2. A can do a piece of work in 8 days which B can perform in 10 days. In how many days can it be done by both working together? Let x = number of days required. 1 Then = what both can do in one day. X 1 Also =what A can do in one day. 8 1 By the conditions, 8 10 X 4 days, Ans. EXAMPLE FOR PRACTICE. A can do a piece of work in a hours, which B can do in b hours. together? ILLUSTRATIVE EXAMPLES. ab hours. a + b Ans. 1. A sum of money, amounting to $4.32, consists entirely of dimes and cents, there being in all 108 coins. How many are there of each kind? 2. The second digit of a number exceeds the first by 2; and if the number, increased by 6, be divided by the sum of its digits, the quotient is 5. Required the number. Then 2 the second. And 2x+2= the sum of the digits. The number itself is equal to 10 times the first digit, plus the second, which is 10x + x + 2, or 11x + 2. Hence, by the The first digit of a number is three times the second; and if the number, increased by 3, be divided by the difference of its digits, the quotient is 16. Required the number. Ans. 93. ILLUSTRATIVE EXAMPLES. 1. Two persons, A and B, 63 miles apart, start at the same time and travels towards each other. A travels 4 miles an hour, and B 3 miles an hour. How far will each have traveled when 2. Separate 41 into two parts such that one divided by the other may give 1 as a quotient and 5 as a remainder. Separate 113 into two parts such that one divided by the other may give 2 as a quotient and 20 as a remainder. Ans. 82, the dividend; 31, the divisor. ILLUSTRATIVE EXAMPLE. At what time between 3 and 4 o'clock are the hands of a watch opposite each other? Let OM and OH represent the positions of the minute and hour hands at 3 o'clock, and OM' and OH' their positions when opposite each other. M M' Let x the arc MHH'M' over which the minute-hand has passed since 3 o'clock. H Then H' X the arc HH' over which the hour Also, the arc MH = 15 minute spaces. Now, arc MHH'M' = arc MH + arc H'M' + arc HH'. EXAMPLE FOR PRACTICE. How A, B, and C together can do a piece of work in 6 days; B's work is one-half of A's, and C's work is two-thirds of B's. many days would it take each working alone? 10. Ans. A, 11 days; B, 22 days; C, 33 days. SIMPLE EQUATIONS. Containing two Unknown Quantities. When a single equation contains two unknown quantities, as x+y=20, the values of x and y cannot be definitely determined; because, if any value be assumed for x, we have a corresponding value for y. Thus, if x 15, then 15+ y = 20, or y = 5, will satisfy a given equation. Equations of this kind are called indeterminate. = But if we have the equations, x+y= 20 and xy 10, it is evident that the values x 15, y = 5, are the only correct ones, since they satisfy both equations simultaneously. Simultaneous Equations are such as are satisfied by the same values of their unknown quantities. Independent Equations are such as can be made to assume the same form. Two unknown quantities require for their determination two independent, simultaneous equations, which may be solved by an operation called Elimination, which combines them so as to form an equation containing but one unknown quantity. The three methods of elimination are: 1. By Addition or Subtraction. 2. By Substitution. 3. By Comparison. ELIMINATION BY ADDITION OR SUBTRACTION. 11. Rule. Multiply the given equations by such numbers as will make the coefficients of one of the unknown quantities equal. Then add or subtract the resulting equations according as the equal coefficients have like or unlike signs. NOTE. Such multipliers should be used as will produce the least common multiple of the coefficients which are to be made equal. Substituting this value in (2), 10x-14= 24 |