OF CANCELLING.

190. When one number is to be divided by another, the operation may often be shortened by striking out or cancelling the factors conimon to both, before the division is made.

1. For example, suppose it were required to divide 360 by 120.

ter the form of a fraction. We next separate both of them into factors, and then cancel the factors which are alike.

2. Divide 630 by 35. We separate the dividend and divisor into like factors, and then cancel those which are common in both.

OPERATION.

630 3 × 5 × 6 × 7
BX7

35

= 18.

་

191. If two or more numbers are to be multiplied together and their product divided by the product of other numbers, the operation may be abridged by striking out or cancelling any factor which is common to the dividend and divisor. For example, if 6 is to be multiplied by 8 and the product divided by 4, we have

QUEST.-190. How may the division of two numbers be often abridged? Explain the example mentally. Also the second example. 191. When two numbers are multiplied together and their product divided by a third how may the operation be abridged?

in the latter case we cancelled the factor 4 in the numera or and denominator, and multiplied 6 by the quotient 2.

1. Let it be required to multiply 24 by 16 and divide the product by 12.

Having written the product of the figures, which form the dividend, above the line, and the product of the figures which form the divisor below it, then

OPERATION.

2

24 × 16

= 32.

12

1

We cancel the common factors in the numerator and denominator, and write the quotients over and under the numbers in which such common factors are found, and if the quotients stili have a common factor, they may be again divided.

2. Reduce the compound fraction of 8 of 2 of 5 to a simple fraction.

3 12

16

=

1 3 2

16; 6 is twice a factor in 12;

3 three times a factor in 9;

and 5, once a factor in the denominator 5.

3. What is the product of 3 × 8 × 9 × 7 × 15 divided by 63 × 24 × 3 × 5?

OPERATION.

= 1.

This example presents a case that often arises, in which the product of two factors may be cancelled. Thus, 3 x 8 is 24: then cancel the 3 and 8 in the numerator and the 24 in the denominator. Again, 9 times 7 are 63; therefore cancel the 9 and 7 in the numerator and the 63 in the denominator. Also, 3 x 5 in the denominator cance s the 15 remaining in the numerator: hence, the quotient is unity.

4. What is the product of 126 × 16 × 3 divided by 7 × 12?

next divide 16 and 2 by 2, giving the quotients 8 and 1. Hence, the result is 72.

EXAMPLES.

1. What is the product of 1 x 6 x 9 x 14 x 15 x 7 x 8 divided by 36 × 128 × 56 × 20?

2. What is the value of 18 × 36 × 72 × 144 divided by 6 x 6 x 8 × 9 × 12 × 8?

3. What is the product of 3 × 9 × 7 × 3 × 14 × 36 divided by 252 × 81 × 2 × 7?

4. What is the product of 19 x 17 x 16 × 8 × 9 × 6 divided by 32 × 4 × 27 × 2?

5. What is the product of 4 × 12 × 16 × 30 × 16 × 48 × 48 divided by 9 × 10 × 14 × 24 × 44 × 40?

192. The process of cancelling may be applied to the terms of a proportion.

If we have any proportion, as

6 : 15 :: 28 : 70,

We may always cancel like factors in either couplet. Thus,

in which we divide the terms of the first couplet by 3, and those of the second by 2, and write the quotients above.

QUEST,-192. How else may the process of cancelling be applied? What may be cancelled in each couplet?

RULE OF THREE.

193. THE Rule of Three takes its name from the circumstance that three numbers are always given to find a fourth, which shall bear the same proportion to one of the given numbers as exists between the other two.

The following is the manner of finding the fourth term :

I. Reduce the two numbers which have different names from the answer sought, to the lowest denomination named in either of them.

II. Set the number which is of the same kind with the answer sought in the third place, and then consider from the nature of the question whether the answer will be greater or less than the third term.

III. When the answer is greater than the third term, write the least of the remaining numbers in the first place, but when it is less place the greater there.

IV. Then multiply the second and third terms together, and divide the product by the first term: the quotient will be the fourth term or answer sought, and will be of the same denomination as the third term.

EXAMPLES.

1. If 48 yards of cloth cost $67,25 what will 144 yards cost at the same rate?

QUEST.-193. From what does the Rule of Three take its name? What is the first thing to be done in stating the question? Which number do you make the third term? How do you determine which to put in the first? After stating the question, how do you find the fourth term? What will be its denomination?

tain $201,75 for the cost of 144 yards of cloth.

2. If 6 men can dig a certain ditch in 40 days, how many days would 30 men be employed in digging it?

less than the third; therefore, 30 men, the greater of the remaining numbers, is taken as the first term. Besides, it is plain that the fourth term must be just so many times less than 40, as 6 is less than 30.

3. If 25 yards of cloth cost £2 3s. 4d., what will 5 yards cost at the same rate?

QUEST.-In the first example which is greater, the third or fourth term? Which number must then be in the first term? How many times will the fourth term be greater or less than the third?