(2) The rule for DOUBLE POSITION' is applied to those cases in which the required quantity is not a multiple, part, or parts, of a given quantity, but furnishes an equation of the form Now supposes to be the value of x, which by substitution does not satisfy the equation, but gives (ac) s+b-d = e........ (2), then subtracting (1) from (2), (a − c) (s − x) = e. Again, supposes' to be the value of x, which by substitution and subtraction, as before, gives which proves the common Rule, namely, Make two suppositions (s and s) for the required quantity; treat each of them in the manner pointed out by the question; and note the errors (e and e'); then the required quantity will be found by dividing the difference of the products es', e's by the difference of the errors e, e'. Ex. What number is that which, upon being increased by 10, becomes three times as great as it was before? 1st. Suppose the number to be 20, (s) then 20+ 10 = 30, but 3 x 20 = 60, .. e = - - 30. 2dly. Suppose the number to be (30), (s) 5. Find the least common mult. of 21, 63, and 5. 33 101 63 640 1 1 1 6 60 420 420 Ans. 95. 5 70 6. Reduce to the least common numerator which is the greatest. (1). Ans. 132 132 132 (2). Ans. The 1st. 15 8. Which is greater 71 or 73; and how much? Ans. 715 by 9 368 9. What is the difference between 7 and 7 × ? Ans. 23. 10. What fraction of £1 is 19s. 103d.? 191 Ans. 17 13. Reduce 5 yards 2 feet to the fraction of a mile. Ans. 5280 11 17. Find the number which exceeds by 1483 the difference between 195 and 95 Ans. 24736 23 18. Multiply 453 by 173, and divide the product by 41. Ans. 190. 19. Divide 2 of 711 by 3 11 31. If two-thirds of an estate be worth £220, what is the value of of the same? Ans. £90. 32. An article which cost 3s. 6d. is sold for 3s. 104d.; what is that per cent. profit? Ans. 10%. 33. How much per cent is 14s. 6d. of £3. 10s. ? Ans. 20 34. How much per cent. is 27 parts out of 36? Ans. 76 alloy, and the rest pure silver. 35. A shilling weighs 3dwts. 15 grs. of which 3 parts out of 40 are How much per cent. is there of alloy, and what weight of pure silver ? 1 360 36. The length of of the Earth's circumference is 691 miles nearly; what is the Earth's diameter, assuming that the diameter of a 7 circle is of its circumference? 22 5 37. There are five numbers, of which the first two are 2, 34; and each number exceeds the preceding one by the same fraction; find the numbers, and the sum of them. 7 38. What is the sum of which is 5s. 3d.? 3 11 Ans. 8s. 3d. 39. Divide into two parts, so that one is greater than the other DECIMAL FRACTIONS. 1. Reduce to vulgar fractions 0.375, 0·8125, 4075, 0·0064. 2. What is 0.003 of 0.27 of 90? 3. Divide 0-27 by 0.003; 0.06 by 60; 600 by 0-06; (1) Ans. 90. (2) Ans. 0·001. (3) Ans. 10000. 2 4. Reduce to their equivalent decimals 40 Ans. 0.0729. and 0.006 by 600. (4) Ans. 0·00001. 1 4000 1 Ans. 0.08, 0.00125, 0.0078125, 15.625. 5. Find the value in shillings and pence of £0.97216. Ans. 19s. 51d. 6. Reduce to vulgar fractions 0-8383...., 4041666..., 0-09009009... 9 What decimal of a square mile is one acre? Ans. 0.0015625. Ans. 0.0000000317. 10. What decimal of a year is 1 second? 11. Divide 0.454545.... by 0.121212...... Ans. 3.75. 333 12. Find the decimal which does not differ from by the ten 106 14. Divide 24+ by 3 - ; and express the result in a decimal Ans. 0.790123. 15. Extract the square root of 1 to 4 places of decimals; and the cube-root of 34 to two places. (1) Ans. 1.1726. (2) Ans. 1:56. 16. Find correct to 7 places of decimals; and |