Separate the dividend into its prime factors, 7, 3, and 7; and the divisor into its prime factors, 7 and 3. By striking out, or cancelling, the common factor 7, and the common factor 3 (Prin. II.), there remain in the dividend the factor 7, and in the divisor 1. Hence, the quotient is 7.

199. Example 2.

SOLUTION.

75

= 37

18 X 54

2

Divide 9 × 75 × 39 by 13 × 54.

EXPLANATION. - Write the dividend above and the divisor below the line. First cancel the common factor 9 from the dividend and from 54 in the divisor, leaving the factor 6 in the divisor.

Next cancel the common factor 13 from 13 and 39, leaving 3 in the dividend.

Next cancel 3 from 3 and 6, leaving 2 in the divisor.

The uncancelled factor in the dividend is 75, and in the divisor is 2, and 75 ÷ 2 = 37, the quotient required (Prin. II.).

200. Rule for Cancellation.

I. Write the numbers forming the dividend above a horizontal line, and the numbers forming the divisor below the line. II. Cancel all factors common to the divisor and the dividend. III. Divide the product of the uncancelled factors of the dividend by the product of the uncancelled factors of the divisor. The quotient will be the result required.

Problems.

Divide by cancellation

1. 255 by 60; 819 by 441; 3744 by 2952.

2. 3765 by 2625; 7056 by 6068; 24375 by 5625. 3. 15 × 6 × 7 by 11 × 8; 36 × 7 by 27 × 11. 4. 3 × 26 × 14 by 84; 4 × 3 × 9 by 10 × 4 × 15. 5. 16 × 5 × 4 by 20 × 8; 21 × 11 × 26 by 14 × 13.

6. 7 x 9 x 15 by 9 x 5; 4 x 5 x 56 by 8 x 7 x 5. 7. 36 × 11 × 7 by 5 × 14 × 9 ; 56 x 7 x 12 x 3 by 6 × 14 × 8; 51 × 11 × 4 by 17 × 8 × 44.

8. 75 × 2 × 10 by 2 × 11 × 5; 81 × 7 × 8 by 27 × 28. 9. 7 × 9 × 45 by 7 × 9 × 9; 5 × 5 × 125 by 25 × 25. 10. Divide 28 × 32 by 7 × 8 × 9 × 16; 560 × 100 by 80 × 25; 26 × 56 × 14 by 910 × 2.

11. What is the value of (306 × 45 × 61) ÷ 200 × 60 × 91? Of 50 × 19 × 8 ÷ (150 × 2 × 95)?

12. How many times is 27 x 35 contained in 90 × 7? 13. Find the quotient of 35 × 51 × 7 ÷ 7 × 102 × 14. 14. The dividend is 3 × 9 × 15 × 21, and the divisor is 15 x 3 x 5 x 7. What is the quotient?

15. The factors of a dividend are 5, 35, and 49, and of a divisor are 25, 7, and 14. What is the quotient?

16. How many pounds of tea at 75 cents a pound must be given in exchange for 5 pounds of coffee at 30 cents?

17. How many barrels of flour at 9 dollars a barrel are worth as much as 18 tons of coal at 7 dollars a ton?

18. How much corn at 90 cents a bushel will pay for 120 yards of muslin at 15 cents a yard?

19. If 7 acres of land are worth $350, what will 150. acres cost?

20. At $21 a suit, how many suits can be made out of 4 pieces of cloth, each containing 24 yards worth $3 a yard?

Review Problems.

1. Find the prime numbers between 100 and 150. The composite numbers between 150 and 200.

2. Find the prime factors of 16580, 19475, and 24992.

3. What are all the prime divisors of 45744 and 62872? 4. Find the prime factors common to 216, 504, 726. 5. Find all the different divisors of 288 feet and $564. 6. Find every divisor common to 288 and 384.

7. Which of the numbers 739, 882, 997, and 8085 are prime? Which are composite, and what are their factors? 8. What is the greatest common divisor of 27924 and 41276? Of 31059, 44631, and 59508?

9. Find the least common multiple of 7, 168, 28, 70, 112, and 224. Of 249, 1245, 1743, and 3984.

10. Find every prime factor of 576 and 384. Their common divisor. Their greatest common divisor.

11. What is the smallest number that can be exactly divided by 92, 434, 14, 62, 186, and 1400?

12. How many times is 34 × 15 contained in 9 × 7 × 3 × 5 × 2? 7 times 217 in 62 times 140?

13. Divide 22932 by 1600, using their factors.

14. Divide the least common multiple of 996, 2490, 17430, and 7968 by their greatest common divisor.

15. Find the least common multiple of the first six even numbers. Of the first six odd numbers.

16. What is the largest number that will exactly divide 195 and 225? The smallest number that can be exactly divided by each of them?

17. Find the least number of feet that can exactly contain 26 feet, 312 feet, and 390 feet; and the greatest number that can exactly divide each.

18. Find the product of the least common multiple of 572, 884, 1336, and 2828, by their greatest common divisor.

19. What is the longest chain that will exactly measure the distances 198 feet, 264 feet, and 495 feet?

20. Find the smallest sum of money for which I could buy an exact number of books at $3, $5, $4, or $6 each.

21. How many days' work at 75 cents a day will pay for 125 bushels of corn at 30 cents a bushel?

22. What is the greatest length of the rails that will exactly lay three side-tracks of a railroad 3013, 2231, and 2047 feet long?

23. What is the least number of acres that can be divided into farms of 150, 200, or 250 acres each?

24. How many tons of hay at $24 a ton can be exchanged for 16 thousand feet of boards at $30 a thousand?

25. A boy desires to cut three balls of twine containing 300 feet, 360 feet, and 600 feet into kite-strings of equal length. What is the greatest length he can make them?

Review Questions.

1. What are the Factors of a number? Define composite number. Explain the difference between a composite and a prime number. Name five composite numbers, and tell what their factors are. Give five examples of prime numbers. Tell the difference between an even number and an odd number. Give five examples of even and of odd numbers.

2. Define Exact Divisor. Give three examples of exact divisors. What is the only difference between the terms factor and divisor, and show it by an example? What is a prime factor, or divisor? Name three prime and three composite factors of 24. What is Factoring? State the principles of factoring. Tell how to find the prime factors, or divisors, of a number. Explain how to find all the factors of a number.

3. Define Divisor, or Measure. Give three examples. What is a common divisor? Define Greatest Common Divisor. Name three common divisors, and also the greatest common divisor of 18 and 24. When are numbers prime to each other? State the principles of common divisors. Tell how to find a common divisor. Tell how to find a common divisor, and how to find all the common divisors of two or more numbers. Repeat the rule to find the greatest common divisor of two numbers; and of more than two numbers,

4. Define Multiple, or Dividend. Give two examples. What is a common multiple? Define Least Common Multiple. Name three common multiples of 4 and 6; and their least common multiple. Tell the difference between multiple and dividend. State the principles of common multiples. Tell how to find the multiple of a number. How to find a common multiple. What is the rule to find the least common multiple of two or more numbers?

5. Define Cancellation. What is the chief use of cancellation? What effect upon a number is produced by omitting one of the factors of that number? If a factor is omitted from the divisor only, is the quotient made greater, or less? What effect upon the quotient is produced by striking out a factor from the dividend only? What is the effect upon the quotient if a common factor is rejected from both dividend and divisor? State the principles of cancellation. Repeat the rule for cancellation.

SECTION VIII.

FRACTIONS.

1. If an apple is cut into two equal parts, what is each part called? What part of the apple is each piece?

2. Into how many halves can an apple be cut? How many halves are in a unit, or in any thing?

3. If an apple is cut into four equal parts, what is each part called? What are three of the pieces called?

4. Into how many fourths can an apple be cut? How many fourths are in a unit, or in any thing?

5. If a cake is cut into three equal parts, what is each part called? What are two of the parts? Three?

6. Into how many thirds can a cake be cut? How many thirds are in a unit, or in any thing?