CHAPTER IV

PROBLEMS INVOLVING LITERAL EQUATIONS

191. Formulae and Rules.-Represent the given number in a problem by letters, then its solution will be an expression involving these letters and will include all problems of its particular form. Such an expression is called a formula, and the translation of this formula into words is called a rule.

Thus for example:

I. Find two numbers whose sum is s and whose difference is d.

then

Let

the smaller number;

x+d = the larger number.

But the sum of the two numbers is s.

Since these formulae hold true for all values of the numbers s and d, the following rule for finding two numbers when their sum and difference are given, can be formulated.

RULE: The greater number is found by adding the difference to the sum and taking one half the result.

The smaller number is found by subtracting the difference from the sum and taking one half the result.

II. A can do a piece of work in a days, and B can do the same work in b days. In how many days can both together do the work?

Let

the number of days required. If A can do the work in a days, he can do of it in one day; and similarly B can do

1

a

1

of

the same work in one day. Therefore working together they can do in one day

x

but if both together can do the work in a days they can do of it in one day.

The translation of this formula will give a rule for finding the time required by any two agents working together to produce a given result, if the time that it takes each separately to produce it, is given.

RULE. The time required by any two agents to produce a given result is the quotient found by dividing the product of the numbers which express the time in units required by each to produce the result, by the sum of the numbers.

REMARK.-Compare problem 26, §187.

III. A person has just a hours at his disposal. How far may he ride in a coach which travels 6 miles an hour, so as to return home in time, walking back at the rate of c miles an hour.

Let

Ꭺ

and

the distance AB in miles which he is to ride in the coach and to return by foot:

= the time required to travel from A to B in the coach,

с

the time required to walk from B to A.

Therefore the total time required to ride out to B and walk back

RULE. Divide the product of the time at disposal expressed in any unit (a), the number of miles the coach can travel in that unit (b) and the number of miles he can walk in the same unit (c) by the sum of the rates of travel (b + c).

IV. One man asked another what time it was and received the answer that it was between n and (n + 1) o'clock and the hour hand and the minute hand pointed in opposite directions. What was the

time?

At n o'clock the minute hand points to 12 and the hour hand to The hour hand is therefore 5n minute divisions in advance of the minute hand.

n.

If n =

1, it was

(35)

38

If n =

3, it was

(45)

=

5, it was 1

11

(55)

If n

In case n is 7

=

49

12

(5n+30).

min. past 1 o'clock.

min. past 3 o'clock.

60 min. past 5 o'clock or 6 o'clock.

or any integer between 6 and 12, then a little

care will show that a will have the value,

If n = 8 it was 10 minutes past 8 o'clock. connection compare problem 49, 187. For the case n = 3, compare problem 52, of the same set of examples.

V. A train, starting from a point A, travels m miles per hour; a second train, starting from a point B, p miles behind A, travels in the same direction n miles daily. After how many hours will the second train overtake the first, and at what distance from B will the meeting take place?

It is assumed that n> m.

Let x = number of hours after which the trains meet; then me number of miles travelled by the first train = and nx = number of miles travelled by the second train,

But BA BC AC,

x

AC,

= BC.

the number of hours after which the trains meet.

The distance travelled by the first train is

mx

их

point of meeting

Solve the following problems:

1. Find a number which added to m gives a sum equal to n times the number.

Let m = 10, n = 11.

2. Divide a into two parts so that of the first plus of the second shall be equal to b.

4. A man's age and his wife's age now have to each other the ratio nm; but r years from now they will have the ratio of pq. How old are they now? (Compare problem 17, 187.)

5. The sum of two numbers is m, and the quotient formed by dividing the less by the greater in. What are the numbers?

6. A and B can do a piece of work in m days, A and C in n days, and B and C in p days. In what time can they do the work all working together?

7. A passenger train, going from Boston to Portland at the rate of m miles an hour, occupies h hours less time than a freight train at 27 miles an hour. Find the distance from Boston to Portland. 187.)

(Compare problem 31,

8. Two towns, A and B, are a miles apart. One person sets out from A and travels toward B at the rate of b miles an hour; at the same time another person sets out from B and travels toward A at the rate of c miles an hour. How many miles from A will they meet? (Compare problem 42, 187.)

9. A merchant adds yearly of his capital to it, but takes from it at the end of each year d dollars. At the end of the third year after deducting the last d dollars he has of his capital left. his original capital. (Compare problem 41, 187.)

Find

10. A asked B what time it was, and received the answer that it was between n and n + 1 o'clock, and the hour hand was directly under the minute hand. What time was it?

11. A was employed a days on these conditions: for each day he worked he was to receive b dollars, and for each day he was idle he was to forfeit c dollars. At the end of a days he received d dollars. How many days did he work? (Compare 48, 187.)

12. A has m dollars and B has n dollars. A gives to B a certain number of dollars and has left 9 times as many dollars as B. How much money did B receive from A?

13. A can do a piece of work in 2 m days, B and A together in n days, and A and C together in m+ days. In what time will they do it working together?

14. A broker invests of his capital in a% bonds, and the remainder in % bonds; his annual income is c dollars. Find the amount in each kind of bond, and the sum invested.

15. A banker has two kinds of coin: it takes m pieces of the first kind to make one dollar, and n pieces of the second kind to make a dollar. A person wishes to obtain pieces for a dollar. How many pieces of each kind must the banker give him?