18. If the multiplicand is 27, and the multiplier 23, what is the product?

19. Divide 621 by 23; 621 by 27.

20. If the product is 621, and the multiplier 23, what is the multiplicand?

21. If the product is 621, and the multiplicand 27, what is the multiplier?

24.

22. Divide 672 by 24; 792 by 24; 912 by 24. 23. Divide 784 by 28; 868 by 28; 980 by 28. 24. Divide 1001 by 11; 1920 by 24; 2304 by

25. How many times will 37 go into 1369? 45 into 2160? 31 into 992 ?

26. If the dividend is 1216, and the divisor 44, what is the quotient?

27. How many times will 39 go into 1209? Into 1755? Into 2496 ?

28. Find the number of times 41 is contained in 2829; in 2501; in 3321.

29. Required the number of times 47 is contained in 2444; in 3666.

30. Find how many times 59 will go into 4012; into 4661.

NOTE. 59 will go 80 times into 4720; 4661 is how much less than 4720?

31. If a train of cars move at the rate of 34 miles an hour, how many hours will it be in moving 1187 miles? 1189 miles?

32. How many days will 3484 gallons of water supply a family, if 67 gallons of it are used daily? 3488 gallons? 5025 gallons?

33. Divide 4212 by 78; 5109 by 78; 6097 by 78.

NOTE. — In any case, when it can be readily done, reduce the fraction formed by writing the divisor under the remainder, to its lowest terms.

34. If 94 bushels of corn can be produced on 1 acre, at the same rate, how many acres will be required to produce 5264 bushels? 5292 bushels?

LESSON LX.

1. Find the factors of 32.*

NOTE. Factors of a number are such numbers as will, by being multiplied together, produce that number. Unity, or 1, is not regarded as a material factor, since multiplying or dividing any number by 1 does not alter its value. It will be disregarded, therefore, when speaking of the factors of numbers. 2 is a factor of any even number. 2 and 16 are factors of 32.

2. Find the factors of 16; of 8; of 4; of 2.

NOTE. When a number has no factor but itself and unity, it is called a prime number. 2 is therefore a prime number. Factors that are prime numbers are called prime factors. The factors of 32, if we take account of them all, are either 5 twos, or 3 twos and 1 four, or 2 twos and 1 eight, or 1 two and 1 sixteen, or 1 two and 2 fours.

3. Find the prime factors of 12; of 6; of 3.

NOTE. 3, having no factors but itself and 1, is a prime number.

4. What prime numbers are there under 10? 5. Required the prime factors of 14.

6. Which are the two factors of 56 which are

nearest equal?

NOTE. 56 = 28 X 2, or 14 X 4, or 8 X 7.

7. Divide 63 into 3 factors.

8. What are the prime factors of 49? Of 77? Of 91?

* The process of resolving a quantity into its factors is called factoring.

9. How many prime numbers are there among the twenties?

NOTE. No even number is prime, for all even numbers can be divided exactly by 2. All whole numbers whose last figure is even, are even. No whole number whose last figure ends in 5 is prime, for all such numbers can be divided by 5. Any number of tens can be divided by 2 or 5.

10. Required the factors of 35; of 39.

11. How many prime numbers among the thirties?

12. How many times does 7 occur as a factor in 98?

13. I have 42 bushels of grain; which are the three smallest sizes of bags, holding an exact whole number of bushels, that will exactly contain the same? How many bags of the smallest size would exactly contain the grain? How many of each of the other two sizes?

14. Multiply 53 by 28.

NOTE. 28 is 7 times 4; 7 and 4 are the factors, therefore, of 28. To multiply 53 by 28, we may use the factors of the 28, multiplying first by 7, and then its product by 4.

In finding the factors of any number less than 100, we need divide only by 2, 3, 5 and 7, since, if it cannot be exactly divided by any of these, it is prime.

15. Multiply 49 by 45, using the factors of 45.

NOTE. 45 can be divided by 5, because its last figure is 5. Its other factor is 9.

16. Multiply 47 by 54.

NOTE. All questions of this kind may be solved by using factors, or by the use of tens and units.

17. What cost 62 cows, at $39 apiece?

18. At $84 an acre, what cost 59 acres of land? 73 acres? 14 acres?

19. Find the product of 18 by 32; 48 by 42; 68 by 64.

20. Multiply 127 by 121, using factors.

NOTE. To find the factors of a number less than 400, we may divide by any prime number not over 20; that is by 2, 3, 5, 7, 11, 13, 17 and 19. If the number is more than 400, and less than 900, we can also divide by 23 and 29; and if between 900 and 1000, also by 31.

21. What are the factors of 127? Of 143? Of 221? Of 323? Of 483?

22. What are the factors of 162? Of 328? Of 492? Of 564?

NOTE. In the last number we find 4 × 141; we then can find the factors of 141, and so on.

23. Multiply 195 by 132; 112 by 154; 311 by 686, by using the factors.

24. Multiply by 10 the numbers 13; 123; 265 and 884.

NOTE. We may use the factors in multiplying by 10, 100, and so on, or we may annex a 0, or cipher, to the multiplicand, to multiply it by 10; two ciphers to multiply it by 100; and so

on.

25. Multiply by 10 the numbers 924; 572; 763; 292 and 29.

26. How many dimes in $ 10? In $129? In $355? In $762?

27. Multiply 253 by 10; by 100; by 1000. 28. Multiply 303 by 6; by 60; by 600.

29. How many cents in $5? In $50? In $72 ? In $125 ?

30. How many dollars in 265 eagles? How many dimes? How many cents? How many

mills?

31. What cost 218 yards of cloth, at $1.30 a yard?

32. How many pounds in 56 tons, at 2000 pounds to a ton?

33. Divide 1200 by 10; by 100.

NOTE.-We may divide by 10, by cutting off a figure from the right of the dividend; by 100, by cutting off two figures, and so on; the figure or figures cut off, if other than a cipher or ciphers, must be considered as a remainder.

34. Divide 1000 by 10; 1500 by 10; 1700 by 100. 35. How many dollars are there in 30 dimes? In 300 dimes? In 900 dimes?

36. How many dollars in 500 cents? In 3000 mills?

37. At $20 an acre, how many acres of land can be bought for $1240?

38. At $300 each, how many horses can be bought for $5400 ?

39. How many times will 130 go into 11800 ? 40. How many village lots, at $159 each, may be bought for $10494?

NOTE.

Resolve 159 into its prime factors, and divide by the factors in succession.

LESSON LXI.

1. What kind of a fraction is?

NOTE.- A fraction whose denominator is 10, 100, 1000, and so on, is called a decimal fraction; other fractions are called common fractions.

2. What kind of a fraction is ? How can you make a decimal fraction of it?

NOTE. When the denominator of any fraction is changed to 10, 100, 1000, or so on, the fraction becomes a decimal.

3. Change to a decimal fraction.

56

NOTE. By multiplying both terms of the fraction by 4 which does not change its value, we have 10, the answer required.

4. Change to a decimal fraction.