7. What is of ? I of ? of? of? 8. What is of ? of? 4 of 3? 4 of 4?

The following are to be written.

9. What is of? To obtain a ninth part of a number, we must divide by 9. But (§ XLI.) multiplying the denominator divides the value. Hence, of 15x15 =135 A. & of ? A. 5. of? A. 195.

35

10. What is of? First find of: then of 7 times as much=7; then This is the ANALYTIC MODE of solution..

of=30% A.

11. What is of 17? A. 597. 18 of 4? A. 57. In every case, the pupil will observe, that the answer is found by multiplying the two numerators together, and likewise the two denominators together. Hence, the rule,

MULTIPLY THE NUMERATORS TOGETHER, FOR THE NUMERATOR OF THE PRODUCT, AND THE DENOMINATORS TOGETHER, FOR THE DENOMINATOR OF THE PRODUCT.

If mixed numbers occur in the multiplier or multiplicand, it is best, when the integral parts are small, to reduce them to improper Fractions; when otherwise, to multiply separately the whole and Fractional parts of the multiplier, into the whole and Fractional parts of the multiplicand.

A case of the unperformed multiplication of two or more Frac tions, that is, a Fraction of a Fraction (like the expressions, of, of of, &c.) is called a COMPOUND FRACTION, A Fraction, not. connected with others in this way, is called a SIMPLE FRACTION. Some of the members of a compound Fraction, may sometimes be whole or mixed numbers. Thus of 31, and of of 2, are compound Frac. tions. Compound Fractions may be reduced to simple ones by multiplying numerators together, and denominators together. Whole numbers are of course made factors in the numerator. this reduction, if the numerator of one Fraction be equal to the denominator of another, both may be neglected. (§ XLI.) Thus, in reducing of of, I may neglect the two 3s, and likewise the two 2s, and the resulting Fraction will be. So, likewise, if any numerator has a common factor, or measure, with any denominator, they may be divided by it before reducing.

In.

Thus, 3 of 21 of 7. 36 and 48 may both be divided by 12; 21 and 75 by 3; and, as 37 measures 74, it may be neglected and 74 divided by it. The Fraction will then be of of 25. 2 and 4 may still be divided, and it becomes of 1 of 2=1.

In the Addition and Subtraction of Fractions, it is evident that compound Fractions should first be reduced to simple ones. It will be observed that the word or between two Fractions signifies the same thing as the sign X.

13. A man owned of a factory, and sold of his

share. What part of the whole capital did he sell? A. . What part did he then own? A. 535.

14. During a storm, the master of a vessel was obliged to throw overboard of the cargo. A. owned of the cargo; what part of the whole cargo was his share of the loss. A. What had he left? A. of

the whole cargo.

15. At $ pr. yd. what cost 132 yds. of cotton cloth?

A. $2199.

133

16. If 1 lb. of cotton cost $1, what cost 15 lbs.?

17. At $3

A. $1.

340

23 pr. qt., what cost 4 gals. 1 qts. of rum?

959

A. $275.

18. A and B bought a cord of wood together; A paid What part of the whole did each

3 dollars, and B, 4. pay? How many solid feet ought each to have? Ans. A. 54 and B. 734. 19. Two men, E and F, traded in company. E furnished $2,000 and F, $1,000 of the capital. What part of the whole did each furnish? They gained $864. What was each one's share? A. E's $576, F's $288. 20, A, B and C traded in company and gained $300. A's capital was $600, B's $400, C's $200. What part of the gain ought each to have? What was each man's A's $150, B's $100, C's 50.

share?

Questions like the last three, belong to what is called FELLOWSHIP. BY FELLOWSHIP IS MEANT A CO-PARTNERSHIP, OR JOINT INTEREST; as when several men are trading together.

Fellowship is not, however, confined to trading companies, but embraces all cases, whatever, in which a distribution of money or property, or an adjustment of gain or loss is to be made among several individuals in a certain proportion. The money or the value of the articles put into the common fund, is called STOCK or CAPITAL

The profit or loss to be shared or divided is called a DIVIDEnd.

A company is said to declare a dividend, when public notice is given of the amount due on each share of stock. After a dividend has been declared, the company is bound in law to make payment of the amount to the owners of stock, on or after the day when it is declared to be due. From the above we have the rule.

FIND WHAT FRACTIONAL PART OF THE WHOLE CAPITAL, EACH PARTNER'S CAPITAL IS, AND MULTIPLY THE WHOLE DIVIDEND BY IT.

It will be seen that many questions in Fellowship have nothing to do with Capital or Stock. Such are the division of prize money among a crew, of estates among heirs, the assessment of taxes, the distribution of a bankrupt's estate among ereditors, &c. &c.-for all which things, see § XCII.

In a

21. A, B, and C freight a vessel with 300 tons. storm, the crew were obliged to throw 100 tons overboard. What was each man's loss, allowing that A owned 90 tons, B, 120, and C, the rest?

ANALYTIC SOLUTION. First find the loss on 1 ton. The loss on 1 ton is of the loss on 300 tons, and is therefore of 100 tons=100=1 of a ton. The loss, then, is of a ton on every ton of freight, and, as A owned 90 tons, he ought to lose 90 times =30=30 tons. B ought to lose 120 times 130=40 tons; and C ought to lose 100-30-40-30 tons.

22. A, B, C and D own, respectively, $300, $500, $900 and $300 stock in trade. They gain $200. What is each man's share? A. $30, $50, $90, $30.

NOTE. To solve it analytically, find the gain first on $1, &c. The pupil will soon find this mode pleasing, simple, and satisfactory.

23. Three men, A, B and C, purchased a hogshead of wine; for which A paid $45, B, $44, and C, $100. How many gallons ought each to receive?

Ans. A 15 gals. B 143 gals. C 331.

24. Four men purchased a field for $6,000. The first paid $1,200, the second, $2,000, the third, $1,000, and the fourth, the rest. The field contained 600 acres. What was each man's share? A. The first, 120, the second, 200, the third, 100, and the fourth, 180 acres.

his

25. A and B trade together. A furnishes $100 of capital, and B, $200. At the end of 2 months B withdraws capital, but A goes on to the end of a year from the commencement of the partnership. The gain is then found to be $32. How much ought each man to have? This case differs from the preceding, because the stocks are in trade different lengths of time.

ANALYTIC SOLUTION. A's capital, $100, is in trade 12 months, and gains a certain sum. 12 times as much capital would have gained the same sum in 1 month: that is $100×12=$1,200. Hence, A's gain is the sum which $1,200 would acquire (gaining at the same rate) in 1 month. B's capital $200 is in trade 2 months. 2 times as much capital would have gained the same sum in 1 month: that is, $200 ×2=$400. Hence, B's gain is the sum which $400 would acquire (at the same rate) in 1 month. Hence, A's and B's gains together are

what $1,200+$400-$1,600 would acquire in 1 month. Now, if $1,600 gain $32, $1 will gain as much= $3-$3. Then $1,200 will gain 1,200 times as much =81200=$24, A's gain; and $400 will gain 400 times as much as $1=84%=$8, B's gain.

Solution by FRACTIONAL PARTS, or RATIOS. We find above, that A's gain is what $1,200 would acquire in 1 month, and B's, what $400 would acquire in 1 month. Therefore, the whole gain, $32, is what $1,200+$400 =$1,600 would acquire in 1 month. A's part of this is 2003 of $32=$24; B's is %% of $32=$8. Hence, the rule,

400=

1600

MULTIPLY EACH PARTNER'S CAPITAL BY ITS TIME, AND ADD THE PRODUCTS. FIND WHAT FRACTIONAL PART EACH ONE'S PRODUCT IS OF THIS SUM, AND MULTIPLY THE WHOLE DIVIDEND BY IT.

26. A and B hire a pasture for $36. A put in 23 oxen for 27 days, and B 21 oxen for 35 days. How much ought each to pay? A. A. $16.48, B. $19.51 4.

13

27. Two merchants traded in partnership. A put in at first $200, and at the end of 8 months put in $100 more. B put in at first $550, but at the end of 4 months, he took out $140. At the end of 18 months they have gained $526. What is each man's share?

A. A's $192.951, and B's $333.041184. 28. Three graziers hire a pasture for $60.50. A puts in 5sheep for 4 months, B, 8 for 5 months, and C, 9 for 61 months. What ought each to pay?

A. A $11.25, B $20 and C $29.25. 29. Three persons make a joint stock. A puts in $185.66, and B $98.50, and C $76.85. They gain $222. What is each one's share? A. A's $114.171T, B's $60.57.0243, C's $47.2538775.

30. A bankrupt is indebted to A $277.33, to B $305.17, to C $152, and to D $105. His estate is worth only $677.50. How must it be divided? A. A $223.813§38, B $246.28, C $122.669339 and D $84.739988.

665

580

31. A, with $1,000 capital, began trade Jan. 1, 1829, He took in B as a partner, with $1,500 capital on the 1st. of March following. Three months after, they admit C, with $2,800 capital, and Jan. 1, 1830 they find the whole gain $1,776.50. What sum belonged to each?

A. A $457.46384, B $571.83238, C $747.19348.

SUBTRACTION.

MENTAL EXERCISES.

L. 1. A boy had of a quart of chesnuts, and gave away. How many eighths had he left?

2. A man bought of a barrel of flour, and gave to some labourers. How much had he left?

3. Two boys together gathered of a peck of strawberries, and one of them took 3 of a peck, as his share. How many had the other?

4. from leaves how many ninths? 5. from leaves how much?

from 1?

from 23 ? from 81?

6,3 % from 47?

from?

8 12

from ?

from 5?

from

But we shall often be required to make subtractions between Fractions whose denominators are different. In this case we may reduce them to a common denominator, and then we can subtract as above.

NOTE. When mixed numbers occur, perform the subtraction of the whole, and fractional parts separately, if it can be done.

13. From 2013 take 63.

14. From 3711 take 204. A. 17.

15. From 30 take 151.

A. 1475.

A. 157.

16. From 211 take 63. A. 14.

NOTE. When the Fraction in the subtrahend is greater than that in the minuend, it will be necessary to borrow a unit. Sometimes, likewise, there will be no Fraction in the minuend. In this case, it will also be necessary to borrow. 17. From 225 take 711, A. 15312. 18. From 621 take 4975. A. 123. 19. From 525 take 326. A. 1981. 13

5

7840

8

20. From 2,983237 take 1,843348. A. 1,13942333. 21. From 2148 take 13817819. A. 811081377777 Hence, to perform Subtraction of Fractions,

28965 7467

REDUCE THE FRACTIONS TO A COMMON DENOMINATOR, Find the DIFFERENCE OF THE NUMERATORS, AND WRITE IT OVER THE COMMON DE NOMINATOR.