QUESTIONS. What does the denominator of a fraction show? What does the numerator show?

What is meant by the expression ? Ans. 5 of the 8 equal parts into which a unit is divided.

What is meant by the expression ? ? ? ?

TO

118. If we compare common fractions with the last expression for division in Art. 55, we shall see that their forms are alike. A fraction implies division, the numerator being the dividend, and the denominator the divisor. Thus, & may be considered either three fourths of 1 or one fourth of 3. The following diagram will show that these are equivalent expressions, of the one line in figure 1 being equal to of the three lines in figure 2.

What does

denote? Ans. It denotes either 5 of the seven equal parts into which 1 is divided, or one seventh of 5.

What does

show?

119.

?1?

GENERAL PRINCIPLES.

NOTE.-The following propositions should be copiously illustrated by the teacher, and frequently referred to, until they are fully comprehended by the pupil.

PROPOSITION I. As the denominator of a fraction shows the number of parts into which a unit is divided, and the numerator shows how many parts are taken, it follows that if we multiply the numerator of a fraction by a whole number, we multiply tha number of parts, and thus increase the value of the fraction; but if we multiply the denominator of a fraction, we multiply the number of parts into which a unit is divided, and thus diminish the size of the parts, and consequently decrease the value of the fraction.

PROPOSITION II. If we divide the numerator of a fraction by a whole number, we divide the number of parts and thus diminish the value of the fraction; but if we divide the denominator of a fraction, we divide the number which shows into how many parts the unit is divided, and thus increase the size of the parts, and consequently increase the value of the fraction.

PROPOSITION III. If we multiply the numerator and denom, inator, each by the same number, we increase the number of parts of the fraction, but diminish their size in the same proportion; consequently the value of the fraction is not altered.

3 4

ILLUSTRATION.

3 X 2 6
4X

Fraction not altered
in value.

PROPOSITION IV: If we divide the numerator and denominator, each by the same number, we diminish the number of parts in the same proportion as we increase their size, consequently the value of the fraction is not altered.

ILLUSTRATION.

4

4÷2 2
6 ÷ 2

Fraction not altered in value.

QUESTIONS. How does multiplying the numerator of a fraction affect the value of the fraction? Why? How does multiplying the denominator affect the value of the fraction? Why?

How does dividing the numerator of a fraction affect the value

67

of the fraction? Why? How does dividing the denominator affect the value of the fraction? Why?

If, then, you multiply the numerator and denominator each by the same number, what is the effect upon the fraction? Why?

If you divide the numerator and denominator each by the same number, what is the effect upon the fraction? Why?

REDUCTION OF FRACTIONS TO LOWEST TERMS. 120. A Fraction is expressed in its lowest terms when the numerator and denominator are prime to each other.

121. ILL. EX.

Reduce to its lowest terms.

OPERATION.

8

4 X 2

4

Ans.

10

5X2

5

8=4X2; 105 X 2. Dividing the numerator and denominator each by striking out the common factor 2, the value of the fraction will not be altered (Art. 119,

Prop. IV.), and will be expressed in its lowest terms. Hence the

RULE. To reduce a fraction to its lowest terms: Remove from the numerator and denominator all their common factors.

NOTE I.— 1, being a factor of all numbers, will remain when all other factors are struck out, as in the numerator of example 13.

NOTE II. In case the factors of the numerator and denominator cannot readily be ascertained, find the G. C. D. of the two terms, and divide each of them by it.

Reduce to their lowest terms,

CANCELLATION.

123. Cancellation consists in rejecting equal factors from dividend and divisor.

124. All arithmetical operations in division may be ex pressed in the form of a fraction, the dividend being the numer ator, and the divisor the denominator; since dividing both terms of a fraction by the same number does not alter its value, it follows, that we may strike out, or cancel, any factors common to the dividend and divisor without changing their relative value.

N. B. All operations upon arithmetical quantities should first be expressed, as far as possible, by signs, that the processes may be clearly indicated to the teacher, and that the work to be done may be reduced, if possible, by cancellation.

ILL. EX. Divide 3 times 4 times 6 times 5 times 7, by 2 times 8 times 6 times 9 times 10.

Express, cancel, and perform the following:

1. Divide 8 × 6 × 3 × 9 × 7 × 4, by 2 × 5 X 7 X 10 X 8.

2. Divide 81 X 42, by 99 × 7.

Ans. 613

3. Multiply 75 X 10, by 3 X 6, and divide that product by 15 X 25 X 12.

4. Divide 7 X 8 X 48, by 63 X 4 × 5 × 17, and multiply the quotient by 51.

5. Divide 99 X 28 X 6, by 5 × 8 × 18, multiply the quo tient by 4X4, and divide by 22 × 27.

6. Spent of $75, which I received for work, for flour at $5 a barrel; how many barrels did I buy?

7. If 25 pounds of lead costs $4.60, what do 8 pounds cost? 8. What will be received for 27 pieces of broadcloth, if 6 pieces bring $864?

9. If it requires 13 bushels of wheat to make 3 barrels of flour, how many bushels will be required to make 78 barrels of flour? Ans. 338 bushels.

10. If a tree 69 feet high casts a shadow of 90 feet, what length of shadow will be cast by a tree 92 feet high?

Ans. 120 feet.

11. A merchant exchanged 561 pounds of sugar, at 9 cents per pound, for eggs at 11 cents per dozen; how many dozen were received?

12. If 12 pieces of cloth, each piece containing 62 yards, cost $372, what cost 24 yards?

13. If a bar of iron 8 feet long weighs 36 pounds, what will a bar of the same size 100 feet long weigh?

14. How many boxes of oranges can be bought for $420, if $28 be paid for 7 boxes?

15. If the work of 7 men is equal to the work of 9 boys, how many men's work will equal the work of 63 boys?

16. If 15 men consume a barrel of flour in 6 weeks, how long would it last 9 men? Ans. 10 weeks.

17* If the interest of $650 for 12 months is $52, what is the interest of three times that sum for eight months? Ans. $104. 18. If 12 men can build a wall in 42 days, how many days will be required for 21 men to build it?

19. If $15 purchase 12 yards of cloth, how many yards will $48 purchase? Ans. 38 yards.

20. A ship has provision for 15 men 12 months; how long will it last 45 men?

21. How many overcoats, each containing 4 yards, can be made from 10 bales of cloth, 12 pieces each, 42 yards in each piece?

22. If 375 barrels of pork, each 200 pounds, cost $6000, what is the cost of 5 barrels, each 195 pounds?

23* Sold 20 barrels of apples at $2.50 per barrel, and spent the money thus obtained for cloth at $.50 a yard, which I sold at $.60 a yard, and bought a horse with the proceeds. What did I pay for the horse?