Let x represent the number of days; then 1 1 is the part the woman would drink in one day. 30 We shall then have the Equation 1 1 From this Equation we shall find x=20 days. represents the part he would drink in one day; and 1 12' 29. If A and B together can do a piece of work in 9 days, and A alone could do it in 15 days, in what time could B alone do the work? Let a represent the number of days; then 1 the part both would drink in 1 day. represents the part B could do in 1 day; and is the part A could do in 1 day. 15 We shall then have the Equation + 1 X 15 This Equation will give x=221 days. 30. The hour and minute hands of a watch or clock are exactly together at 12 o'clock. When are they next together? 1 the part both could do in 1 day. 9' The dial face of the clock is divided into 60 equal minute spaces. Let x represent the number of these spaces passed over by the hour hand before the two hands are next together. Then, since the minute hand moves 12 times as fast as the hour hand, 12x represents the number of spaces passed over by the minute hand before the two are next together. But 60+ also represents the number of these spaces passed over by the minute hand in the same time. We shall then have the Equation 12x=60+x; which will give x=5 spaces; hence they will be together 55 minutes past 1 o'clock. 31. It is required to divide $1000 between A, B, and C, so that A shall have half as much as B, and C half as much as A and B together. which will give a $2223, A's share; then 2×2223 = $444, B's share; and 2223=$333, C's share. 32. A person being asked the hour, answered that the time past noon was of the time till midnight. What was the hour? Let x represent the number of hours past noon; then 12- represents the number of hours till midnight; and, by the conditions of the problem, we have the Equation 2 3 x= (12-x); which will give x=44 hours=4h. 48m. 33. It is required to divide the number 60 into two such parts, that their product shall be equal to the square of the less. What are the parts? Let x represent the less part; then 60-x represents the greater part; 3r2 represents three times the square of the less. The Equation will then be 60x-x23x2. Dividing both sides by x, we have 60-x=3x; from which x will be found 15, the less part; hence 60-15=45, the greater part. 34. How much wine at 90 cents a gallon, and how much at $1.50 a gallon, will be required to form a mixture of 20 gallons which shall be worth $1.25 a gallon? Let x represent the number of gallons at 90 cents a gallon; then 20- — represents the number of gallons at $1.50 a gallon; 90x represents the value of the 1st kind in cents; 150(20-x) represents the value of the 2d kind in cents; and 20 x 125, or 2500 cents, is the value of the mixture. The Equation, then, will evidently be 90x+150(20-x)=2500; which will give x= =83 gallons, the quantity at 90 cents a gallon; hence 20-8311 gallons, the quantity at $1.50 a gallon. 35. A cistern is supplied with water by three pipes which would severally fill it in 4, 5, and 6 hours. In what time would the three pipes running together fill the cistern? Let x represent the number of hours. Then, since the first would fill of the cistern in 1 hour, the second of it, and the third of it, x represents the part the 1st would fill in x hours; 4 x represents the part the 2d would fill in a hours; and represents the part the 3d would fill in x hours. 5 x 6 Then, we shall have the Equation X 4 from which x will be found X + + 5 = x 6 13 hours. 36. If $1000 be divided between A, B, and C, so that B shail nave as much as A and half as much more, and C as much as B and half as much more, what will be the portion of each? =1, the entire cistern, 3x 1 3x 2 9x + of or 4' -, represents C's share. The Equation will then be 9x 3x 4 from which x will be found equal to $21019, A's share; hence of 21018 $31515, B's share; and 2 of 21018 = $47313, C's share. = 37. A person has a lease for 99 years, and of the time which has expired on it is equal to of that which remains. Required the time which remains on the lease. Let x represent the time remaining; then 99-x represents the time expired. Then, by the conditions of the problem, we shall have 7x 5 11x = 1000; 4x 112 7 from which we shall find x= 5 38. A merchant bought cloth at the rate of $7 for 5 yards, which he sold again at the rate of $11 for 7 yards, and gained $100. How many yards were thus bought and sold? Let x represent the number of yards. Then, as of a dollar is what he paid per yard, and of a dollar what he received per yard, represents what he paid for x yards; and represents what he received for x yards. 7 Since he gained 100 dollars, we shall have the Equation 7x = +100; 583 yards. 39. A and B together possess the sum of $9800; and five-sixths of the sum owned by A is the same as four-fifths of the sum owned by B. What is the sum owned by each? Let x represent the sum owned by A; then 9800- represents the sum owned by B. By the conditions of the problem, we shall have 5x 4 (9800-x); 6 5 which will give x = $4800, A's sum of money; hence 9800-4800 =$5000, is B's sum of money. 2x 3 40. The assets of a bankrupt, amounting to $5600, are to be divided among his creditors, A, B, and C, according to their respective claims. A's claim is of B's and C's is of B's. What sum must each of the VYZVY creditors receive? Let x represent B's claim; then Xx represents A's claim; and represents C's claim. X 2x The Equation will then be x+ + 2 3 which will give x=$2584, B's claim; then 258432-$12943, is A's claim; and of 2584-$1723, is C's claim. PROBLEMS In Simple Equations of one, two, &c., unknown (85) 1. Find two numbers such, that of the first with of the second shall be equal to 9, and of the first with of the second shall be equal to 5. 82 214 Let x represent the first and y the second number; then, by the conditions of the problem, we shall have =5600; 9; and =5. These Equations will give x=8, and y=15. 2. Divide the number 100 into two such parts that of the first and of the second part shall together make 30. |