being the reverse of Multiplication, the reason of the Rule is evi dent. Note, A fraction is best divided by an integer, by dividing the numerator by it; but if it will not exactly divide, then multiply the denominator by it. MAKE the necessary preparations as before directed; then multiply continually together, the second and the third terms, and the first with its parts inverted as in Division, for the answer.* EXAMPLES. 1. If of a yard of velvet cost of a pound sterling; what will of a yard cost? 5 8 2 B =6s 8d, Answer. 2. What will 33 oz of silver cost, at 6s 4d an ounce ? Ans. Il 1s 44d. 3. If of a ship be worth 2731 2s 6d; what are of her worth? Ans. 2271 12s 1d. 4. What is the purchase of 12301 bank-stock, at 108 per cent.? Ans. 1336 1s 9d. 5. What is the interest of 273 15s for a year at 31 per cent. ? Ans. 81 17s 11дd. what part of her is Ans.. 7. What length must be cut off a board that is 73 inches broad, to contain a square foot, or as much as another piece of 12 inches long and 12 broad? Ans. 18 inches. 6. If of a ship be worth 731 1s 3d; worth 2501 10s? 8. What quantity of shalloon that is of a yard wide, will line 94 yards of cloth, that is 24 yards wide? Ans. 313 yds. This is only multiplying the 2nd and 3rd terms together, and dividing the product by the first, as in the Rule of Three in whole numbers. VOL. I. 10 9. If 9. If the penny loaf weighs 6 oz, when the price of wheat is 5s the bushel; what ought it to weigh when the wheat is 8s 6d the bushel? Ans. 4 oz. 10. How much in length, of a piece of land that is 1111 poles broad, will make an acre of land, or as much as 40 poles in length and 4 in breadth Ans. 13 poles. 11. If a courier perform a certain journey in 35 days, travelling 13 hours a day; how long would he be in performing the same, travelling only 11 hours a day? Ans. 4091 days. 952 12. A regiment of soldiers, consisting of 976 men, are to be new cloathed; each coat to contain 2 yards of cloth that is 15 yard wide, and lined with shalloon yard wide: how many yards of shalloon will line them? Ans. 4531 yds 1 qr 2 nails. DECIMAL FRACTIONS. A DECIMAL FRACTION, is that which has for its denominator an unit (1), with as many ciphers annexed as the numerator has places; and it is usually expressed by setting down the numerator only, with a point before it, on the left hand. Thus, is 4, and 24 is 24, and is 074, and 124 is 00124; where ciphers are prefixed to make up as many places as are ciphers in the denominator, when there is a deficiency of figures. A mixed number is made up of a whole number with some decimal fraction, the one being separated from the other by a point. Thus, 3.25 is the same as 3,35, or 325. Ciphers on the right hand of decimals make no alteration in their value; for 4 or 40, or 400, are decimals having all the same value, each being or. But when they are placed on the left hand they decrease the value in a ten-fold proportion: Thus, 4 is, or 4 tenths: but 04 is only or 4 hundredths, and 004 is only, or 4 thousandths. The 1st place of decimals, counted from the left hand towards the right, is called the place of primes, or 10ths; the 2d is the place of seconds, or 100ths; the 3d is the place of thirds, or 1000ths; 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 ten-fold proportion; as in the following Scale or Table of Notation. SET the numbers under each other according to the value of their places, like as in whole numbers; in which state the decimal separating points will stand all exactly under each other. Then, beginning at the right-hand, add up all the columns of numbers as in integers; and point off as many places, for decimals, as are in the greatest number of decimal places in any of the lines that are added; or place the point directly below all the other points. EXAMPLES. 1. To add together 29-0146, and 3146-5, and 2109, and -62417, and 14.16. 29.0146 8146.5 2109. .62417 14.16 5299-29877 the Sum. Ex. 2. What is the sum of 276, 39-213, 72014-9, 417, and 5032? 3. What is the sum of 7530, 16-201, 3·0142, 957.13, 72119 and ⚫03014. 4. What is the sum of 312.09, 3.5711, 7195-6, 71·498, 9739-215, 179, and ⚫0027 ? SUBTRACTION |