10. Divide each of the following numbers, namely, 75, 254, 36, 51, 28, 60, 3ğ, 72, 92, 46, and 316, severally by 7, by 15, by 17, by 8, by 2 (1-4), by %, by 1, by 76 (1—16), by 16 (16+16), by 48, by 24 (25—0), by 26 (25+30), by 24, and by 2. Prove each by division by the old method.

9

8

11. Perform problems from 1 to 10, immediately above, by inspection; that is, omit all superfluous steps, as follows:

No. 1. 32X1=30. No. 2. 54X-463.

No. 3. 78x=51%•

No. 4. 26X=1019g, &c.

The above rapid and concise methods will furnish excellent exercise for the pupil, giving employment both to his thinking and active faculties. Some few of the computations may be found more operose than by the usual methods; but a little practice will enable the student at a glance to tell which will be the simplest mode, and to choose accordingly.

Involution and Evolution of Common Fractions.

1. What is a square? See Involution, Def. 2, p. 162. How much is of ? Is, then, the square of ? If a fraction, then, be squared by multiplying each of its terms by itself, how can the square root of a fraction be found? Ans. By dividing each of its terms into equal factors, or finding the square root of each term. What, then, is the square root of?

If a

2. What is a cube? See Involution, Def. 4, p. 163. How much is of of ? Is 7, then, the cube of? How is a fraction involved to the third power, or cubed, then? fraction, then, be involved to the third power, or cubed, by multiplying each of its terms twice by itself, how can the cube root of a fraction be found? Ans. By dividing each of its terms into equal factors, or by finding the cube root of each term. What, then, is the cube root of 27?

Remark.-Fractions should always be placed in their most simple form before attempting to find their roots; that is, compound fractions should be changed to simple ones, mixed numbers to improper fractions, and every fraction should be in its lowest terms, as any other course would unnecessarily multiply figures. If either term has no exact root, an approximation may be found by putting the common fraction in a decimal form.

3. Find the square root of each of the following fractions: and prove by involution.

16 36

1, 49, 121,

4. Find the cube roots of, prove by involution.

4, 5, and 16, and

2744 13824, 125,

5. Find the square roots of § of ; also of of 2%, and prove by involution.

6. Find the cube root of 21, 1725, and 21, and prove by involution.

Remark. Sometimes the exact square or cube root of a common fraction, both of whose terms are surds, can be found by changing their form. Thus, += ~542=1=}; and

128

1

7. Find the exact square root of and the exact cube root of, and prove by involution. 21 Ans. and 3. 6, and the exact cube root

8. Find the exact square root of

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Practical Exercises on Fractional Quantities.

1. A tradesman, taking an account of stock, desired his clerk to ascertain the amount of the following remnants of calico: 3 yards, at 61 cents per yard; 24 yards, at 83 cents; 5 yards, at 9 cents; and 2 yards, at 7 cents. What was the amount? Ans. $1.001. How much would Ans. 31. of his share for

2. A lady bought 57 yards of cotton. remain after using 23 yards?

3. A merchant, owning of a ship, sold $4500. What portion of the ship did he sell? and what portion remained in his possession? Ans. ; and.

4. If of of a ship be worth $4500, what is of the vessel worth? and what is the value of the whole ship at that rate? Ans. to the last question, $12,000.

5. The purchaser of the part of the vessel mentioned above, wishing to have the whole in his own hands, offered the owners to take the remainder of the ship at the same rate (that is, at $4500 for 3 of 3). What would be the amount of this second purchase? Ans. $7500.

6. If 4 yards of cloth cost a certain sum, what portion of that sum will 1 yard cost? If 1 yard cost of the sum, what portion of it will 7 yards cost? By what fraction, then, must the price of 4 yards be multiplied to ascertain the price of 7

yards? Then, if 4 yards of cloth cost $12, what will 7 yards of the same cloth cost? Of what cancellation is of 12 susceptible? Cancel, and ascertain the result by inspection.

7. If 7 yards of cloth cost a certain sum, what portion of that sum will 1 yard cost? What portion, then, will 4 yards cost? By what fraction, then, must the price of 7 yards of cloth be multiplied to ascertain the price of 4 yards? If 7 yards of cloth, then, cost $21, what will 4 yards cost? Of what cancellation is 4 of 21 susceptible? Cancel, and ascertain the result by inspection.

In the last two examples, 1 yard requires LESS (that is, costs less), than 4 or 7 yards, and is represented by or 4. But it frequently happens that 1 requires MORE than a larger number, and consequently is represented by an improper fraction, as or, as will plainly appear from the two following examples. As this inversion, as it is called by mathematicians, frequently occurs in computations of this nature, it is requisite that pupils, when forming the factor fraction, should ask themselves LESS or MORE? at least, until the subject has become very familiar to them. This question is inserted into a few of the examples that follow, to show how and where it should be introduced. The pupil himself should introduce it into all the others.

8. If a piece of work can be finished in a certain number of days by 5 men, in what time can it be done by 1 man? In LESS or MORE time; that is, in or of the time? If 1 man require times longer to finish it than 5 men, how much time will 6 men require? LESS or MORE; that is, or the time? What is of? By what fraction, then, must the time required by 5 men be multiplied to give the time required by 6 men? If, then, 5 men can do a piece of work in 10 days, in what time will 6 men perform it? Ascertain the result by inspection, as follows:

(10)=81 days.

9. If 6 men can do a piece of work in a certain number of days, in what time can 1 man do it? LESS or MORE? in for ? If 1 man require longer than 6 men, what time will be necessary for 5 men? LESS or MORE?or? What is of

? By what fraction, then, must the time required by 6 men be multiplied to give the time required by 5 inen? If, then,

6 men can do a piece of work in 8 days, in what time can 5 men perform it? Cancel, and ascertain the result by inspection, as follows:

(X3)=2X5=10 days.

10. If 8 men, in a certain time, can make 24 rods of wall, how many men will be required for 18 rods in the same time? LESS or MORE? 1 or 2 of 8? Cancel, and ascertain the result by inspection.

11. If 4 lbs. of tea cost $2′50, what will be the cost of 24 lbs. ? LESS or MORE?

12. If 24 lbs. of tea cost $15, what will be the cost of 4 lbs. ? LESS or MORE?

13. If 16 lbs. of sugar cost $128, what will 54 lbs. cost? 14. If 54 lbs. of sugar cost $432, what will 16 lbs. cost? 15. If 6 bushels of turnips cost $150, what will 33 bushels cost?

16. If 33 bushels of turnips cost $825, what will 6 bushels cost?

17. How many men must be employed to finish a piece of work in 8 days, if 4 men can do it in 24 days?

18. If 12 men can finish a piece of work in 8 days, how many men will be able to finish it in 24 days?

19. If 4 men can do a piece of work in 24 days, in how many days can 12 men do it?

20. If 12 men can perform a piece of work in 8 days, in how many days can 4 men do it?

21. If 15 cords of wood cost $50, what would 27 cords of the same wood cost?

22. If 27 cords of wood cost $90, what would be the cost of 15 cords of the same wood?

23. If 60 bushels of potatoes can be exchanged for 25 bushels of rye, how much rye can be had for 200 bushels of potatoes?

24. If 83 bushels of rye can be exchanged for 200 bushels of potatoes, how much rye can be had for 60 bushels of potatoes?

25. If 6 men can cut 24 acres of grain in 5 days, in how many days could 4 men have cut the same field?

26. If 4 men take 7 days to cut a certain field of grain, in what time could 6 men cut it?

27. If 5 days be required for 6 men to reap a certain field, how many men could reap it in 7 days?

28. If of a bushel of grain cost $75%, what will 18g bushels cost? Simplify the money term by performing the division indicated, and the other two terms by multiplying each by 5, and dividing by 4. Why?

29. If 183 bushels of grain cost $171, how much will of a bushel cost? Simplify as above.

30. Bought 5000 planks, of 15 feet long, 1 foot wide, and 2 inches thick. To how many planks of 12 feet long, 1 foot wide, and 1 inches thick, are they equivalent? Make the fractional quantities disappear, by quadrupling the length and thickness of the planks. Why?

31. A carpenter exchanged 85714 planks, each 12 feet long, 1 foot wide, and 1 inches thick, for some that were 15 feet long, 1 foot wide, and 2 inches thick. How many ought he to receive?

32. If 8 men, in a certain time, make 24 rods of wall, how many men will be required to build 18 rods in the same time? LESS or MORE? 1 or 2 of 8? Again; if 8 men can make the 18 rods in 6 days, how many men can make it in 3 days? LESS or MORE? or ?. Now, if a change in the length of wall requires the number of men, and the change of time the number, what are the factors of both changes, as in the following statement?

If 8 men can build 24 rods of wall in 6 days, how many men can build 18 rods in 3 days? Resolve into primes, and cancel as follows:

Observe here that the number of men depends upon two circumstances,-the number of rods, and the number of days. 33. If 12 men can build 18 rods of wall in 3 days, what number of rods can be built by 8 men in 6 days? Resolve and cancel. The number of rods will be affected by what fraction? To know which term is numerator, ask, for each fraction, MORE or Less rods?