10. What is the smallest sum of money for which you can purchase a number of pears at either 5 cents, 10 cents or 12 cents each?

SECTION XII.

CANCELLATION.

145.-1. What is the quotient of 60 divided by 15? 2. What is the quotient of one-third of 60 divided by one-third of 15?

3. What is the quotient of one-fifth of 60 divided by one-fifth of 15?

4. What common factors have 60 and 15?

5. If you take out from 60 the factors common to 15, what factor will remain?

6. What is the quotient of 2 × 7 × 3 divided by 2 X 7?

you

7. If strike out from 42 and 14 all commo factors, and divide the remaining factors, what is the quotient?

DEFINITION.

146. Cancellation is the process of shortening computations by striking out equal factors from the dividend and divisor, and using only the remaining factors.

147. Principle.-Striking out equal factors from both dividend and divisor, or dividing them by the same number, does not change their quotient.

WRITTEN EXERCISES.

148.-1. Divide 11 × 3 × 2 by 11 × 2.

out those common factors in both, leaving 1×3×1÷1X1=3. When a factor is canceled, 1 remains, and if not written is understood.

2. Divide 150 by 30.

SOLUTION. Since the dividend and divisor

30)150 have 10 as a common factor, cancel it by striking off the cipher from the right of both, leaving 3)15, which equals 5.

5

3. Divide 13 X 11 X 7 by 13 X 3. 1001 by 13 X 3. 4. Divide 2500 by 500. 360 by 10 × 9.

149. Rule for Cancellation.-Cancel in the dividend and divisor factors common to both, and then divide the product of the remaining factors of the dividend by the product of the remaining factors of the divisor.

PROBLEMS.

150. Divide1. 48 X3 X5 by 8 X 7 X 5. 2. 50 X 5 X 3 by 15 X 10. 3.75×11×2 by 11X5X2.

4. 31 X 25 by 7 × 5 × 5. 5. 4 × 13 × 3 by 39 × 2. 6. 81 X 8 X 7 by 27 × 28.

7. How many tons of coal at 9 dellars a ton can be exchanged for 81 barrels of apples at 3 dollars a barrel ?

4

8. How many tons of hay at 24 dollars a ton can be exchanged for 8 thousand feet of boards at 15 dollars a thousand?

9. Divide 11461 by 1400, using the factors 100 and 14. 1400)11461(8261

112

261

1400

SOLUTION.-Strike off two orders of figures from the right of the dividend and divisor, which divides each by 100, and gives for a new dividend 114 hundreds, and 61 remaining, and for a new divisor, 14 hundreds. The new dividend 114, divided by the new divisor 14, gives a quotient 8 and a remainder 2, which is 2 hundreds. Annexing to this remainder the first remainder, 61, gives 261, the entire remainder, and the entire quotient is 82%.

261

10. Divide 450 by 70, using the factors 10 and 7. 11. Divide 11400 by 600, using the factors 100 and 6. 12. If you should earn 1350 dollars in 300 days, how much would you earn each day?

151. TEST QUESTIONS.-1. What are the factors of a number? What is a prime number? A composite number? What are the prime factors of a number? What is factoring? A principle of factoring?

2. What is a divisor, or measure, of a number? A common divisor of two or more numbers? The greatest common divisor of two or more numbers? A principle of the greatest common factor or divisor?

3. When is a number a multiple of another? A common multiple of two or more numbers? The least common multiple of two or more numbers? What is a principle of the least common multiple of two or more numbers?

4. What is cancellation? What is a principle in cancellation?

NOTATION OF FRACTIONS.

152.-1. When a cake is divided into two equal parts, what is each of the parts called?

2. Into how many halves can a cake be cut?

3. When an apple is divided into three equal parts, what is each of the parts? What are two of the parts? How many thirds in an apple?

4. When an orange is cut into four equal parts, what is each of the parts? parts? Three of the parts? orange?

What are two of the How inany fourths in an

5. If a cake be divided into five equal parts, what is each of the parts? What are two of the parts? Three of the parts? Four of the parts?

6. Into how many halves can any thing be divided? Into how many thirds? Fourths? Fifths? Sixths?

7. What is meant by one half of any thing? By one third of any thing? By two thirds? By one fourth? By two fourths? By three fourths?

8. Which are the smaller parts of any thing-halves or thirds? Halves or fourths? Thirds or fourths? Halves or sixths? Thirds or sixths? Fourths or sixths? 9. Halves, thirds, fourths, etc., are expressed by figures, as follows

How

10. How many halves of 1 does express? many thirds of 1 does express? How many fourths

of 1 does express?

11. What does the figure 4 under the dividing line in the expression denote?

The number of equal parts into which the unit 1 is divided. 12. What does the figure 3 above the dividing line in the expression & denote?

The number of equal parts of the unit 1 which are taken. 13. How is the expression 3 read?

14. In the expression 34, what expresses the integer? What the fraction?

15. In the expression 7, what expresses the integer? What the fraction?