(ART. 69.) A number may be considered as part of another number; for instance, 3 is one-half of 6, and 2 is one-fourth of 8; and we may mechanically find the and, by dividing the first number by the second; thus, =:=1. Or, the comparison remains the same, if we divide both numbers, (the numbers we compare,) by the smallest number; thus, 3 compares with 6, as 1 to 2; 2 to 8, as 1 to 4.

3

What part of 3 is 2? Certainly. What part of 2 is 3? Certainly

Now, wherever this phraseology applies, (What part of ... is,) the number after the word is, must be taken for a numerator, and the other for a denominator; and reduce the fraction to its most simple form, and we shall have the most simple relation of one number to the other. (See Art. 46.)

Make complex fractions of the two last examples, and reduce them.

(ART. 70.) Before numbers can be compared in this way, they must be of the same denomination. If, for example, we would compare 5 feet with 2 yards, and asked what part of 5 is 2 yards, we could not say ; for we all know that 2 yards is more than five feet.

We

must first reduce the yards to feet; and then the question is, What part of 5 feet is 6 feet?

Ans.

7. What part of 3 hundred weight is 1 hundred weight 3 quarters? Reduce both to quarters; then compare.

Ans.

8. What part of 6 shillings is 8 shillings 6 pence

Ans.. Ans. To

9. What part of 4,6 is ,46? 10. What part of 3 gallons is 2 quarts 1 pint?

11. What part of 1 yard is 2 feet 6 inches?

12. What part of 5 dollars is 35 cents? What part of 12 is 107?

6 ?

What part of 3 is TT

What part of 61 is 67?

4

Ans.

?

Ans..

Ans. •

The relations of numbers and things compared in this way, may be applied to the solution of a great variety of practical problems.

Example. If 3 men can build 7 rods of wall in a day, how many rods can 9 men build? Here we compare 9 men with 3 men; and we say, what part of 3 is 9? Ans. 3 times. Now, the rods of wall must compare, as the men that build them. Hence 7X3=21 rods of wall, for the answer to the question.

Example 2. If three horses eat 8 bushels of oats in 2 weeks, how long will it take them to eat 40 bushels? 8 compares to 40 bushels, as 1 to 5; hence 2X5=10 weeks.

Example 3. If 41⁄2 tons of hay keep 3 cattle over the winter, how many tons will it require to winter 25 cattle? 3 cattle compare with 25, as 1 to 25; hence, 25 X 175 =37 tons, Ans.

Example 4. If 19 gallons of molasses cost 12 dollars, what will 3 quarts cost? Here we cannot directly compare gallons with quarts; one or the other must be reduced. The quarts are 1. But to reduce quarts to gallons, we divide by 4; therefore, our 34 quarts is 1, expressed in gallons. Now, 19 compares with 1, as 1 to; hence 12X dollars, or 60 cents the Ans.

19

Example 5. If 16 bushels of oats cost $6,75, what will 320 bushels cost? Oats compares with oats, in the supposition and demand; as, 16 to 320; or, as 1 to 20. Hence, 6,75×20=135 dollars, Ans.

Example 6. If 1 acre and 20 rods of ground produce 45 bushels of wheat; at that rate, how much will 9 acres produce? One acre and 20 rods is 180 rods. Nine

As

acres is 9X 160 rods. 180 compares with 9×160. 20 compares with 160, by dividing both by 9; or, as 1 to 8, by dividing both by 20. Hence, 45X8=360 bushels, the Ans.

Example 7. If of a yard of cloth be worth 4 dollars, what will be the worth of 7 of a yard? Observe, 28. Compare to, which is the same as 6 to 7. But 6 is to 7 as 1 to 7. Hence 7×4=43 dollars, Ans.

Example 8. If 3 yards of cloth cost 12 dollars, how many yards may be bought for 102 dollars? Observe, 12351; 51 is to 102 as to But is to 2 as I to 8. Hence 3X8=24 yards, the Ans.

9. Boarding at 12 shillings 6 pence per week, how long will 32 pounds 10 shillings last me?

Ans. 52 weeks. 10. If a barrel of beef last 10 men 95 days, how long will it last 25 men? Ans. 38 days. 11. If I lend a man $200 for 60 days, how long ought he to lend me $275 to requite the favor?

Ans. 437 days. 12. How many years will it require for 5 cents to gain the same interest that $100 does in 1 year?

Ans. 2000 years. 13. If it be required to line cloth of a yard wide, with lining wide, what must be the relative quantity of the lining?

Ans..

14. If 22 cents buy 3 pounds of coffee, how many pounds will 44 dollars buy? Ans. 600.

15. If 3 pounds are purchased for 25 cents, what will 135 pounds cost? Ans. $11,25. 16. At 4 dollars per ton, how many ton can be purchased for $900 ? Ans. 200.

PROPORTION.

(ART. 71.) From the comparison of numbers we derive proportion, one of the most fruitful and valuable principles in arithmetic, the mathematics, and natural philosophy. Too much cannot be said of its importance and utility.

Definitions.

1. When two numbers are directly compared, by dividing one by the other, as in Article 69, the quotient expresses their relation.

2. If two other numbers are compared in the same way, and we find the same quotient the same relation, the four numbers may constitute a proportion.

3. Two numbers, thus compared, are called a couplet. 4. The quotient, or relation, is technically called the ratio.

For example: If we compare 2 to 4, by dividing 4 by 2 we find the ratio, 2. Also, 3 compares with 6 by the same ratio. Therefore, 2 is to 4 as 3 is to 6.

The words between the terms need not be written, as points, thus, : :: : or, : = : express the same as the written words, (Definition, Art. 5). Hence,

2 : 4 :: 3 : 6,

is a proportion. The ratio to this proportion is 2. Observe, that the product of the extreme terms is equal to the product of the mean or middle terms; that is 2×6 =4X3, and this is true in every proportion. If, in any proposed case, this is not true, the proportion is false, and is not, in fact, a proportion.

(ART. 72.) This property of extremes and means, results from the equality of the ratio between the couplets.

Hence,, because, things equal to the same thing

equal one another.

6X2

Multiply both terms by 2, and 4=

3.

Now mul

tiply by 3, and 3×4=6×2; that is, the product of the extremes equals that of the means. 12: 48 compare, as 1 to 4; that is, the ratio between the terms of this couplet is 4.* 3: 12 compares, as 1 to 4, same ratio; therefore 12: 48 :: 3 : 12 must be a perfect proportion; hence, 12X12=48 X 3.

(ART. 73.) The terms of this proportion may be changed, and still constitute a perfect proportion; but will have a different ratio between the couplets.

The ra

1. Invert the means, and 12 : 3 :: 48: 12. tio is now ; but the product of the extremes and means is the same as before.

2. If we treat both couplets exactly alike, no matter what we do, the result will still be a proportion. Let us subtract; the second from the first is to the 2d, as the fourth from the third is to the 4th; that is, 9: 3 :: 36: 12. These couplets, we perceive, have the equal ratio,, and the product of the extremes is equal to the product of the Again the second from the first is to the first, as the fourth from the third is to the second. That is, 9: 12 :: 36: 48. ratio; Also, ratio. Hence this proportion is true.

means.

4

(ART. 74.) Two or more proportions may be multiplied together, term by term, and constitute a new pro

* Some mathematicians, particularly the English, reduce the second term to unity, by dividing both terms by the second. In that case, the ratio of 12 to 48, would be 4, in place of 4. Either way is correct, if we are uniform and consistent. We prefer the French method, of reducing the first term to unity, conceiving it more simple than the other.