31. The denominator of a certain fraction is greater by 2 than the numerator. If 1 is added to both the numerator and denominator, the fraction becomes. Find the original fraction.

32. Out of a certain sum a man paid a bill of $30, loaned of the remainder, and finally had left $56. How much had he at first?

33. The largest of three consecutive odd numbers is divided into the sum of the other two, the quotient being 1 and the remainder 9. Find the numbers.

34. The sum of the ages of a father and son is 80 years, but if each were 2 years older the son's age would be § the father's age. How old is each ?

35. A certain number is decreased by 12 and the remainder is divided by 4. If the resulting quotient is increased by 7, the sum is the same as if the original number had been increased by 7 and then divided by 3. Find the number.

36. A man gave of a certain sum to relatives, to each of two churches, to a library, and the remainder, $ 6000, to a hospital. What was the total bequeathed?

37. In a certain baseball game a total of 13 runs was made by both teams. If the winning team had made 2 more runs, and the losing team 3 less, the quotient obtained by dividing the winning runs by the losing runs would have been 5. How many runs did each team make?

38. The distance around a rectangular field is 96 rods, and the length of the field is the width. Find the length and the width of the rectangle, and the number of square feet it contains.

39. In eight games a certain fielder made 2 less runs than hits. If 5 times the number of hits he made is divided by the number of runs increased by 3, the quotient is 4. How many hits and how many runs did he make?

40. Find three consecutive even numbers such that 3 less than one half the first, plus 2 less than one half the second, plus 1 less than one half the third equals 15.

41. In going a certain distance an automobile moving 20 miles an hour required 3 hours less time than a second automobile making 16 miles an hour. What was the distance in miles?

42. A number is composed of two digits whose difference is 4. If the digits are reversed, the resulting number is the original number. Find the number.

43. A can run 10 yards in 1 second, B 8 yards in 1 second. If A gives B a start of 3 seconds, in how many seconds will A overtake B?

44. The length of a rectangle is 9 rods more than its width. If the length is increased by 6 rods and the width decreased by 3 rods, the area is unchanged. Find the length and breadth of the rectangle.

45. An automobile going 25 miles an hour is 40 minutes ahead of one going 30 miles an hour. In what time will the second automobile overtake the first?

46. At what time between 8 and 9 o'clock do the hands of a clock point in opposite directions?

47. A freight train goes from A to B at 15 miles per hour. After it has been gone 4 hours an express train leaves A for B, going at a rate of 45 miles per hour, and the express reaches Bhour ahead of the freight. How many miles is it from A to B?

48. In traveling a certain distance a train going 45 miles an hour requires 5 hours less time than an automobile going the same distance at 27 miles per hour. What is the distance between the two points?

CHAPTER XV

APPLICATIONS OF GENERAL SYMBOLS. REVIEW

STATEMENTS.

PHYSICAL FORMULAS.
EXPRESSIONS

DERIVED

THE GENERAL STATEMENT OF A PROBLEM

178. From the following illustrations it will be seen that when the given numbers of a problem are literal quantities, the statement and the solution result in a formula or general expression for that particular kind of problem.

Illustrations :

1. If A can mow a field in m days, and B can mow the same field in p days, in how many days can both together mow the field?

Let

Then

Also

Hence, +

and

Solving,

x= the number of days both working together require.

х

m

1

Ρ

1

m p

1 1

+ m p

= the portion of the work both together can do in 1 dạy.

= the portion of the work that A alone can do in 1 day.

= the portion of the work that B alone can do in 1 day.

=

the portion of the work both together can do in 1 day;

This expression is, therefore, a formula for finding the time in which two men whose individual ability is known, can, work

ing together, accomplish a given task. By substituting in this formula, any problem involving the same condition can be solved. For example:

A requires 4 days to do a certain task, and B requires 5 days for the same work. In how many days can both working together complete the work?

Here we have A's time alone (or m) = 4; B's time alone

(or p) 5. In the formula x = =

=

=

20

4.5
тр
m + p 4+5 9

the time in which both together can do it.

2. Divide the number a into two parts such that m times the smaller part shall be contained q times in the larger part.

To use this formula: Suppose we are required to divide 60 into two parts such that twice the smaller part shall be contained 7 times in the larger part. We have

Exercise 55

1. Divide the number c into two parts such that m times the larger part shall equal n times the smaller part.

2. Divide a into two parts such that the sum of th of the

1.

larger part and th of the smaller part shall be q.

r

n

3. The sum of two numbers is s, and if the greater number, 9, is divided by the less number, the quotient is q and the remainder r. Find the numbers.

4. If A and B can together mow a field in t days and B alone can mow the same field in b days, find the number of days that A working alone will require to do the work.

5. Show that the difference of the squares of any two consecutive numbers is 1 more than double the smaller number.

6. m times a certain number is as much above k as d is above c times the same number. Find the number.

7. A and B are m miles apart, and start to travel toward each other. If they start at the same time and A goes at a rate of k miles an hour while B goes at the rate of s miles, how far will each have gone when they meet?

8. When a certain number is divided by a, the quotient is c and the remainder m. Find the number.

9. The front wheel of a wagon is m feet in circumference, and the rear wheel n feet in circumference. How far has the wagon gone when the rear wheel has made r revolutions less than the front wheel?

10. A and B can together build a barn in r days, B and C the same barn in s days, and A and C the same in t days. In how many days can each alone build it ?