7. There is a number, consisting of two digits, the first of which is greater than the second, and if the number be divided by one more than the sum of the digits, the quotient will be 6; but if the digits be inverted, and that number be divided by the sum of the digits, the quotient will be 4. Required the number. Let a denote the first digit, or that in the ten's place, and y denote the 2nd. digit, or that in the unit's place; then x+y=the sum of the digits, by transposition, 6y-3x=0, or 2y-x=0; Multiplying this equation by 4, 8y-4x=0; but 4x-5y=6; by addition, 3y = 6, or y = 2; .. x=2y=4; whence, the number is 42. NOTE. Inverting the digits, is placing the y, which is in the unit's place, in the ten's place, and placing the x, which is in the ten's place, in the unit's place. Thus, in the above example, 10x+y denotes the number; but when the digits are inverted, 10y+a denotes the number. EXAMPLES FOR PRACTICE In Simple Equations of two unknown quantities. 1. A draper bought two pieces of cloth, one at 18s. per yard, and the other at 16s. per yard, for £43; and three times the length of the piece at 18s. per yard, exceeded four times the length of the piece at 16s. per yard by ten yards. Required the length of each piece. Ans. 30 yards at 18s., and 20 yards at 16s: 2. A bill of £27. 5s. was discounted in half guineas and half crowns, and three times the number of half guineas exceeded twice the number of half crowns by 20. Required the number of each. Ans. 40 half guineas, and 50 half crowns. 3. A mercer bought two pieces of silk for £8. 12s. the first 18 yards and the second 20 yards, and 7 yards of the first piece cost 3s. more than 5 yards of the second piece. Required the price per yard of each piece. Ans. First 4s. and second 5s. per yard. 4. A gentleman bought a quantity of brandy at 5s. a bottle, and a quantity of rum at 3s. 9d. a bottle, and paid twice as much for the brandy as for the rum. Had he bought as many bottles of brandy as he bought of rum, and as many bottles of rum as he bought of brandy, he would only have paid 5s. more for the rum than for the brandy. How many bottles of each did he buy? Ans. 12 of brandy, and 8 of rum. 5. A farmer sold 30 bushels of wheat and 40 bushels of barley, and received for the whole £25: he also sold 50 bushels of wheat and 30 bushels of barley, and received for this bargain £7, 10s. more than he did the former time. Required the price of each per bushel. Ans. Wheat 10s., and barley 58. I 6. A certain sum of money put out to interest, amounted to £297. 12s. in eight months, and in fifteen months to £306. Required the sum and rate per cent. Ans. £288 at 5 per cent. 7. Required that fraction, whose numerator increas ed by 2, its value becomes one-half, but if one be added to its denominator, its value becomes one-third. 8. A butcher bought 12 sheep and 7 oxen for £150; he afterwards bought 21 sheep and 6 oxen for the same sum. Required the price of each. Ans. Sheep £2 each, and the oxen £18 each. 9. A gentleman left his son and daughter each a legacy, part of which was to be paid at six months, and the remainder at twelve months after the father's death; at six months the executors paid them £800, giving to the son five-sixths of his legacy, and to the daughter three-fourths of her legacy: on winding up the affairs, at the end of twelve months, they found that there only remained for the children £100, of this they gave to the son three-fifths of what remained due to him, and to the daughter two-fifths of what remained due to her. How much did the father leave to each ? Ans. £600 to the son, and £400 to the daughter. 10. There is a number, consisting of two digits, the first of which is greater than the second; now, if the number be divided by twice the sum of the digits +2, the quotient will be 3; and if the digits be inverted, and that number be divided by 7 times the difference of the digits +2, the quotient will also be 3. Requir ed the number. Ans. 96. 11. There is a certain number, if to the product of whose digits you add twice the right-hand digit, the result will be seven times that digit; but if to six times the sum of the digits you add 5, it will be the number required. What is that number? Ans. 53. 12. A and B played at marbles, B won three-fourths of what A had, and then lost one-half of his stock; then he had only two-thirds as many marbles as A; but when A gave him 10, they had each the same number. How many had each at first? Ans. A had 80 and B 20 13. Find two numbers, the greater of which shall be to the less, as their sum is to 21, and as their difference is to 3. Ans. 16 and 12. 14. Find two numbers, whose sum, difference, and quotient, shall be as the numbers 4, 2, and 1, respectively. Ans. 9 and 3. 15. A person measured a rectangular court yard, and found that if 5 feet were added to each of its dimensions, its area would be 250 feet more; but if 6 feet were added to the length, and 4 to the breadth, its area would be increased by 244 feet. Required its length, breadth, and area. Ans. Length 25, breadth 20, and area 500. 16. A farmer bought calves at £2. 10s. each, and sheep at 30s. each, which together cost him £125, and found that the number of calves was to four-fifths of the number of sheep more than the number of calves, as 5 to 6. Required the number of each. Ans. 20 calves, and 50 sheep. 17. After A had been dispatched 16 hours on his journey from Leeds to London, a distance of 192 miles, B started to overtake him as he entered the city; in order to do this, he found that the time it took him to travel 33 miles, added to the time it took A to travel 22, must be exactly 11 hours. How many miles did each travel per hour? Ans. A 4 miles, and B 6 per hour. 18. A farmer mixed 28 bushels of barley at 2s. 4d. per bushel, with rye at 3s. per bushel, and wheat at 4s., so that the whole mixture may consist of 100 bushels, and be worth 3s. 4d. per bushel. How many bushels of rye and wheat did he mix with the barley? Ans. 20 of rye, and 52 of wheat. 19. A person measured a rectangular field, and found if the length was increased by 4 chains and the breadth by 3, the adjacent sides would be in the ratio of 4 to 3; but if each side was decreased by 3, the ratio of the adjacent sides would be as 3 to 2. Required the area of the field. Ans. 10a, 3r, 8p. 20. Find two numbers in the proportion of 4 to 5, from which two other numbers, in the proportion of 6 to 7, being respectively deducted from them, the remainders shall be in the proportion of 2 to 3, and their sum equal to 20. Ans. 2 first numbers, 32 and 40, and the others, 24 and 28. 21. A draper sold two pieces of cloth, one 20 and the other 30 yards, for £45: he received for 8 yards of the shorter piece £2 more than he did for 4 yards of the other. Required the price of each piece per yard. Ans. The shorter piece 15s., and the other 20s. per yard. |