20. A class is to be formed having twice as many boys as girls. If there are to be 30 pupils in the class, find the number of girls.

21. The sum of three numbers is 80. The second is twice as large as the first and the third is five times as large as the first. Find the three numbers.

47. What every pupil should know and be able to do. In the unit just completed the meaning of formula and equation has been developed. The pupil is now expected:

1. To use correctly the expressions: polynomial, monomial, binomial, trinomial, term, similar terms, value of a literal number, coefficient, formula, equation, polygon, triangle, quadrilateral, pentagon, hexagon, vertex, side, perimeter.

2. To understand the arithmetical, geometrical, and algebraic forms of representing numerical facts, i.e., the representation by table, graph, and formula.

3. To know the meanings and names of the symbols (), [], { }.

4. To find the value of expressions of the form 6x; a+b+c 4-2a

a+b+c; 2x+3x+x;

; where the values of the

literal numbers are whole numbers, common fractions, or decimal fractions.

5. To solve equations of the form 5x = 12; 2x+7x=4.

6. To solve simple problems leading to equations of the forms given in 5.

7. To know the following laws:

a. Law of order in addition.

b. Division axiom.

48. Typical problems and exercises. Pupils should be able to work problems and exercises of the type given below:

1. The price of oranges of a certain size is 50 cents a dozen. Find the price of 2, 3, 4, ...., 10 dozen.

Represent these facts in tabular form; in graphical form; in a formula.

Show how to obtain the facts in the table from the formula. From the graph determine the price of 42 doz., 6 doz., 8 doz.

2. Translate into words the statement i= 12ƒ.

3. The perimeter of an equilateral pentagon is 35 inches. By means of an equation find the length of each side.

4. John solved twice as many problems as Mary, and James solved three times as many as Mary. Together they solved 36 problems. How many did each solve?

5. Draw a triangle and find the perimeter by measuring each side and then adding the results; by drawing the sum and then measuring the sum.

Find the value of each of the following, if a=6, b, c = 1.2. 6. ja+a+a.

Solve the following equations:

10. 45x120.

11. 2.12n=62.1.

12. 5x+4x-x=70.

13. State the following laws:

The division axiom.

The law of order in addition.

14. Write a paper on one of the following topics: a. The value of the formula.

b. The equation as a tool for solving problems.

CHAPTER IV

A STUDY OF ANGLES

HOW ANGLES ARE USED

49. Meaning of angle. After studying the line segment, which is the simplest geometric figure, we are now able to extend our study of geometry to more complicated figures made up of several line segments. A figure formed by two lines (Fig. 70) is called an angle. The word comes from the Latin

angulus, meaning corner.

More precisely, we may say that an angle is a figure formed by two straight lines, as BA and BC, meeting

FIG. 70

C

at the same point (Fig. 70). The straight lines are the sides, arms, or legs of the angle; the point, B, where the sides meet, is the vertex.

EXERCISES

1. Angles are found all about us. Point out some angles in the classroom, and the sides and vertex of each.

2. Point out several angles outside of the classroom.

3. Draw an angle on the blackboard.

4. Make an angle by folding a sheet of paper.

5. Draw a polygon and point out the angles in it.

50. Uses of angles. A knowledge of angles is important. Builders and architects use angles in plan

ning and constructing our homes. We shall learn how the surveyor finds unknown distances which he cannot

measure directly, such as the height of a tree (Fig. 71), the height of a tower or mountain, or the width of a river (Fig. 72). It will be shown how the astronomer, by a knowledge of angles, determines the position of the stars and planets, and determines the exact time, and how a knowledge of angles enables the navigator guiding his ship along the coast, to avoid hidden and dangerous rocks (Fig. 103). Engineers, designers, artists, and many others, make use of angles in doing their work.

FIG. 71