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16. Note the number of decimal places in each of the following expressions:

1. .4 = 4 tenths. (1 decimal place.)

2. .27 = 27 hundredths. (2 decimal places.)

3. .346 = 346 thousandths. (3 decimal places.)

4. .2758 = 2758 ten-thousandths.

5. .07286 = 7286 hundred thousandths.

6. .000896 = 896 millionths. (6 decimal places.)

7. .000,468,275 = ■ billionths. (9 decimal places.)

8. .000,000,000,462 = trillionths.

$. .000,000,000,000,527 = quadrillionths.

10. In any number of thousandths there are decimal

places.

11. In any number of millionths there are decimal

places.

12. In any number of billionths there are decimal

places.

13. In any number of hundredths there are decimal

places.

14: In any number of ten-thousandths there are

decimal places.

15. In any number of hundred thousandths there are decimal places.

Algebra—Notation.

17. Letters are used to represent numbers; thus, the letter a, b, or c may represent a number to which any value may be given.

18. Known numbers, or those that may be known without solving a problem, when not expressed by figures, are usually represented by the first letters of the alphabet; as, a, b, c, d.

Illustrations.

(a) To find the perimeter of a square when its side is given.

Let a = one side.* Then 4 a = the perimeter. Hence the rule: To find the perimeter of a square, multiply the number denoting the length of its side by 4.

(b) To find the perimeter of an oblong when its length and breadth are given.

Let a = the length. Let b = the breadth. Then 2a + 26, or (a + b) X 2 = the perimeter. Hence the rule: To find the perimeter of an oblong, multiply the sum of the numbers denoting its length and breadth by 2.

19. Unknown numbers, or those which are to be found by the solution of a problem, are usually represented by the last letters of the alphabet; as x, y, z.

Illustration. (a) There are two numbers whose sum is 48, and the second is three times the first. What are the numbers? Let x = the first number.

Then 'Ax- the second number,

and x -4- 3 x = 48. 4 x = 48. x = 12. 3a= 36. * That is, the number of units in one side. The letter stands for the number.

20. The sign of multiplication is usually omitted between two letters representing numbers, and between figures and letters; thus, a x b, is usually written ab; bx 4, is written 4 6. 6 ab, means, 6 times a times b, or 6 x a X b.

21. Exercise.

Find the numerical value of each of the following expressions, if a 8, b = 5, and c - 2:

1. a+b+c= 5. 2ab =

2. a + b c = 6. 3abc =

3. 2a + b + c= 7. 2ab + 5c =

4. a + 6-2c= 8. aZ> + 6e =

(a) Find the sum of the eight results.

22. Exercise.

Find the numerical value of each of the following expressions if a = 20, 6 = 5, and c = 2:

1. 3(o + 6) = « 5. a+ 6 =

2. 2(a - 6) = 6. (a + 6) + c = f

3. 4(a + 6 + c) = 7. (a + b) -*- 3c =

4. 2(a + b-c)= 8. (a + 26) + 2c =

(b) Find the sum of the eight results.

23. Exercise.

1. If 2x = 20, 2. If 5x = 40, 3. If 6x = 72,

a;=? a; =? a;=?

4. If 2a; + 3a; = 60, 5. If 3a; + 4a; = 56,

x = 1 x =?

* This means, 3 times the sum of a and 6.

t This means, the sum of a and b, divided by e.

[graphic]

24. A geometrical line has length, but neither breadth noi thickness.

Note.—Lines drawn upon paper or upon the blackboard are not geometrical lines, since they have breadth and thickness. They represent geometrical lines.

25. A straight line is the shortest distance from one point to another point.

26. A curved line changes its direction at every point.

27. A broken line is not straight, but is made up of straight lines.

1. The line AB is a .

2. The line CD is a .

3. The line EF is a .

4. The line FG is a .

5. The line EK is a .

6. The perimeter of a square is a line made up of

equal lines.

7. The perimeter of a regular pentagon is a line

made up of equal lines.

8. The circumference of a circle is a line, every

point in which is equally distant from a point called the center of the circle.

9. Imagine a straight line drawn upon the surface of a stovepipe. Can you draw a straight line upon the surface of a sphere?

28. Miscellaneous Review.

1. If a equals one side* of a regular pentagon, the perimeter of the pentagon is .

2. If b equals the perimeter of a square, the side of the square equals b s .

3. If a equals a straight line connecting two points and b equals a curved line connecting the same points, then a is than b.

4. Find the difference between two hundred seven thousandths, and two hundred and seven thousandths.

5. How many zeros in 1 million expressed by figures? 1 billion? 1 trillion?

6. How many decimal places in any number of millionths? billionths? trillionths?

7. How many decimal places in 25 thousandths? in 275 thousandths? in 4346 thousandths?

8. A figure in the second integral place represents units how many times as great as those represented by a figure in the second decimal place?

9. If a 6, b = 2, and d = 8, what is the numerical value of the following? 12 a + 3b 5 d.

10. John had a certain amount of money and James had 5 times as much; together they had 354 dollars. How many dollars had each?

Let x = the number of dollars John had.

Then 5 x - the number of dollars James had,
and x + 5 x = 354 dollars.
6 x = 354 dollars.

x =? 5 x =?

* The expression " o equals one side" means that a equals the number of units in one side. Remember that in this kind of notation the tetters employed stand for numbers.

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