17. What is the quotient of 35 times the product of 48 and 119, divided by 14 times the product of 21 and 17 ?

18. If a dividend is the product of 68, 91, and 95, and the divisor is the product of all the prime numbers between 12 and 20, what is the quotient?

19. If we divide the product of 16, twice 16, and three times 16, by the product of the square of 2 taken three times as a factor, what is the quotient?

20. How many times is the product of all the prime numbers between 20 and 30 contained in the product of 203 and 230 ?

21. A man worked 4 weeks, 6 days a week, for a market gardener, and received for his pay 24 crates of berries valued at $2 a crate. How much did he earn a day?

22. How many dozen quart bottles of imported pickles at $6 a dozen must be given in exchange for 3 cases of olives, each containing 2 dozen bottles at $3 a dozen?

23. A grocer exchanged 16 boxes of soap, each containing 60 cakes at 4 cents a cake, for a certain number of cases of preserved fruit, 2 dozen jars to a case at 40 cents a jar. How many cases did he get ?

24. A man exchanged 9 firkins of butter weighing 56 pounds each at 30 cents a pound for several rolls of matting, 40 yards to the roll worth 42 cents a yard. How many rolls did he get ?

25. If 24 men, working 9 hours a day, can do a piece of work in 12 days, how many days will it take 18 men, working 8 hours a day, to do the work?

26. If 15 men can do a piece of work in 16 days, working 9 hours a day, how many days will it take 20 men working 12 hours a day to do the work?

27. A clerk working 48 weeks in a store, at $12 a week, receives the same amount that another clerk receives in 36 weeks. What is the weekly wages of the other clerk?

28. A dealer sells a shipment of 36 horses for $2700. He sells another shipment of 25 horses for the same amount. What is the average price per horse in each of the two shipments?

29. We find the volume of a rectangular box by taking the product of its three dimensions expressed in the same unit of measure. The dimensions of one box, expressed as abstract numbers, are 4, 5, and 6, and two dimensions of another box of the same volume are 3 and 10. What is the other dimension of the second box?

30. Expressed as abstract numbers, the dimensions of one box are 8, 12, and 15, and two dimensions of another box of the same volume are 6 and 30. What is the other dimension of the second box?

31. Expressing the dimensions as abstract numbers, a cellar is 20 long, 14 wide, and 6 deep. How deep must another cellar be that is 16 long and 15 wide, in order that the capacity of both cellars may be the same?

32. How many times as much earth must be removed from a cellar excavation 175 feet long, 80 feet wide, and 15 feet deep, as from a cellar excavation 150 feet long, 40 feet wide, and 5 feet deep?

33. How many times as much earth must be removed from a cellar excavation 160 feet long, 40 feet wide, and 9 feet deep, as from a cellar excavation 60 feet long, 40 feet wide, and 6 feet deep?

34. How many times as much will a box contain whose dimensions are 8, 27, and 35, as one whose dimensions are 7, 12, and 18?

CHAPTER X

COMMON FRACTIONS

183. Fraction. A number that shows what part or what number of parts of a unit is taken is called a fraction.

Fractions have already been studied in the Fourth Year. They are now reviewed and are studied more carefully.

184. Common Fraction. A fraction expressed by two numbers, one under the other with a line between them, is called a common fraction.

185. Terms of a Fraction. In a common fraction the number above the line is called the numerator; the number below the line is called the denominator; the two together are called the terms of the fraction.

In the fraction & the numerator is 3 and the denominator is 4. The denominator (namer) shows into how many equal parts the unit has been divided, and the numerator (numberer) shows how many of these equal parts have been taken.

186. Unit Fraction. A fraction whose numerator is one is called a unit fraction.

Thus is a unit fraction. It is also called a fractional unit.

187. Proper Fraction. A fraction whose numerator is less than the denominator is called a proper fraction.

For example, is a proper fraction.

188. Improper Fraction. A fraction whose numerator is not less than the denominator is called an improper fraction. For example, and are improper fractions.

189. Fraction as an Expressed Division. In this figure we see that of 1 inch equals of 2 inches, or the result of

dividing 2 inches by 3; that is, is the same as of 2, or 2 divided by 3. Therefore,

A fraction is an expressed division, the numerator being the dividend and the denominator the divisor.

190. Reduction of a Fraction to Lowest Terms. Since we may divide both terms of a fraction by the same number without changing the value of the fraction, we may continue this division until the terms are prime to each other.

If the terms of a fraction are prime to each other, the fraction is expressed in lowest terms.

For example, is expressed in lowest terms, but is not. Therefore, to reduce a fraction to lowest terms, cancel all the factors common to both numerator and denominator. For example, reduce to lowest terms.

Canceling 2, we have 1; canceling 3, we have .

191. Reduction of a Fraction to Higher Terms. By multiplying both terms of a fraction by the same number we may reduce a fraction to higher terms.

For example, reduce to sixtieths.

Here 60125, so that the denominator must be multiplied by 5 to make 60. Therefore we multiply both terms by 5, and }} = {{.

Therefore, to reduce a fraction to a fraction having a higher denominator, divide the required denominator by the given denominator and multiply both terms of the fraction by the quotient.

192. Reduction of an Integer to an Improper Fraction. An integer may be expressed as an improper fraction with any required denominator.

Reduce 3 to an improper fraction with denominator 7.

We may simply think of 3 as equal to . We have, then, the same case as in § 191. Multiplying both terms by 7, we have = 21.

Therefore, to reduce an integer to a fraction with a given denominator, multiply it by the given denominator and under this product write the denominator.