104. Change the following mixed numbers to improper fractions:

1. 21, 31, 5, 83, 44, 78, 93.

2. 12, 88, 10, 9, 11, 133, 77.

3. 9, 15, 11, 20, 331, 12, 93.

4. 161, 251, 22, 305, 20§, 154, 1241.

106. To Change Fractions to Equivalent Fractions having a Common Denominator.

6

1. Change, t, fo, fe, f4, 1, and to halves. How did you do it? Did you change the value of any of the fractions? Why not?

2. Can you change

back to 6ths? To 8ths? To 10ths? To 12ths? To 14ths? To 16ths? To 18ths? How is it done? Does it change the value of the fraction? Why not?

3. Change, f1⁄2, 18, 18, 14, and to thirds.

4. Now change back to 9ths; to 12ths; to 15ths; to 18ths; to 21sts; to 24ths. How did you do it?

5. Change, Tx, H, 15, 1, and to fourths. How did you do it?

6. Now change back to 8ths; to 12ths; to 16ths; to 20ths; to 24ths; to 28ths. How did you do it?

7. Change to 10ths; to 15ths; to 20ths; to 25ths. 8. Change and to 12ths; to 18ths; to 24ths; to 30ths.

9. Change and to 6ths; to 12ths; to 18ths; to 24ths.

10. Change and to 12ths; to 24ths; to 36ths. 11. Change and to 15ths; to 30ths; to 45ths. 12. Change,, and to 18ths;,,, and to 20ths.

Fractions which have the same denominators are called similar fractions.

Fractions which have not the same denominator are called dissimilar fractions.

The denominator of similar fractions is called a common denominator.

A common denominator of two or more fractions is any common multiple of their denominators.

The least common denominator of two or more fractions is the least common multiple of their denominators. (How is the least common multiple of several numbers found? See p. 97.)

EXERCISES.

107. 1. Change,,, and to equivalent fractions having the least common denominator.

EXPLANATION: The Least Common Multiple of the denominators=2×3×5×7=210. The denominator 15 of the first fraction contains the factors 3 and 5 only. Put with them the other factors 2 and 7. Why? Put these same factors with the numerator 7. Why? The first fraction then becomes. Proceeding

in the same manner with the other fractions, we find that the factors to be put with the numerators are 3 and 7, 2 and 5, 3 and 5, respectively, making the new fractions 53 40 1,1%, and 7%, respectively.

Change the following to equivalent fractions having the least common denominator:

27. Into what parts of the same size may,,,

,, and 19 of an apple be divided?

parts would there be in each case?

How many such

DECIMAL FRACTIONS.

109. 1. Define fraction. Numerator. Denominator.

2. Express the following in figures: One tenth; one hundredth; one thousandth; one ten-thousandth; two tenths; two hundredths; two thousandths; two ten-thousandths.

3. Review Sections 16-27, pp. 13-17.

4. What is the use of the decimal point? For what is the zero used? What is the effect upon the value of a figure when it is moved one place to the left?

5. Moving a figure one place toward the left makes its value ten times what it was before. Moving it two places toward the left makes its value one hundred times what it was before, and so on.

6. Express the following numbers in figures in a column, using the decimal point: One thousand; one hundred; one ten; one. The value of the 1 standing in hundreds' place is what part of the value of the 1 standing in thousands' place? The value of the 1 in tens' place is what part of the value of the 1 in hundreds' place? The value of the 1 in ones' place is what part of the 1 in tens' place? What, then, is the effect upon the value of the figure when it is moved one place to the right?

7. Moving a figure one place to the right makes its value one tenth of what it was before. This is just as true on the right-hand side of the decimal point as it is on the left-hand side. Moving the 1 to the first place at the right of the decimal point, thus, .1, makes its value one tenth of 1.; moving it to the second place, thus, .01, makes its value one tenth of .1; moving it to the third place, thus, .001, makes its value one tenth of .01, and so on.

8. If the value of .1 is one tenth of 1., how shall we read .1? Why? If the value of .01 is one tenth of .1, how many like .01 are required to make 1.? How, then, shall we read .01? If the value of .001 is one tenth of .01, how many like .001 are required to make 1.? How, then, shall we read .001? How shall we read .0001? Why?

A

9. A figure in the first place at the right of the point signifies tenths, because it signifies relative units of which it takes ten to make one primary unit. figure in the second place at the right of the point signifies hundredths, because it signifies relative units, of which one hundred are required to make one primary unit. In the same way, the name of the third place at the right is thousandths, of the fourth place is ten-thousandths, etc.

10. In how many ways, then, can one tenth be expressed in figures? One hundredth? One thousandth?

11. Express in both these ways, two tenths; two hundredths; two thousandths; two ten-thousandths; two hundred-thousandths; seven tenths; seven hundredths; seven thousandths; seven ten-thousandths; seven hundredthousandths.