§ XVI.

MISCELLANEOUS EXAMPLES IN MULTIPLICATION AND DIVISION OF COMPOUND NUMBERS.

1. Bought 30 boxes of sugar, each containing 8cwt. 3qr. 20lb., but having lost 67cwt. 3qr. 12lb., I sold the remainder for 1£. 17s. 6d. per cwt.; what sum did I receive?

Ans. 375£.

2. A company of 144 persons purchased a tract of land containing 11067A. IR. 8p. John Smith, who was one of the company and owned an equal share with the others, sold his part of the land for 1s. 94d. per square rod; what sum did he receive? Ans. 1101£. 12s. 11d.

3. The exact distance from Boston to the mouth of the Columbia River is 2644m. 3fur. 12rd. A man, starting from Boston, travelled 100 days, going 18m. fur. 32rd. each day; required his distance from the mouth of the Columbia at the end of that time. Ans. 746m. 7fur. 12rd.

4. James Bent was born July 4, 1798, at 3h. 17m. A. M.; how long had he lived Sept. 9, 1807, at 11h. 19m. P. M., reckoning 365 days for each year, excepting the leap year 1804, which has 366 days? Ans. 3353da. 20h. 2m.

5. The distance from Vera Cruz, in a straight line, to the city of Mexico, is 121m. 5fur. If a man set out from Vera Cruz to travel this distance, on the first day of January, 1848, which was Saturday, and travelled 3124rd. per day until the eleventh day of January, omitting, however, as in duty bound, to travel on the Lord's Day, how far would he be from the city of Mexico on the morning of that day?

Ans. 43m. 4fur. 8rd. 6. Bought 16 casks of potash, each containing 7cwt. 3qr. 18lb., at 5 cents per pound. I disposed of 9 casks at 6 cents per pound, and sold the remainder at 7 cents per pound; what did I gain? Ans. $203.78.

7. A merchant purchased in London 17 bales of cloth for 17£. 18s. 10d. per bale. He disposed of the cloth at Havana for sugar at 1£. 17s. 6d. per cwt. Now, if he purchased 144cwt. of sugar, what balance did he receive?

Ans. 35£. Os. 2d. 8. A and B commenced travelling, the same way, round an island 50 miles in circumference. A travels 17m. 4fur. 30rd. a day, and B travels 12m. 3fur. 20rd. a day; required how far they are apart at the end of 10 days.

Ans. 1m. 4fur. 20rd.

9. Bought 760 barrels of flour at $5.75 per barrel, which I paid for in iron at 2 cents per pound. The purchaser afterwards sold one half of the iron to an axe-manufacturer; what quantity did he sell ? Ans. 48 tons 15cwt. 1qr. 22lb.

10. Bought 17 house-lots, each containing 44 perches, 200 square feet. From this purchase I sold 2A. 2R. 240ft., and the remaining quantity I disposed of at 1s. 24d. per square foot; what amount did I receive for the last sale?

Ans. 5914£. 19s. 51d.

11. J. Spofford's farm is 100 rods square. From this he sold to H. Spaulding a fine house-lot and garden, containing 5A. 3R. 17p.; and to D. Fitts a farm 50rd. square; and to R. Thornton a farm containing 3000 square rods; what is the value of the remainder, at $ 1.75 per square

rod?

Ans. $6235.25.

§ XVII. CANCELLATION.

ART. 112. CANCELLATION, as commonly used in arith.netic, signifies erasing or striking out any factor or factors common to the divisor and dividend.

It can be employed in most rules involving multiplication and division of whole numbers, but is more especially important in abridging operations in multiplication and division of vulgar fractions, and in simple and compound proportions.

ART. 113. If the dividend and divisor are both divided by the same number, the quotient is not altered. Thus, if the dividend is 20 and the divisor 4, the quotient will be 5. Now, if we divide the dividend and divisor by some number, as 2, their proportion is not altered, and we obtain 10 and 2 respectively; and 10 ÷ 2 = 5, the same as the original quotient.

ART. 114. If a factor is cancelled in any number, the number is divided by that factor. Thus, if 15 is the dividend and 5 the divisor, the quotient will be 3. Now, since the divi

QUESTIONS.-Art. 112. What does cancellation signify? In what rules is it most advantageously employed? - Art. 113. What is the effect on the quotient when the dividend and divisor are both divided by the same number? -Art. 114. What is the effect of cancelling a factor of any number?

sor and quotient are the two factors, which, being multiplied together, produce the dividend (Art. 50), it is plain, if we cancel the factor 5, thus 5 x 3 = 15, the remaining factor 3 is the quotient, and the dividend 15 has been divided by 5.

ART. 115. Method of cancelling, when there is one factor, or more, common to the dividend and divisor.

Ex. 1. A man sold 25 hundred weight of iron at 5 dollars per hundred weight, and expended the money for flour at 5 dollars per barrel; how many barrels did he purchase?

OPERATION.

Dividend Divisor

× 25
5

Ans. 25 barrels.

The price of the iron per hundred 25 weight must be multiplied by the number of hundred weight sold, to obtain the value of the whole; and

this product, being divided by the cost of the flour per barrel, will give the number of barrels bought. But we may indicate this multiplication and division by their signs, without actually performing the operations. In this example, 5 is a common factor of the divisor and dividend; therefore, we divide the divisor and dividend by this factor, or, which is the same thing, cancel it in both, and obtain 25 for the quotient.

2. Divide the product of 15, 2, 4, and 6, by the product of 4, 3, 2, and 5, and find the quotient. Ans. 6.

In this example we cancel the common factors in the divisor and dividend, and divide the product of the remaining factors in the dividend by the product of those in the divisor, and obtain the quotient 6.

-

RULE. 1. Write the dividend above the divisor, with a horizontal line between them, as in division. (Art. 47.)

2. Cancel the factor or factors common to the dividend and divisor, and, if there is only one factor remaining in the dividend and one

QUESTIONS. -Art. 115. How do you arrange the dividend and divisor for cancellation? How do you then proceed? Is the factor 5, in Ex. 1, reduced to 0, or 1, by being cancelled? If all the factors, both in the dividend and divisor, are cancelled, what will the quotient be? If more than one factor remain in the dividend and divisor after cancelling, how do you obtain the quotient? What is the rule for cancellation?

in the divisor, divide the factor of the dividend by the factor of the divisor.

3. If there are two or more factors remaining in the dividend, and two or more in the divisor, divide the product of the factors of the dividend by the product of the factors of the divisor.

NOTE. When a factor is cancelled, it is not reduced to 0, but to unity or 1. Therefore, when all the factors are cancelled, either in the divi dend or divisor, the factor 1 remains, and must be used as a factor of the divisor or dividend, as the case may be.

EXAMPLES FOR PRACTICE.

3. Divide 27 × 16 by 27.

4. Divide 42 x 19 by 19.

Ans. 16.

Ans. 42.

5. Divide the product of 8, 6, and 3, by the product of 6, 3, and 4.

Ans. 2. 6. Divide the product of 17, 6, and 2, by the product of 6, 2, and 17.

Ans. 1.

7. Sold 15 pieces of shirting, and in each piece there were 30 yards, for which I received 10 cents per yard; expended the money for 10 pieces of calico, each containing 15 yards; what was the calico per yard? Ans. 30 cents.

ART. 116. When a number in the dividend and another in the divisor have a common factor.

8. Divide the product of 12, 7, and 5, by the product of 2, 4, and 3. Ans. 17.

It will be seen, in this example, that 4 in the divisor is a factor of 12 in the dividend. Therefore we divide 12 by 4, cancelling these numbers, and use the quotient 3 instead of 12. The operation may be carried still further by cancelling this factor 3, and a similar one in the divisor.

9. Divide the product of 20, 13, and 9, by the product of 13, 16, and 1. Ans. 111.

QUESTIONS. Art. 116. How do you proceed when a number in the dividend and another in the divisor have a common factor? Is the common factor always one of the two numbers?

In this example, 20 in the dividend and 16 in the divisor may be We therefore cancel these numbers and use their quo

divided by 4.

tients in the operation.

RULE. When there is a factor in the dividend and another in the divisor which may be divided by the smaller factor, or by some other number, without a remainder, cancel these factors, and use the quotients arising from the division instead of them.

EXAMPLES FOR PRACTICE.

10. Divide the product of 9, 8, 2, and 14, by the product of 3, 4, 6, and 7. Ans. 4. 11. Divide the product of 16, 5, 10, and 18, by the product of 8, 6, 2, and 12. 12. Divide the product of 22, 9, 12, and 5, by the product of 3, 11, 6, and 4.

Ans. 12.

Ans. 15. 13. Divide the product of 25, 7, 14, and 36, by the product of 4, 10, 21, and 54. Ans. 117.

14. Divide the product of 26, 72, 81, and 12, by the product of 36, 13, 24, and 54.

Ans. 3.

ART. 117. When the product of two or more factors of the dividend is equal to the product of two or more factors of the divisor, or conversely.

15. Divide the product of 8, 5, 3, 16, and 28, by the product of 10, 4, 12, 4, and 7. Ans. 4.

Dividend
Divisor

OPERATION.

4

X 5 × 3 × 16 × 28 10 × 4 × 12 × 4 × 7

= 4 Quotient.

The product of the factors 8 and 5 in the dividend is equal to the product of 10 and 4 in the divisor; therefore we cancel these factors. Again, the product of the factors 3 and 16 is equal to the product of

QUESTIONS. What is the rule for cancelling when a number in the dividend and another in the divisor have a common factor? - Art. 117. How do you proceed when the products of two or more factors in the dividend and divisor are alike?