PROBLEMS FOR COMPLETION

1. A man bought three Treasury Savings Certificates and paid $253.80 for them. Complete the problem and solve it.

It usually happens that a problem may be completed in various ways. The pupil should be encouraged to make any reasonable problem out of what is given, so long as he solves that problem correctly.

2. Thrift Stamps cost 25¢ each. With 16 Thrift Stamps and 23¢ it is possible to buy one Government Savings Stamp in a certain month. A man has $5 to invest. Complete the problem and solve it.

Complete each of the following problems and solve it:

3. A man has $1950. He wishes to buy Treasury Savings Certificates at $846 each.

4. A man takes his family on an automobile trip. The total distance to be covered in 4 da. is 548 mi. The first day they go 142 mi.; the second day, 131 mi.; the third day, 1363 mi.

5. The expenses of a certain school last year amounted to $1583.15. This included four items: the teacher's salary, $146.90 for books and supplies, $139.50 for fuel and lights, and $96.75 for miscellaneous expenses.

6. A certain school is in session from a quarter before nine to noon, and from a quarter past one to a quarter before four. 7. Kate is knitting a sweater. She needs 2 hanks of yarn. For a smaller sweater she needs 21 hanks.

8. A certain baseball team has won 16 games and lost 13, and another team has won 14 games and lost 11.

Teachers should encourage the frequent use of Problems for Completion, particularly such as relate to local conditions. The pupils should be asked to make such problems for use in class contests.

1. Ralph's father explained the meaning of grading cows, keeping their records for milk, and disposing of those with poor records. The following is a record of twelve herds of the same size:

Ralph calculated to the nearest cent the average annual cost of the food and the profit per cow for group I. He then did the same for group II. Find the results in each of the two cases.

WASTE AND SAVING

1. Mrs. Stevens finds that her cook is wasting one slice of bread every day. If a loaf of bread is cut into 12 slices, in how many days will she waste 3 loaves of bread?

2. Mrs. Campbell finds that if she buys potatoes during the winter, she pays an average of $3 a bushel for them. By buying 10 bu. in the fall she can get them for $1.50 a bushel. Find how much she saves by buying this amount in the fall, even if she loses 1 bu. by their rotting during the winter.

3. Mrs. Brown decided to save money by buying in larger quantities. She made a list of a few items, showing what she had been paying and what the grocer would charge for the same goods in the quantities stated. If she can use the larger amounts, find how much she can save on the following order:

4. By the use of a fireless cooker Mrs. Brown reduced her gas bills for January, February, and March from a total of $8.95 for these months in the previous year to $7.25. At this rate how much will she save if she uses the cooker all the year?

5. In July a coal dealer telephoned Mr. Field that he could buy a winter's supply of coal for $11.60 a ton. Mr. Field waited until October, however, and then paid $12.85 a ton for the same grade of coal. He required 16 tons. How much could he have saved by buying his coal in July?

The pupils are not yet prepared to consider the question of interest on the money, but should simply solve in accordance with what is given.

CHAPTER II

I. COMMON FRACTIONS

USING WHAT YOU HAVE LEARNED

1. In a relay race the first boy of one team ran his lap in 14 sec., the second boy in 138 sec., the third in 14 sec., and the fourth in 13 sec. What was the total time of the race?

2. Miriam has a piece of goods 2 yd. long to dress her two dolls. She uses yd. for one doll and yd. for the other. With the rest she makes a jacket for the larger doll. How much does she use for the jacket?

3. Making no allowance for doors and windows, how many feet of picture molding will be needed to go around a classroom 22 ft. long and 181 ft. wide?

4. Malcolm makes a box to put under a shelf in a space 211 in. high. The wood he has is in. thick. If he needs a space of 31 in. between the cover and the shelf, what must be the inside height of the box?

5. A man spent $60 in visiting a certain city. Of this amount $22 was for railroad fare, $8 for sleeping-car tickets, and $30 for hotel bills and other expenses. What part of the money did he spend for each?

In all such examples the results must be expressed in lowest terms. This page affords a brief review of the addition and subtraction of fractions as studied in Chapter I. We shall now take up the multiplication and division of fractions.

Cancellation. On page 48 the work in canceling common factors was sufficient to meet the need at that time. It will, however, be found helpful to learn a little more about canceling.

Greatest Common Divisor. The greatest factor common to two or more numbers is called their greatest common divisor (G.C.D.). For example, although 3 is a common divisor of 12, 18, and 48, the greatest common divisor is 6.

The greatest common divisor of such numbers as we have met or shall meet in fractions is easily found. For example, consider 12 and 30. We see that

12= 2 × 3 × 2,

30= 2 × 3 × 5,

and that 2 and 3 are the only common factors. Therefore 2 × 3, or 6, is the greatest common divisor.

The treatment of fractions has changed so greatly with the omission of those having large terms that the subject of G.C.D. has become practically obsolete in connection with this kind of work. The subject is included here for those who may care to use it, but it may safely be omitted.

Indeed, the whole question of the divisibility of numbers has no longer the practical value that it once had. The work on pages 74-76 should therefore be taken up as presenting some interesting properties of numbers rather than as being of great practical value.

It should be explained informally to the class that when we speak of a factor of a number, or of a common divisor of two or more numbers, or of the divisibility of numbers, we always refer to exact division of integers.

Even Number. A number that is divisible by 2 is called an even number. For example, 20, 38, and 1112 are even numbers.

Odd Number. A number that is not even is called an odd number. For example, 21, 39, and 127 are odd numbers.

Such definitions as these are given as matters of general information rather than as being essential to the reduction of fractions. They are related to the divisibility of numbers now to be considered.