4. From ten millions ten thousand and ten take seven hundred and eighty-eight thousand and eight. Also from one million take one.

5. Multiply two millions fourteen thousand and eight by seven hundred and sixty-nine thousand and seventy, and prove by division.

6. Divide sixty-two millions sixteen thousand one hundred and fiftytwo by fourteen thousand and sixty-nine, and prove by multipli

cation.

7. What two factors make 35; also 27? And what three factors make 60; also 24?

8. Resolve into elementary factors (i.e. prime numbers) 12, 32, 90, 48, 72.

9. What is a common measure of 21 and 42 (i.e. a factor common to each); and what is the greatest common measure (i.e. the greatest common factor) of those two numbers?

10. State the greatest common measure of 12 and 24; of 9, 12, and 15; of 16, 24, 48, and 72..

11. Find the greatest common measure of 1296 and 1512.

12. Explain the phrase "least common multiple," and illustrate by showing the least common multiple of 4 and 6.

13. State a simple rule for finding the least common multiple of two numbers, and show how it may be applied to three. Then find the least common multiple of 18 and 24, and of 18, 24, and 108.

14. State a simple practical method of finding the least common multiple of several numbers; and find the least common multiple of 12, 16, 40, 54, 72; and the least common multiple of all the numbers from 1 to 10.

""differ

15. Explain the words "prime number," "factor," "sum," ence," "product," "quotient," "dividend."

16. Work the following sum:

A. A manuscript, printed in duodecimo, makes 324 pages, each page containing 24 lines, and each line having on the average 35 letters, how many letters are there? And how many pages would there be if the book were printed in octavo, each page then containing 36 lines, and each line 48 letters?

B. How many bricks may be taken away in 24 carts, each taking 500 bricks? And how many in as many carts again each larger?

C. Two years ago Tom's father was twice as old as Tom, and

the united ages of Tom and his father are now seven times the age of Tom's youngest brother; now the father is just sixty, and his age is equal to the ages of his two sons and one daughter; required the ages of the children.

D. What is the difference between six dozen dozen, and half a dozen dozen? Also between four times twenty-five and four times five and twenty?

3. FRACTIONS, VULGAR AND DECIMAL.

1. Explain the signs + "vulgar," "decimal."

A.

× ÷ =, and define the words "fraction,"

2. Explain the words "numerator" and "denominator," as applied to vulgar fractions.

3. Explain and illustrate, by an example of each, "proper fraction," "improper," "compound," "complex," and "mixed number."

4. Show that an improper fraction is a mixed number. Also that a vulgar fraction is not altered in value by its numerator and denominator being similarly dealt with.

5. When is a fraction said to be in its lowest terms; and how may a fraction be reduced to its lowest terms when it is not so?

6. Before fractions can be added or subtracted, what must generally be done with them? Give a familiar instance of the impossibility of adding unlike things.

7. Place the fractions, 2, in a condition to be added or subtracted; also add the fractions,,, ; and from 3 take .

8. Multiply,, together, and show the reason of what is called cancelling; also divide by and prove the truth of the rule, "invert the divisor and multiply."

9. Multiply 21 by 33, and divide 2 by 14.

31

10. Reduce the complex fraction to a simple form, working the sum at full length.

41

11. Divide by 3, and also divide by 3; and show the truth of the statement "multiplying the denominator is equivalent to dividing the numerator."

12. Divide 31. 2s. 7gd. by 3; also divide 24. 10s. 111d. by 6s. 91d. and

prove each sum by multiplication, working out accurately the fractions in each case.

13. Reduce 13s. 4d. to the fraction of 17.; and 2 weeks 3 days to the fraction of a year.

B.

14. State the value of each of the threes in the following,—33333; also in the following,—33333.

15. Express 1525 as single vulgar fractions and as one vulgar fraction; also '025.

16. Arrange for addition 25, 2500, 717, 14012; and also arrange 25, 2.500, 71-7, 140·12.

17. Find the difference in value between 250 and 025, and prove by vulgar fractions.

18. What effect does multiplying by 10 have upon whole numbers, and upon decimals? Also, in the latter case, how is division by 10 effected? Multiply 234 by 10, then by 100, and by 1000; also divide 25 by 10, 100, 1000.

19. State the rule for marking off decimal places after multiplying, and prove its truth. Illustrate by multiplying 21.45 by ·82 and proving by vulgar fractions; also multiply 12:41 by 008, and prove.

20. State the rule for marking off decimal places after division, in every case that can occur; and divide 21:45 by 5, and 515 by '005; also 2814 by 0014, and prove the first by vulgar fractions.

21. What are recurring or circulating decimals? Show how the exact value of 3 may be found; also 45; and show that 3333, &c., will equal and not.

22. Find the value in vulgar fractions of 285714; also of 3·16.

23. Find also the value of 243, and of 14, and of 2.83; and show the reason of the method by which mixed decimals (i. e. partly circulating only) are reduced to fractions.

24. Show that these rules for finding the value of circulating decimals are approximative only, from the instance 9.

4. EXAMPLES OF VULGAR AND DECIMAL FRACTIONS.

A.

1. What is of 2? Also of?

2. Multiply the fractions,,,, by 2, 3, 5, and 7, respectively; and divide the fractions, †,§, 14, by 2, 3, 5, and 7, respectively.

6 5

3. Multiply the fractions, 2, §, respectively by 3; also divide the

same fractions respectively by 3.

4. Add together 3,2,, ; and also add 13, 22, 71, together.

5. From 1 take ; also from 21 take 11.

6. Multiply, 2, 4, §, together; also multiply 23, 32, 44 together. 7. Divide by 2, and 3 by 11.

8,7%

; also 2+38.
also X
32 4+51' 2327
{+} +1

8. Simplify 23

9. Simplify

11. Multiply the sum of 3%, 43, and 44, by the difference of 74 and 5, and divide the product by the sum of 941 and 933.

12. Reduce 33 of 17 half-crowns to the fraction of 10 guineas; and express as a fraction of 57. the difference between 5 moidores and 5 guineas.

B.

13. Express the following both in vulgar and decimal fractions :(075 ×·25)+(·075÷·25)

1—25025+10-100+001 +101.101.

14. Multiply and verify each result by division.

15. Divide the same numbers (as in 14), and verify each statement by multiplication.

16. Add together 1, 1, 1, 1, both as vulgar and decimal fractions, and show that the results coincide.

17. Find the value of

A. 3·5+2·83+6+1·175.
B. 11.73-10-916.

C. 3.375 x 1.6 x 4.8.

D. 1.83 of 954 of 428571 of 4.5.

18. Reduce all the simple fractions from to inclusive to decimals,

and verify by reducing the decimals again to vulgar fractions.

19. Find the value of 025 of a sovereign+571428 of a guinea—·4 of a moidore+125 of a mark,

5. CONCRETE NUMBERS.

1. What is reckoned as the standard of money in England; and how much pure gold is there in a sovereign, and how much alloy ? Also, what is the fixed value of an ounce of gold?

2. What is meant by "decimal coinage," and in what countries is it in use? Name the standard of money in France and in the United States, and the coins mostly in use.

3. Show some of the advantages that are obvious from a decimal system of money, weight, and measure, in simplification of computa tions, &c.

4. Taking the sovereign as the base of the English coinage, and assuming that a coin were struck equal to of a florin, and that the farthings were slightly diminished in value, so that 1,000 farthings were equal to a sovereign, show that this would be a decimal system, and state the value of these coins in present money.

5. Taking 2 d. to be in a decimal form by 51. 14s. 7 d.

of a florin, show that 21. 7s. 81d. is expressed 2.3817., and that 5.737. is in present money

6. Express in a decimal form 87. 12s. 93d., and in common money 17.4617.; also 29.3677.

7. Upon this system reduce 7147. 5s. 8d. to decimal money, and then exchange it into French money, exchange 25.75 (i.e. 25·75 francs for a sovereign).

8. Bring 1552-44575 francs into English decimal money, exchange at 25.25; and express the result in common money.

9. What American money is equivalent to 217. 4s. 10d., exchange 4.5 dollars for one sovereign? And what English money to 18.724 dollars, exchange at 4.25 ?

10. What is the standard of measure in England; its exact length; and how may the exact length be at any time ascertained? Also, what is the standard of weight, and how may the exact weight be at any time found?

11. Write out the three tables of measure (Long, Square, and Cubic). 12. Illustrate by figures that a square yard has 9 square feet, and a cubic yard 27 cubic feet.

13. Write out the three tables of weight (Troy, Avoirdupoise, and Apothecaries').