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Let x denote the number of miles per hour.
to make the direct journey, and
denote the number of hours required
required to make the return journey.
Since he travelled slower on the return journey, he will take a longer time; and from the conditions of the problem,
whence, by solving the equation, we find,
'7, and x" =
x + 2
The first solution is the answer to the given problem, as may be shown by verification. The second solution is the answer to another problem which gives the same equation, and which may be enunciated thus: A person travelled 105 miles, at a uniform rate. On his return, he travelled two miles per hour faster, and made the journey six hours sooner.
Denote the rate of travel on the outward journey, by then the rate, on the return journey, will be denoted by x+2; and, since he travelled slower on the outward journey,
= 6; or,
105 x 2
the same equation as before: hence, its solution should give both roots, x = 7, and x"=
On account of the greater generality of algebraic language, both of the above problems are expressed by the same equation. Hence, the solution of the equation ought to give the answers to both.
When, therefore, two answers are found to a problem, both of which will not satisfy the conditions of the problem, that one only must be taken which does satisfy them. The other one belongs to a similar problem, in which certain of the given quantities are to be consid ered in a contrary sense.
3. A and B set out from two points at the same time, and travel towards each other. On meeting, it appears that A has travelled 30 miles more than B, and that A could travel B's distance in 4 days, and that B could travel A's distance in 6 days. How far apart were they when they started?
Let x denote the number of miles that B travelled; then will x + 30 denote the number that A travelled. Since A can travel B's distance in 4 days,
will denote the number of miles that A travels in
the distance he travels in one day, or
x + 30
1 day. In like manner, denotes the number of " 6 miles that B travels in 1 day. The whole distance that A travels, divided by the distance he travels in one
6x , gives the number of days that he tra (888)
vels. The whole distance that B travels, divided by
the number of days that B travels. But they travel the same number of days each. Hence,
The first result only satisfies the conditions of the problem. Hence, B travels 150 miles, A, 180 miles, and both together, 330 miles, which is the answer to the question proposed.
4. A, B, and C, can together perform a piece of work, in a certain time. A alone can perform the same work in 6 hours more, B alone in 15 hours more, and C alone in twice the time. In how many hours can they all perform it, working together?
Let x denote the number of hours for all to perform it; then will
x+6 denote the number of hours for A to perform it: x + 15
In 1 hour, A can perform a portion of the work de
x +- 6
In 1 hour, B can perform a portion of the work de
x + 15
In 1 hour, C can perform a portion of the work do noted by :
In 1 hour, all can perform a portion of the work denoted by
Since the sum of the parts which each can perform, is equal to the part they can all perform, we have,
Whence, by solving,
Multiplying both members by 2x(x + 6) (x + 15), and reducing,
x' 3, and x" =
The value of x' alone satisfies the conditions of the problem. Hence, the answer is 3 hours.
5. There are two numbers whose sum is 40, and the sum of their squares is 818. What are the num bers?
x2 + (40-x)2 .'. x' = 23, x" = The numbers are 17 and 23.
17, and 40
x' = 17, 40-x' = 23.
6. The difference of two numbers is 9; and their sum, multiplied by the greater, is equal to 266. What are the numbers ? Ans. 14 and 5
7. The sum of two numbers is 73, and their pro duct 732. What are the numbers? Ans. 61 and 12.
8. The sum of two numbers is a, and their pro duct is 5. What are the numbers?
9. A person travels 48 miles at a uniform rate; a second person travels the same distance two hours sooner, and travels 2 miles per hour more than the first. At what rate did the first travel? Ans. 6 miles.
10. A and B start at the same time to travel 150 miles. A travels 3 miles an hour faster than B, and finishes his journey 8 hours before him. IIow many miles does A travel per hour. Ans. 9 miles.
11. A hollow cubical box, whose sides are three inches thick, requires for its construction 27 cubic feet of material. How many cubic feet of water will it con tain? Ans. 64 cubic feet.
12. Two square courts are paved with stones a foot square; the larger court is 12 feet longer than the smaller one, and the number of stones in both pavements is 2120. How long is the smaller pavement? Ans. 26 feet.
13. A person distributes 120 dollars amongst a certain number of people. The next day he distributes the same sum amongst a number of people greater by 2. Each of the latter receives 2 dollars less than each of the former. How many were there in each case? Ans. 10 and 12. 14. A and B set out to meet each other, being