58. Given (x+(y+z)=17, y+}(x+2)=17; and z+1(x+y) =17}, or {x+(y+z)=1000; y+(x+2)= 1000; and z+ (x+2)=1000}; to find x, y and z, in both the above cases. If a 2; b 3; c = = Ans. x-5, y=11, z=13. A general theorem for all questions of a similar nature: = 4; (abc+b+c-2bc--a)n 67n abc+2-a-b-c 107 (abc+a+b-2ab—c)n 83n 107 C's part In this question n is to one thousand. EXPRESSION OF QUESTIONS. As it is sometimes difficult for learners to know how to express the conditions of a question algebraically, the following remarks may be found useful: If the question be concerning one unknown number, or quantity; it may be represented by x or y. If the unknown quantity is to be multiplied by 5, that condition is expressed thus: 5x, or 5y. If 4 be added to that product, and the sum is equal to 14, then, 5x+4=14; but if 4 is to be subtracted, then 5x-4-14. If the unknown quantity is to be divided by 3, that condition may be expressed thus: or and if 7 be subtracted from that X y quotient, and the remainder is equal to 10, those conditions are expressed thus: x —7—10. If the third, fourth, and fifth of a number be added to itself, and the sum is equal to 35, that condition is expressed thus: If the question is concerning two numbers or quantities, they may be called x and y. If it be required that the sum of the two numbers sought be 60, that condition is expressed thus: x+y=60. If their difference must be 24, then x-y=24. If their product is 156, then xy=156. x If their quotient must be 6, then=6, or x÷y=6. y If their ratio is as 3 to 2, then x: y::3: 2, and hence 2x-3y. If the sum of their squares is 100, then x2+y=100. If the difference of their squares is 27, then 2-y2=27. Then = 16, or x2÷y2=16. y2 If their sum added to their difference be 10, and x be the greater number and y the less, then 2x=10; For x+y+x-y=2x. If the difference of their sum and difference be 6, and x be the greater and y the less, then 2y=6; For x+y-(xy)—2y. If the product of their sum, multiplied by their difference, must be 64, then (x+y)(x-y)=64; whence a-y-64. If the square of their sum be 36, Then (x+y), or x2+2xy+y2=36. If the square of their difference be 16, Then (x-y), or x2-2xy+y=16. If the sum of the squares of their sum and of their difference be 100, then 2x2+2y=100. For x+2xy+y' added to x2-2xy+y=2x2+2y. If the difference of the squares of the sum and difference equal 64, then 4xy-64. As may be seen by subtracting, x2-2xyy from x2+2xy+y3. If their sum, divided by their difference, be 3, then x+y x-y -3. If two-thirds of one, and four-sevenths of the other, make 60, If two-thirds of one, subtracted from four-sevenths of the other, If their sum must be four times their difference, Then x+y=4(x-y), or x+y=4x-4y. If the sum of their squares is five times the sum of the num bers, then 2+y=5(x+y)=5×(x+y.) Or x2+ y2=5x+5y. If their product is six times their sum, Then xy=6(x+y), or xy=6x+6y. 9x If their product is nine times their quotient, then xy= y. If one number must be as much above 20 as the other wants of 20, then x-20—20—y. If one number must be three times as much above 20 as the other wants of 20, then x-20-3(20—y)=3× (20—y). Or x-20-60-3y. If their difference and sum are to each other as the numbers two and three, then x-y: x+y::2:3; or, Multiplying extremes and means, 3x-3y=2x+2y. If their sum and product are to each other as the numbers three and five, then x+y: xy::3:5; or, Multiplying extrems and means, 5x+5y=3xy. If one number ought to be as many times contained in 20 as the other contains the number 4, then Whence xy=80. 20 y ; or 20 xy: 4; x 4 If the number 20 must be a mean proportional between the two numbers sought, then x: 20 :: 20: y; whence xy=400. If the greater being divided by the less, and again the less by the greater, the first quotient must be to the second as 5 to 3; If one number increased by 2, and multiplied by the other diminished by three, produce 40, then (x+2) (y-3)=40, or xy-3xc +2y-6-40. If three numbers, x, y, and z, must be in continued proportion, then xy::y z. Whence, xz=y3. It is sometimes easier to employ fewer letters than there are unknown quantities. Thus, the solution becomes more easy and elegant. There are some examples of this kind of notation. These and the preceding are some of the relations which are easily expressed; many others occur which are less obvious, but as they cannot be described by particular rules, their expression is best explained by examples, and must be acquired by experience. SOLUTION OF QUESTIONS. To solve a simple equation containing but one unknown quantity. RULE. Clear the equation of fractions by rule 3, and of radicles by rule 4; second, transpose the unknown terms or quantities to one side of the equation, and the known terms to the other, by rule 1. Collect each side into one term, and the unknown quantities, with a known coefficient, will form one side of the equation, and a known quantity the other side. Divide each side by the coefficient of the unknown quantity, and the value of the unknown will be exhibited. 1. What number is that, from the treble of which if 18 be subtracted, the remainder is 6? Ans. 8. Let x the number; ... 3x-18-6, or 3x=24, and x=8. 2. What number is that, the double of which exceeds four-fifths of its half by 40? 4 x Ans. 25. 40, and 10x-2x-200, or 3. In fencing the side of a field, whose length was 450 yards, two workmen were employed; one of whom fenced 9 yards, and the other 6 per day. How many days did they work? Ans. 30. Let x, 9x and 6x the number of days and of yards fenced by each respectively; .. (9x+6x)=15x-450, and 30, Ans. 4. A mercer bought 4 pieces of silk, 50 yards; the second was twice, the fourth four times as long as the first. lengths of the pieces? which together measured third three times, and the What was the respective Ans. 5, 10, 15, 20 yards. Let x, 2x, 3x, 4x, and 10x=50, be the number of yards in the first, second, third, fourth, and the equation, respectively, and x =5. 5. A farmer sold 13 bushels of barley at a certain price; and afterwards 17 bushels at the same rate; and at the second time received 36 shillings more than at the first. What was the price of a bushel ? Ans. 9 shillings. Let x= the price of a bushel, 13x and 17x the sum received for the first and second; .. (17x-13x=4x)=36, and x=9, Ans. 6. A person bought 198 gallons of beer, which exactly filled 4 casks; the first held twice as much as the second, the second twice as much as the third, and the third three times as much as the fourth. How many gallons did each hold? Ans, 108, 54, 27, and 9 gallons. Let x, 3x, 6x, and 12x the number of gallons the fourth, third, second and first, respectively; .. 22x-198, and x=9, and 108, 54, 27 and 9 were the answers. 7. A silversmith has 3 pieces of metal, which together weigh 48 ounces. The second weighs 12 ounces more than the first, and the third 9 ounces more than the second. What were their respective weights? Ans. 5, 17, and 26 ounces. Let x, x+12, x+21= the number of ounces the first, second, and third weighed, respectively, and 3x+33—48, or x+11=16, and x=5, and 17, 26 ounces, Ans. 8. A vinter fills a cask, containing 96 gallons, with a mixture of brandy, wine, and water. There are 20 gallons of water more than of brandy, and 17 more of wine than of water. How many are there of each ? Ans. 13 gallons of brandy, 33 of water, and 50 of wine. Let x, 20+x, 37+ be the number of gallons of brandy, of water, and of wine, respectively; 57+3x-96, or 19+x-32, and 13, and the number of gallons of brandy, water, and wine, were, 13, 33, 50, respectively. 9. A gentleman buys 4 horses; for the second of which he gives $12 more than for the first; for the third $6 more than for the second; and for the fourth $2 more than for the third. The sum paid for all was $230. How much did each cost? Ans. 45, 57, 63, and 65 dollars. Let x, x+12, x+18, x+20 denote the price of the first, second, third, and fourth; . 4x+50-230, or 4x-180, and x-45. 10. A poor man had 6 children, the eldest of which could earn 7d. a week more than the second; the second 8d. more than the third; the third 6d. more than the fourth; the fourth 4d. more than the fifth; and the fifth 5d. more than the youngest. They altogether earned 10s. 10d. a week. How much could each earn a week? Ans. 38, 31, 23, 17, 13, and 8 pence per week. Let x, x+5, x+9, x+15, x+23, and x+30 be the sum earned by the youngest, fifth, fourth, third, second, and eldest; .. 6x+ 82-130, or x=8. 11. An express set out to travel 240 miles in 4 days, but in consequence of the badness of the roads he found that he must go 5 miles the second day, 9 the third, and 14 the fourth day, less than the first. How many miles must he travel each day? Ans. 67, 62, 58, and 53 miles. Let x, x-5, x—9, x— -14- the number of miles travelled the first, second, third, and fourth day; .. 4x-28—240, or x—7— 60, or x=67. 12. There are 5 towns, in the orders of the letters A, B, C, D, E. From A to E is 80 miles. The distance between B and C |