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19. If 1 third of a pie cost 4 cents, what will 2 thirds cost? What will a whole pie cost?

20. If 1 fourth of a pound of ginger cost 3 cents, what will 2 fourths of a pound cost? 3 fourths? What will a whole pound cost?

21. İf 1 eighth of a yard of cloth cost 2 shillings, what will 3 eighths cost? 5 eighths? 7 eighths? What will a whole yard cost?

22. If 1 third of a barrel of flour cost 3 dollars, how much will a whole barrel cost? How much will 5 bar

rels cost? 8 barrels ?

23. If 1 sixth of a hogshead of molasses cost 5 dollars, what will be the cost of a hogshead? Of 4 hogsheads? Of 10 hogsheads?

24. If 1 pound of sugar cost 12 cents, what will 1 half a pound cost?

Suggestion. If 1 pound cost 12 cents, it is plain that 1 half of a pound will cost 1 half of 12 cents. 1 half of 12 cents is 6 cents. Ans. 6 cents.

25. If one yard of ribbon cost 15 cents, how much will 1 third of a yard cost?

26. If one pound of tea cost 4 shillings, how much will 1 fourth of a pound cost? How much will 2 fourths cost?

27. If a ton of hay cost 15 dollars, how much will 1 fifth of a ton cost? How much 2 fifths? 3 fifths?

28. What will 1 tenth of an acre of land cost, at 30 dollars per acre? 2 tenths? 6 tenths?

29. What will I eighth of a ton of iron cost, at 48 dollars per ton? 3 eighths? 5 eighths? 7 eighths?

30. If bushel of corn cost 1 half a dollar, what will 2 bushels cost? 4 bushels?

Suggestion. If 1 bushel cost 1 half a dollar, 2 bushels will cost twice as much. 2 times 1 half are 2 halves, or a whole dollar. 4 bushels will cost 4 times 1 half, or 2 whole dollars.

31. If one man eats 1 half of a loaf of bread at a meal, how many loaves will 3 men eat?

32. How many whole ones are 4 halves equal to ? 5 halves 6 halves? 8 halves? 9 halves?

33. If I burn 1 third of a ton of coal in a week, how much shall I burn in 3 weeks? 4 weeks? 6 weeks? 10 weeks? 12 weeks?

34. How many whole ones in 4 thirds, and how many over? In 6 thirds? 8 thirds? 11 thirds? 14 thirds?

35. If a horse eat 1-fourth of a bushel of oats a day, how many will he eat in 6 days? In 8? In 10? In 12?

36. If a boy saws 1 eighth of a cord of wood in a day, how much will he saw in 6 days? In 12 days? In 15 days? In 24 days?

37. If 12 oranges were divided equally among 4 boys, what part of them would each boy receive; and how many oranges would each have?

Suggestion. 1 is 1 fourth of 4; hence, 1 boy must receive 1 fourth part of the oranges. 1 fourth of 12 oranges is 3 oranges.

38. A builder employed 6 men to do a job of work, for which he gave them 24 dollars: what part of the money did 1 man receive? What part did 2 receive? What part did 4 receive? How many dollars did one man have? How many did two have? Three? Four?

39 If 5 yards of cloth cost 40 dollars, what part of 40 dollars will 1 yard cost? 3 yards? 4 yards? How many dollars will one yard cost? 2 yards? 3 yards? 4 yards?

40. 2 is 1 third of what number?

Solution. If 2 is 1 third of a number, 3 thirds, or the whole number, is three times as many.

Or thus, 2 is a third of 3 times 2; and 3 times 2 are 6.

41. 4 is 1 fifth of what number? 1 sixth of what number? 1 third? 1 eighth? 1 fourth? 1 seventh?

42. 6 is 1 third of what number? 1 fourth? 1 seventh? 1 tenth 1 ninth ? 1 twelfth ?

43. 5 is 1 fourth of what number? 1 sixth? 1 eighth? 1 eleventh? 1 twelfth?

44. 8 is 1 seventh of what number? 1 sixth? 1 tenth? 1 ninth 1 twelfth ?

45. 4 is two thirds of what number?

Solution. First find 1 third. Now if 4 is 2 thirds, 1 third is 1 half of 4, which is 2; and 3 thirds is 3 times 2, or 6. Ans. 6.

46. 9 is 3 fourths of what number?
47. 8 is 4 fifths of what number?
48. 16 is 4 ninths of what number?
49. 20 is 5 eighths of what number?
50. 32 is 8 twelfths of what number?

The number

105. When a number or thing is divided into equal parts, these parts are called FRACTIONS. thus divided is called an Integer.

106. Fractions are divided into two classes, Common and Decimal. (For the illustration of Decimal Fractions, see Section VIII.)

107. Common Fractions are expressed by two numbers, one placed over the other, with a line between them. One half is written thus ; one third, ; one fourth, ; nine tenths,; thirteen forty fifths, 13, &c.

The number below the line is called the denominator, and shows into how many parts the number or thing is divided.

The number above the line is called the numerator, and shows how many parts are expressed by the fraction. Thus in the fraction, the denominator 3, shows that the number is divided into three equal parts; the numerator 2, shows that two of those parts are expressed by the fraction.

The denominator and numerator are called the terms of the fraction.

QUEST.-105. What are fractions? What is an integer? 106. Of how many kinds are fractions? 107. How are common fractions expressed? What is the number below the line called? What does it show? What is the number above the line called? What does it show? What are the denominator and numerator, taken together, called?

OBS. 1. The word fraction is of Latin origin, and signifies broken, or separated into parts. Hence fractions are sometimes called broken numbers.

2. Common fractions are often called vulgar fractions. This term, however, is very properly falling into disuse.

3. The number below the line is called the denominator, because it gives the name or denomination to the fraction; as halves, thirds, fifths, &c.

The number above the line is called the numerator, because it numbers the parts, or designates how many are expressed by the fraction.

108. When the numerator is less than the denominator, the fraction is called a proper fraction. Thus, 1⁄2, }, , &c., are proper fractions.

When the numerator is equal to, or greater than the denominator, it is called an improper fraction. Thus, 3, 4, ,, &c., are improper fractions.

12

A compound fraction is a fraction of a fraction; as of.

of

A mixed number is a whole number and a fraction expressed together; as 43, 2511.

A complex fraction is one which has a fraction in its

numerator or denominator, or in both; as

24 2

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109. Fractions, it will be seen, both from the definition and the mode of expressing them, arise from division, and may be regarded as expressions of unexecuted division, the numerator answering to the dividend, and the denominator to the divisor. (Arts. 67, 105.) Hence,

110. The value of a fraction is the quotient of the numerator divided by the denominator. Thus the value of is two; ofis one; ofis one third, &c. Hence,

111. If the denominator remains the same, multiply

QUEST. Obs. What is the meaning of the word fraction? What are common fractions sometimes called? Why is the lower number called the denominator? Why is the upper one called the numerator? 108. What is a proper fraction? What, an improper fraction? What, a compound fraction? What, a mixed number? What, a complex fraction? 109. From what do fractions arise? 110. What is the value of a fraction?

ing the numerator by any number, multiplies the value of the fraction by that number.

Since the numerator and denominator answer to the dividend and divisor, multiplying the numerator is the same as multiplying the dividend. Now multiplying the dividend, we have seen, multiplies the quotient, (Art. 83,) which is the same as the value of the fraction. (Art. 110.) Thus the value of 2. Multiplying the numerator by 3, the fraction becomes 18, whose value is 6, and is the same as 2×3.

112. Dividing the numerator by any number, divides the value of the fraction by that number. For, dividing the dividend, divides the quotient. (Art. 84.) Thus, =2. Now dividing the numerator by 2, the fraction becomes , whose value is 1, and is the same as 2÷2. Hence,

OBS. With a given denominator, the greater the numerator, the greater will be the value of the fraction.

113. If the numerator remains the same, multiplying the denominator by any number, divides the value of the fraction by that number. For, multiplying the divisor is dividing the quotient. (Art. 85.) Thus, 244. Now multiplying the denominator by 2, the fraction becomes , whose value is 2, and is the same as 4÷2.

114. Dividing the denominator by any number, multiplies the value of the fraction by that number. For, dividing the divisor, multiplies the quotient. (Art. 86.) Thus, 244. Now dividing the denominator by 2, the fraction becomes 24, whose value is 8, and is the same as 4x2. Hence,

OBS. With a given numerator, the greater the denominator, the less will be the value of the fraction.

QUEST-111. What is the effect of multiplying the numerator, while the denominator remains the same? Explain the reason. 112. What is the effect of dividing the numerator? Obs. With a given denominator, what is the effect of increasing the numerator? Explain the reason. 113. What is the effect of multiplying the denominator? Why? 114. What is the effect of dividing the denominator? Why? Obs. With a given numerator, what is the effect of increasing the denominator?

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