antecedents, and 15 and 45 the consequents; also, 16 and 15 make the first couplet, and 48 and 45 the second.

185. If 4lb. of tea cost $8, what will 127b. cost at the same rate?

It is evident that the 4th term, or cost of 127b. of tea, must be as many times greater than $8, as 127b. is greater than 4lb. But the ratio of 4lb. to 127b. is 3; hence, 3 is the number of times which the cost exceeds $8: that is, the cost is equal to $8x3=$24. But instead of writing the numbers

and as the same may be shown for every proportion, we conclude,

That the 4th term of every proportion may be found by multiplying the 2d and 3d terms together, and dividing their product by the 1st term.

EXAMPLES.

1. The first three terms of a proportion are 1, 2, and 3: what is the fourth? Ans. 6. 2. The first three terms are 6, 2, and 1: what is the 4th? Ans. 3. The first three terms are 10, 3, and 1: what is the 4th ? Ans.

185. Explain this example orally. How may the fourth term of every proportion be found?

186. The 1st and 4th terms of a proportion are called the two extremes, and the 2d and 3d terms are called the two means.

Now, since the 4th term is obtained by dividing the product of the 2d and 3d terms by the 1st term, and since the product of the divisor by the quotient is equal to the dividend, it follows,

That in every proportion the product of the two extremes is equal to the product of the two means.

Thus, in the following examples, we have

1 : 6 ::

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

1. For example, suppose it were required to divide

them into factors, and then cancel the factors which are

186. What are the first and fourth terms of a proportion called? What are the second and third terms called? In every proportion, what is the product of the extremes equal to?

187. How may the division of two numbers be often abridged? Explain the example orally. Also the second example.

188. 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

6×8 48
4

4

=12; or,

6x8
4

=6×2=12:

in the latter case we cancelled the factor 4 in the numerator 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

12

1

-32.

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 still have a common factor, they may be again divided.

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

Beginning with the first numerator, we find it is once a factor of itself and 4 times in 16; 6 is twice a factor in 12;

3, three times a factor in 9;

OPERATION.

1

9 12

3

and 5, once a factor in the denominator 5.

188. When two numbers are multiplied together and their product divided by a third, how may the operation be abridged?

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, 3×8 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, 3x5 in the denominator cancels the 15 remaining in the numerator: hence, the quotient is unity.

4. What is the product of 126 × 16 × 3 divided by 7x12?

We see that 7 is a factor of 126, giving a quotient 18, which we place over 126, crossing at the same time 126 and the 7 below. We then divide 18 and 12 by 6, crossing them both and writing down the quotients 3 and 2. We

3

18

OPERATION.

8

72.

126×16×3

7x12

2

1

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

EXAMPLES.

63

Ans. 12

1. What is the product of 1×6×9×14×15×7×8 divided by 36 × 128 × 56 × 20? 2. What is the value of 18 × 36 × 72 × 144 divided by 6×6×8×9x12x8 ?

Ans. 27. 189. 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,

or

6 15
2 5

28: 70,

14 35:

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

189. How else may the process of cancelling be applied? What may be cancelled in each couplet?

RULE OF THREE.

190. 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 their 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?

190. 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? 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?