14. The greatest common divisor of several numbers is the product of all the prime factors common to those numbers.

15. Find the greatest common divisor (G. C. D.) of 140, 112, and 98.

41. A fruit-grower has 105 trees of one kind, 90 of another kind, 60 of another kind, and 45 of another kind. He wants to plant them at an equal distance apart, in rows of equal length. What is the largest number he can put in a row? How many rows will there be in each case?

42. A man has one lot 96 feet long and 48 feet wide, and another lot 144 feet long and 64 feet wide. He wishes to inclose them with a fence of boards of equal length, and the longest that can be used without cutting. What must be the length of the boards? How many boards will it take in each case if he makes the fence 5 boards high?

43. A farmer has 231 bushels of rye, 273 bushels of wheat, and 315 bushels of oats which he wishes to put into the least number of bins containing the same number of bushels without mixing the grain. How many bushels must each bin hold? How many bins must there be?

44. Three men having $434, $465, and $620, respectively, bought cows at the highest price per head that allowed each man to invest all his money. How many dollars did they pay per head? How many cows did each buy?

FRACTIONS.

91. 1. What is a primary unit? What is a relative unit? (Review Sections 2, 3, and 4, pp. 9 and 10.)

Write a figure 1 to stand for a pint. Draw a short horizontal line beneath it. Under the line write the figure which tells how many pints make a quart. How many ones does the writing stand for? Is the 1 a primary or a relative unit? How many of this kind make a larger one of another kind? What is the primary unit to which the 1 is related? What part of a quart is the 1 pint?

3. Write a figure 2 to stand for feet. Beneath it, with a line between, write the figure which tells how many feet make a yard. How many ones does this writing stand for? Are they relative or primary ones? How many of this kind make a larger one of another kind? What is the primary unit to which they are related? What part of a yard are the 2 feet?

4. Fold a sheet of paper into two equal parts. Now fold each part into two equal parts. Write a 3 to stand for three of the equal parts. In the proper place write the figure which tells into how many equal parts the whole sheet is folded. How many ones have you expressed by the figures? What kind? What is the primary unit? What part of the whole sheet are the 3 parts expressed?

5. Write a 7 to stand for dimes, and beneath it write the figures which tell how many dimes make a dollar. What is the primary unit? How many relative units? What figure tells you? What part of the primary unit have you expressed? How do you know?

6. Write 15 to stand for hours, and in the proper place write the figures which tell how many hours make a day. How many units have you expressed? What kind? What is the primary unit? What part of the primary unit does the writing express?

tions.)

What are such expressions as these called? (Frac

8. A written fraction is an expression for one or more relative units smaller than some primary unit to which they are related.

9. The units of a fraction may be called fractional units because they may be considered as parts of their related primary unit; but, as units, they are just as whole and complete as any others.

NOTE: A number composed of fractional units does not differ in any essential particular from the same number composed of any other kind of units. For instance, seven is the same number, possessing the same properties and subject to the same uses, whether it be seven apples, seven hundreds, seven thirds, or seven tenths.

10. In the fractions,,, and, what are the figures which tell the number of units in each case? What are they called? What are the figures which tell how many of the relative units in each case make the primary unit? What are they called?

11. The numerator (numberer) is the figure or figures which tell the number of fractional units.

12. The denominator (namer) is the figure or figures which tell how many relative units of this kind are required to make the primary unit to which they are related.

The numerator and denominator are the terms of the fraction.

13.

14. A fractional expression may also signify an unexecuted division; the numerator expresses the dividend, and the denominator expresses the divisor. Thus, 9÷3, or of 9; 12-19÷6, or of 19, etc.

EXERCISES.

92. Read the following fractions. In each case tell: First. How many relative units are expressed.

Second. How many of these relative units make the

1. Four ninths. (How many? Four.

Ninths. Therefore, write 9 under the 4- thus,

what? .)

2. Two fifteenths; eleven twelfths; twelve nineteenths;

one twentieth. (How many?

what?)

3. Ten thirty-seconds; fourteen fifty-sevenths; six elevenths; twenty-one twenty-fifths.

4. Seventeen seventy-thirds; eight ninety-firsts; thirtythree eighty-sixths; fifty one hundred seventeenths.