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21. A city merchant agrees to sell for a country farmer whatever produce the latter may send to him, on condition that he shall receive for his trouble 5 per cent of the money he receives for the produce. Under this arrangement he sells produce to the value of $1768.37. What ought he to receive for his trouble ?

Note. – A merchant who makes it his business to buy and sell goods for others is called a COMMISSION MERCHANT. The money he receives for his services is called his COMMISSION. For instance: In the last example, the merchant receives a commission of 5 per cent on the value of the produce he sells. He will deduct this commission from the amount of the sales before paying the fariner.

22. A commission merchant sold cloth for a manufacturer to the amount of $7643.79, receiving a commission of 3 per cent. What did his commission amount to? How much money ought he to pay the manufacturer ?

23. A commission merchant buys goods for me to the amount of $387.46, for which he charges a commission of 2 per cent. What will his commission amount to ? How much money must I send him to pay for the goods and commission ?

24. Mr. Moore borrowed some money of Mr. Boyden, agreeing to pay him, for each year's use of it, a sum equal to 6 per cent of the money he had borrowed. He borrowed $125.63, and kept it just one year. How much ought he to pay for the use of it? How much, then, ought he to pay Mr. Boyden in all ?

Note. — Money paid, as in the above example, for the use of money, is called INTEREST. The money used is called the PRINCIPAL. Interest is usually reckoned at a certain per cent of the principal for each year that it is used. The percentage paid for each year is called the RATE PER CENT. The interest and principal added together form the AMOUNT.

25. What is the interest of $137.84 for 1 year at 6 per cent ?

26. What is the interest of $487.31 for 6 months at 6 per cent per year?

Suggestion. — Since the rate for 1 year is 6 per cent, the rate for 6

months must be į of 6 per cent, which is 3 per cent. The interest for 6 months will, therefore, be 3 per cent of the principal.

27. Mr. Adams borrowed of Mr. Wales $718.63, for which he agreed to pay interest at 6 per cent. At the end of 6 months he paid the principal and interest. How much did he pay?

28. What is the interest of $47.83 for 4 months at 6 per cent?

134. Multiplication and Division of the Numerator.

The foregoing illustrations have shown that multiplying the numerator of a fraction multiplies the fraction, and that dividing the numerator divides the fraction. The same thing may be demonstrated more rigidly, by considering the nature of fractions, thus :

Since the numerator of a fraction shows how many parts are considered, multiplying or dividing the numerator multiplies or divides the number of parts considered, without affecting their size, and hence multiplies or divides the fraction.

1. Explain the effect of multiplying the numerator of the fraction by 3.

Answer. – Multiplying the numerator of the fraction by 3 gives 1$ for a result, which expresses 3 times as many parts, each of the same size as before, and is, therefore, 3 times as large. Hence, the fraction i' has been multiplied by 3.

See Note after solution of example 8th. Explain the effect of multiplying the numerator 2. Of Bag by 2. 4. Of g by 4. 6. Of by 9. 3. Of 13 by 8.

5. Of 37 by 11. 7. Of is by 8. 8. Explain the effect of dividing the numerator of the fraction f by 4.

Solution. - Dividing the numerator of the fraction of by 4 gives for a result, which expresses & as many parts, each of the same size as before, and is, therefore, è as large. Hence, the fraction g has been divided by 4.

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NotE. — The work explained above is really equivainnt to 3 times B=14, (just as 3 times 5 apples are 15 apples ;) to ļof ģ = els (just as 4 of 8 apples = 2 apples.) The above forms are, however, necessary, and should therefore be mastered. Explain the effect of dividing the numerator

9. Of by 8. 11. Of is by 9. 13. Of if by 6. 10. Of ii by 2. 12. Of ++ by 3. 14. Of jy by 12.

135. Multiplication of the Denominator. (a.) Since the denominator of a fraction shows into how many parts the unit is divided, or how many parts equal the unit, multiplying the denominator must (121, b. and c.) divide each part, and therefore must divide the fraction.

1. Explain the effect of multiplying the denominator of the fraction i by 2.

Answer. – Multiplying the denominator of the fraction by 2 gives f for a result, which expresses the same number of parts, each as large as before. Therefore, š = of , or multiplying the denominator of 1 by 2, has divided the fraction by 2.

Explain the effect of multiplying the denominator 2. Off by 3. 5. Of i by 7. 8. Of by 7. 3. Of by 4. 6. Of by 2. 9. Of by 10. 4. Of by 4. 7. Off by 3. 10. Of by 6.

(6.) Hence, multiplying the denominator of a fraction divides the fraction, by dividing the size of each part, without affecting the number of parts considered.

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136. Division of the Denominator. Since the denominator of a fraction shows into how many parts the unit is divided, or how many such parts are equal to the unit, dividing the denominator must (121, b. and c.) multiply each part, and therefore multiply the fraction.

1. Explain the effect of dividing the denominator of the fraction it by 4.

Answer. — Dividing the denominator of the fraction 13 by 4 gives

4 for a result, which expresses the same number of parts, each 4 times as large as before. Therefore, ** = 4 times 13, or, dividing the denominator of 13 by 4 has multiplied the fraction by 4.

Explain the effect of dividing the denominator 2. Of by 3. 5. Of is by 5. 8. Of ji by 6. 3. Of by 4. 6. Of is by 12. 9. Of do by 5. 4. Of is by 2. 7. Of by 36. 10. Of z by 3.

Hence, dividing the denominator of a fraction multiplies the fraction, by multiplying the size of each part without affecting the number of parts expressed.

137. Recapitulation and Inferences. (a.) Multiplying the numerator multiplies the fraction, by multiplying the number of parts considered without affecting their size.

(6.) Dividing the numerator divides the fraction, by dividing the number of parts considered without affecting their size.

(c.) Multiplying the denominator divides the fraction, by dividing each part without affecting the number of parts considered.

(d.) Dividing the denominator multiplies the fraction, by multiplying each part without affecting the number of parts considered.

(e.) Hence, 1. A fraction may be multiplied either by multiplying the numerator or by dividing the denominator.

2. A fraction may be divided either by dividing the numerator or by multiplying the denominator.

3. Multiplying both numerator and denominator of a fraction by any kumber both multiplies and divides the fraction by that number, and, therefore, does not alter its value.

4. Dividing both numerator and denominator of a fraction by the same number both divides and multiplies the fraction by that number, and, therefore, does not alter its value.

138. Multiplication and Division of both Numerator and

Denominator by the same Number.

1. Explain the effect of multiplying both numerator and denominator of the fraction by 6.

Answer. – Multiplying both numerator and denominator of the frac

tion 4 by 6 gives for a result, which expresses 6 times as many parts, each š as large as before. Therefore, the value of the fraction is unaltered, and $=.

Explain the effect of multiplying both numerator and denominator of

2. The fraction by 2. 6. The fraction ý by 3. 3. The fraction { by 9. 7. The fraction To by 8. 4. The fraction 10 by 10. 8. The fraction if by 9. 5. The fraction by 7. 9. The fraction 19 by 4.

10. Explain the effect of dividing both numerator and denominator of the fraction ii by 3.

Answer. - Dividing both numerator and denominator of the fraction is by 3 gives for a result, which expresses } as many parts, cach part 3 times as large as before. Therefore, the value of the fraction is unaltered, and =

Explain the effect of dividing both numerator and denominator of

11. The fraction by 8. 15. The fraction 1933 by 7. 12. The fraction by 9. 16. The fraction 84 by 41. 13. The fraction de

8. 17. The fraction şi by 5. 14. The fraction to by 18. 18. The fraction by 39.

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139. Lowest Terms.

(a.) The numerator and denominator are called TERMS of the fraction.

(b.) A fraction is said to be reduced to its LOWEST TERMS when its numerator and denominator are the smallest entire numbers which will express its value.

(c.) From the preceding explanations, it follows that,

1. A fraction may be reduced to lower terms by dividing hoth numerator and denominator by any number which will divide both without a remainder.

2. A fraction may be reduced to its lowest terms by divid. ing both numerator and denominator by any number which

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