16. Make a scale drawing of the floor plan (Fig. 58).

17. A department store reports the sales and profits for a period of five years as follows:

18. On a normal business day the average number of passengers carried in and out of Chicago by the suburban trains of the Chicago and North Western Railway is approximately 60,000.

The number of trains necessary to handle this business is 166. On August 1st, the first day of the street car strike, the suburban trains of this company carried 109,810 passengers.

The number of trains required for this business was 225.

On August 2nd the number of passengers carried was 134,704. The number of trains operated, 245.

On August 3rd the number of passengers carried was 135,619. The number of trains operated, 243.

On August 4th the number of passengers carried was 134,822. The number of trains operated, 242.

Make tabulated statements of these facts as follows:

Represent by line graphs the passengers' increase and the train increase. Represent by a bar graph the number of passengers per train.

19. The following table shows the loss from absences in a high school for one year. Represent the facts in the table graphically.

Per cent of daily absences

2.1 2.5 3.3 5.4 4.7 7.1 5.8 3.5 3.1 4.34

28. What every pupil should know and be able to do. Having made a study of this chapter the pupil should understand how numerical facts may be represented arithmetically as in tables, or geometrically as in graphs. He should be able to tell what a given bar graph or line graph shows, and he should know how to make a graph from a given table of numerical facts.

29. Typical exercises. One who understands this second chapter should be able to work the following problems:

1. Draw a bar graph representing these numbers: 2.50, 3.41, 4.23, 1.68.

2. Some of the longest rivers we read about in the study of geography of the United States expressed in thousands of miles are approximately of the following lengths.

Missouri-Mississippi 4.2, Yukon 2.2, St. Lawrence 2.1, Arkansas 2, Rio Grande 1.8, Columbia 1.4, Colorado 1.4. Make a bar graph representing these approximate lengths.

3. The temperature record for a certain day reads as follows:

Hour.

Temperature.

6

a.m.

7 8 9 10 11

12 2 3 4 5 6

1 p.m.

16° 18° 19° 19° 21° 22° 24° 24° 20° 18° 18° 16° 12°

Make a line graph showing the changes in temperature on that day.

4. Write a paper on the meaning and uses of graphs.

CHAPTER III

REPRESENTING NUMERICAL FACTS BY

FORMULAS

HOW TO MAKE A FORMULA

30. A third method of representing numerical facts. In Chapters I and II we have studied two methods of representing numerical facts, the arithmetical by which facts were arranged in the form of tables, and the geometrical by which they were represented in geometric figures, e.g., in graphs. There is a third method, which we shall study in this chapter and which is sometimes the most convenient of the three. It represents numbers by means of letters. It is used not only in mathematics but in science, shop work, and engineering. The following exercises introduce this new method:

EXERCISES

1. On a sheet of centimeter squared paper lay the edge of a ruler, which is divided in inches, along one of the heavy lines. Using the centimeter as unit, measure carefully segments equal to 1 in., 2 in., 8 inches and tabulate the results as follows:

.....

Find the ratio of each number in the second row to the corresponding number in the first row. If this is done accurately, you

will see that the number of centimeters in a segment is approximately 2.54 times the number of inches.

The equation c=2.54i, stated in words, means the number of centimeters is 2.54 times the number of inches.

2. The price of oranges of a certain size is 30c a dozen.

a. Complete the following table which gives the price of oranges from 1 to 10 dozen.

Number of Dozen . .

1 2 3 4 5 67 8 9

10

Price in cents..

30 60

b. Make a line graph illustrating the facts given in the table above. To represent the price, let one side of a large square on the graphing paper represent 30.

c. From the graph determine the price of 42 doz.; 71⁄2 doz.; 81⁄2 doz. d. From the graph determine how many oranges can be bought for $0.60, $1.80, $2.10, $2.70.

e. In the table above compare by means of ratios the numbers in the second row with the corresponding numbers in the first row. What relation do you find between price and the number of dozens bought?

f. Express the results of (e) in one general statement, giving the price in terms of the number of dozens.

g. Let the number n represent the number of dozens, and the number p the price in cents. Show that the statement in (ƒ) may be expressed briefly in the form p=nX30, or p=30×n.

h. The statement p=30×n in words means: the number of cents is equal to 30 times the number of dozens.

i. When n=1, show that p=30X1=30.

j. Let n=2, and show that p=60.

k. Show how to obtain all of the facts stated in the table and in the graph from the brief statement p=30×n.

31. What is meant by the value of a literal number. In statements like p=30Xn, the literal number n may stand for any number, as, 1, 2, 3, etc. The