Problems G. C. D. and L. C. M.

1. One boy's blocks are 2 inches thick, another's 3, and another's 5. The three boys build "towers" of equal heights. How high at least are they? How many blocks does each one use?

2. What is the least sum that can be paid in either 2, 3, 5, 10, 20, or 25¢ pieces?

3. William has 274, Mary 364, and Harry 514, not in one-cent pieces, yet all in coins of one denomination. What is it?

4. In one grammar school there are 504 girls, in another there are 324 boys. It is desired to divide them into classes of equal size. How many pupils will there be in each class, if as large as it can be made?

5. The four sides of a play-ground measure, respectively, 464, 672, 368, and 240 ft. in length. How long must the boards used in fencing it be cut, so that they shall be of equal length, and as long as possible?

6. On the same day a merchant sends out traveling salesmen with instructions to return, respectively, in 1, 2, 3, 4, and 5 weeks. When any one returns he is sent out again at once for the same period as before. In how many weeks will they be together again?

7. In how many weeks would they come in together if sent out for 6, 9, 12, 18, and 36 weeks, respectively?

8. What is the least sum a dealer in live stock must have to be able to invest equal sums in horses at $105, mules at $68, and beeves at $30 per head? How much if he pays $105 per head for horses, $70 for mules, and $30 for beeves?

9. A court-yard 42 ft. 6 in. long, and 31 ft. 8 in. wide, is to be paved with square tiles of equal size, and as large as possible. How long and wide must each tile be?

10. A man having on deposit $695, $417, and $1251, respectively, in three different banks, wishes to draw out the whole in as large equal sums as possible. What is the greatest sum for which he must draw his checks?

124. Draw on slate or paper twelve lines of equal length, and about one half inch apart. Divide and subdivide the lines as required by the questions.*

1. If the first line were divided into two equal parts, what would you call each part? How many such parts in a line? How many in 2, 5, 7, 9, 12 lines?

2. How many

halves in 2 lines and a half? In 3 lines? In

4 lines and a half? In 11 lines and a half?

3. If each half were divided into two equal parts, how many of the new parts would there be in a whole line? What would you call one part? Two parts? etc. What part of a half is a fourth? 4. If the other lines were divided in the same way, how many fourths in each line? In 3 lines? In 8 lines? etc.

5. How many fourths in 2 lines and 1 fourth? In 5 and 1 fourth? In 6 and 3 fourths? In 7 and 1 fourth? etc.

6. Are 2 halves less or greater than a line? 5 fourths? etc. 7. How many whole lines in 2 halves? 4 halves? etc. many in 3 fourths? 5 fourths? 9 fourths? etc.

How

8. If each fourth were divided into two equal parts, how many parts would there be in a line? What would you call them? (Other questions should here be asked, similar to those on fourths, as above.)

*Slips of paper of any uniform length, paper squares, etc., etc., are convenient

materials for these exercises.

Draw 12 other lines. It would be well if these could be just 12 inches long.

1. If each line were divided into three equal parts, what would the parts be called? Why?

2. How would you express 1 part in figures? 2 parts? 5 parts? (For reading and writing simple fractions see Art. 73, p. 77.)

3. Are greater or less than a whole line? How much? How many thirds in 3, 5, 7, 12 lines?

4. How many thirds in 23, 41/3, 52/3, 71/3, 83, 103?

5. How many whole lines, or how many lines and what parts of a line, are needed to make 7/3, 9/3, 14/3, 21/3, 15/3, 30/3, 32/3?

6. If each third were divided into two equal parts, how many of the new parts would there be in a whole line? What would you call them? How would you express five of them in figures?

7. How many of these smaller parts are there in a third? In

2 thirds? In 3 thirds? How many sixths in 3 thirds?

8. Which is greater, 2 or 2% of one of these lines? Why? What part of a third is 1 sixth? Is a part or the whole of ? 2% 9. How many sixths in 1/3 of a line? In ? In ? In 2

lines? In 6, 8, 10, 11 lines?

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10. How many sixths in 1? In 2%? In 4? In 7% ? In 8%? In 8%? In 73 ?

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11. How many lines, or lines and parts of a line, are needed to make 18%, 216, 24/6, 36/6, 39/6, 42/6, 49/6, 54/6, 58/6 ?

12. If each sixth were divided into two equal parts, what would the new parts be called? How would you express one or more of them? If the line is 12 inches long, what is the length of each part?

13. How many of them in 1% of a line? In 3, 6, 1/6, 1/3, 3/6? 14. How many twelfths in 2 lines and 12? In 3/12? In 4/12? In 712? In 10/12? In 111/12? How many twelfths in 1, 21/6, 45/6, 7/12, 923 lines?

Questions upon the Rules in the Margin.

125. Note. The following questions are designed to be only suggestive of exercises that may be given. A foot-rule or a yard-stick will afford many others.

1. Into how many parts is the first of these two measures divided by the horizontal line in the middle? What do you call the parts? Into how many parts is the measure at the right divided by the longest horizontal lines? What do you call these parts? Why?

2. Which is greater, 1⁄2 or ? How can you tell without seeing or measuring the parts?

3. What parts of the whole do you get by dividing into 2 equal parts? Why?

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4. How many sixths in 1/3, 3, 33? What part of 12 do you get by dividing it into 2 equal parts ? What part of the whole is 1/2 of 1/2 ?

5. How many parts do you get by dividing each of the fourths into 2 equal parts? What are these new parts called? Why?

6. Which is longer, 1% or 1% ? Suppose you could not see, nor measure, would you know which is the greater, 3% or 3? How?

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7. The sixths in the second measure are divided each into 2 equal parts. What is their name? Why?

8. Are these twelfths as large as the eighths in the other measure?

9. What are the smallest parts of the second measure? How many are there?

10. Are the smallest parts of the first measure as large as the smallest parts of the second measure? Can you tell by counting them?

ORAL EXERCISES.

1. Explain how it is that 1/2 is equal to

Note.-Divide any whole thing or number into fourths, and show that one half is equal to 2 fourths.

2. Is

equal to 3%, to %, to /12, to 12/24? State why.

3. Name some other parts equal to 4; also parts equal to %, to 1/8, to 1/3, to 1/12, to 2/3, to 3/4, to 5%, to 5%, to %, to 3⁄48.

4. If you had a line divided into sixths, how could you change the sixths into twelfths? The twelfths back to sixths?

6. How many sixths in 1/3, 3?

6. How many twelfths in 16, 13,

How many eighths in 4, 1⁄2?

1/4, 1/2? In 2/2, 2/3, 2/492/6?

7. How many twenty-fourths in ? In %? In? How many in 2%, 4/6, 36, 56? How many in 199 3⁄4?

Note.—The following representation of a fraction rule will suggest other exercises. The figures at the left show how many parts each side is divided into.

12. How many wholes and ninths are in 159, 29, 21/9, 30/9, 47/9 P 13. Which makes the larger parts, dividing an apple into 10ths or 12ths? Which is the greater, 1% or 1/12 of a thing? 1/20 or 1/16? 1/30 or 1/31? 1/4 or 1?

14. Which is the greater, 24 or 17/24? / or /? 73/145 or 29/145? 5/8 or 3/8? 7/18 or 10/18? Why?

16. Draw two lines of equal length. Divide one into thirds, the other into fourths, and find how many more twelfths there are in 1% than in 14.