15. When we wish to multiply a given quantity by a fraction, we multiply the given quantity by the numerator of the fraction and divide by the denominator. Thus-Find the value of 3 of £2. × 1 6 This equals £== 158. Od. Ans. When necessary, we reduce the given quantity to one denomination, thus:-Find the value of of £5 12s. 6d. Or, without reducing the given quantity, thus:- In dividing the given quantity by a fraction we invert the divisor, and proceed as in multiplication. Examples on 15. Find the value of— 1. of £9; of £8 6s. 9d.; 1 of £5 12s. 8d. 9 11 2. £9 6s. 2d x 15; £3 2s. 4d. × 31; £9 2s. 21d. × 15. 4 14 3. £30 2s. 6d. ; £6 5s. 4d.; £8 12s. 6÷1 ÷ 11 4. of a cwt.; of a lb. troy; of 3 a. 2 r. 9 p. 5. 21 of 1818.; 1 cwt. 2 qrs. 9 lbs. × 2; 2s. 91d × 15. 29 6. 1 m. 6 fur. 91 yds. × 27; 2 ft. 3 in. × 13; 144 yds. ÷ 15. 7. 1 of 10s. 6d. ; 24 of 81s.; § of £9 15s. 9d. 8. £3 + 48. + d. of 22s.; 4 cwt. + lb. × 92 oz. × 31. 9. £58. of 144d.; £-of 21s.; of 140d. 10. of 93d.; 14 lbs. 9 oz. + lb. x 14. 2 11 FRACTIONS. 16. Miscellaneous Examples. 1. Add 45 + 14144 + 78. Add 4.5+ 1913 + 15 + 142 x- X 4 + 14 3 8 3. Simplify + 11411 × 51 + 21 × 3 × 11 3 - 15. 13 + 5. Multiply 381 by 14, and divide the product by 21. 4 6. What fraction multiplied by the sum of 24, 11, 11, will give the product 19? 3 7. What part of a guinea is of 7s. 6d. ? 8. Subtract of 3 guineas from of £2 19s. 9d. 9. Add together of a crown, of a shilling, and 2 of a guinea. 3 10. Express £335 2s. 9d. as a fraction of £394 2s. 11d. 11. Express the value of of an acre of a rood -- 4 of a perch, in square yards. 12. Reduce 109,7 153. 91-84 × 14,1% +886 × 3; also, 91 13. Add together the sum, difference, product, and quotient of 35 and 23. 3 14. Multiply of £1 2s. 6d. by 8, and divide 63 of 15 of by of 91. 21 15. Find the continued product of,, 11, 94, f, 211, 921. 16. What number multiplied by of 11 will produce 9 ? 17. Find the value of 371 of £140 16s. 2d. 3 18. Reduce of a crown + of a shilling ÷ 4s. 11d. to its simplest form. 19. Add 111-92 + 717-418 - 314. 84 20. Reduce 39 25 of of £5 23 39 £42 6s. 334 21 + 31 × 52. 151 + 4–8,2 19 22. Find the value of 3 of 31 of 33 of 1 of 2 cwt. 1 qr. 13 lbs. 23. Express in troy weight the difference between 3 lb. tr. and lb. av. 183 24. A person dies worth a million, and leaves of the money to his wife, to his son, and the rest to his daughter. The wife at her death leaves of her money to the son, and the rest to her daughter, but the son adds his fortune to his sister's, and gives her of the whole. What will the sister gain by this, and what fraction will be her gain of the whole? DECIMAL FRACTIONS. 14 100. A decimal is a fraction of which the numerator only is expressed; the denominator (which is always understood) being 1 followed by as many ciphers as there are figures in the numerator. Thus, 30, 14 = Ciphers added to the right of a decimal do not alter its value; thus, 30 is the same as 3, or 3 Ciphers added to the left of a decimal diminish its value tenfold; thus, 03 = 100, 003 = τοσ· 3 3 10. Thus it will be seen that we may express a decimal fraction as a vulgar fraction, and conversely a vulgar fraction as a decimal. 1. Express as fractions— 1. 14, 49, 82-431, 9.14, 81-3611, 9.0031. 24 91 1. 2010, 132, 29114, 1989, 14203, 21. 18362 7 761 21 31 21 14 2 3. 10, 10000 J' 100046, 827, 59, 381 191′ 11. 8 2411 91 211, 3821, 10000000 5. 11 tenths + 8 thousands + 11 hundred-thousandths. 6. of 9-3 × 9,1 × of 15, 18 of 14 of 148 × 19. ADDITION OF DECIMALS. 2. In adding decimals we must be careful to keep the points under each other in the same vertical line, so that the units may be under each other; we then add as in simple addition, setting the point in the result in the same line with the other points. Find the value of— 1. 2.41 19.8752 + 13·0076 + 9.3213 + 2·001. 4. 13.24976 + 14.98761 + 3456 + ·898769. SUBTRACTION OF DECIMALS. 3. In subtracting decimals we proceed as in simple subtraction, taking care to keep the points under each other, and the point in the result in the same line with the other points. Find the value of 1. 8246- •2102, 24.3216 — 20.00021. 2.321.982 3. 29.841 4. 32.001 5. 21.325 12.000021, 89.8909-21.0002798. 1.0000001, 35.3296 — 14.0101. 6. 231·00125 — 14′00379, 28′36 — 2·0007. 8. 321.142·321, 321·9 — 2.3213. 9. 298-1601·0001, 19·0002 — 3·0010019. 10. 8.90-20000, 3·91 — 2·0001. MULTIPLICATION OF DECIMALS. 4. In multiplying decimals we proceed as in ordinary multiplication, marking off in the product as many decimal places as there are in the multiplier and multiplicand together. 3.24 × 2.5 = 3.24 Thus, The reason of this is seen: 3:24 × 2·5 = = 324 × 5 324 81 = 20 × 10 40 10 Find the value of― 1. 22·5 × 8·9, 19·6 × 3·15, 39·24 × 8·19, 2·003 × 8·6. 2. 0003 × 001, 18·006 × 18, 18·001 × 18·001, 19.54 × 89. 3. 327 × 26, 221.75 × 1891, 384.25 × 2·19, 32.86 × 14. 4. 27·1 x 111, 14·62 × 1·003, 17.94 × 2·001, 18.8 × 3.2. 5. 21.525 × 8008, 192.8 × 0198, 2365 × 0024, 0078 × 19. 6. 17232 × 91, 14.6 × 0243, 18.9 × 2361, 007 × 3·1. DIVISION OF DECIMALS. 5. In dividing decimals we proceed as in ordinary division, marking off in the quotient as many decimal places as are equal to the difference of the number in the dividend and divisor. The reason of this is seen, as it is the reverse of the rule for multiplication. Or if the divisor is not a whole number, we may remove the decimal point altogether, and move the point in the dividend as many places to the right as there are decimals in the divisor; then we divide, and when we arrive at the decimal point in the dividend, we place a decimal point in the quotient. Ciphers may be added to the right of the dividend when there is a remainder, and the division prolonged to any extent. A whole number ending in ciphers when used as a divisor may |