INTRODUCTION TO ARITHMETIC,

PART VIII.

CONTAINING

PROPORTION AND INTEREST.

PROPORTION, is commonly called the Rule of Three; because a question or problem in this rule, has three numbers given to find a fourth; and in ancient times it was called "the golden Rule of Three," by reason of its usefulness: For, by this rule, we can solve many crafty, critical, and astonishingly curious questions, which could not be done without knowing this method in calculation.

EXAMPLE,

If 2 bushels of corn cost 6 shillings, what will 4 bushels cost?

RULE.

Multiply the second and third numbers together, and divide their product by the first; the quotient will be the fourth number or answer.

This answer can be calculated mentally, that is, by the mind, and very soon too; but such easy examples are most likely to inspire a learner with the knowledge of new rules, whereby he can solve problems of a more difficult nature.

These numbers are called terms; the first term, the second term, the third term, and the fourth term.See Lesson A. Card 31.

In stating a question, we place a colon between the first and second term, two colons between the second and third, and if the fourth number be known, we place one colon between the third and fourth, thus, 2 6 4 12 and we read them as follows; as 2 are to 6; so are 4 to 12, that is, 2 bear the same proportion 6, as 4 do to 12.

Let us scan another example: If 2 bushels of oats cost 4 shillings, what will 7 bushels cost?

Here 2 bushels cost 4 shillings, which is at the rate of 2 shillings a bushel; consequently, 7 bushels will cost 14 shillings, because 7 times 2 are 14; thus the first term 2, bears the same proportion to the second, as the third term 7, does to the fourth. See Lesson B. Card 31.

How to know which is the first, second, and third term, before a question is stated.

The first term is a supposition, as, "If 2 bushels cost," or "Suppose 2 bushels cost."

The second term is that which bears the name of the answer, as 4 shillings."

The third term is of the same name of the first, and expresses a demand, by asking a question, as "What will 7 bushels cost ?"

Here our ideas are naturally led to shillings for the fourth term or answer, because the second term was in shillings.

And if the second term had been in pence, or gallons, or quarts, or feet, or inches, &c. the fourth term would be of the same name as the second

EXAMPLE.

If 2 shillings will buy 4 quarts of vinegar, what quantity will 8 shillings buy?

.

In this case the second term is in quarts, and the answer must be likewise in quarts.

Many more particulars which belong to the rule of three, must be taken into consideration; but we will pass on with a few lessons before the mind is confused with too much criticism.

KEY TO CARD No. 31.

Lesson 1.-If 5 pounds of butter cost 80 pence, what will 9 pounds cost? Ans. 12 shillings. Operation lb. d.

lb.

5 O: 80 :: 9 to a fourth number. That is to say, as 5 lb. are to 80d.; so are 9 lb to a fourth number. Multiply the second and third terms together and divide by the first. The quotient or answer will be in pence.

Divide those pence by 12 to reduce them inte shillings.

80 the second term,

9 the third term.

First term,

5)720

Answer,

12)144 pence or fourth term.
12 Shillings.

KEY TO CARD No. 31.

Lesson 2.-If 5 pounds of butter cost 80 pence, how much will 1 pound cost?

lb.

d.

lb.

Ans. 16 pence.

5 : 80 :: 1 to a fourth number. That is to say, as 5 lb. are to 80 pence; so is 1 lb. to a fourth number or answer.

Here, the third term being 1, it makes multiplication useless; therefore only divide the second term by the first, the quotient will be the answer.

Operation.

5)80
16d.

As the second term was in pence, so the answer is in pence.

Lesson 3.-If 1 lb. butter cost 16 pence, what will 5 lb. cost? Answer, 80 pence.

In this case division is unnecessary, because the first term is 1; therefore a multiplication of the second and third term, will give the answer.

1 : 16 :: 5 to a fourth number.

That is, as 1 lb. is to 16d. so are 5 lb. to a fourth number; and, as the second term is in pence, so the answer must be in pence also.

16 second term.

5 third term.

80 fourth term, or answer.

KEY TO CARD No. 31.

Lesson 4.-Bought 8 pieces of chints for £57 12s. what must I pay for 12 pieces at that rate?

Answer, £86 8s. Reduce the sum £57 12 into shillings, viz. 1152s.

Then say,

8 1152 :: 12 to a fourth number. Read the statement thus; as 8 pieces are to 1152 shillings, so are 12 pieces to a fourth number. Operation. 1152 second term.

12 third term.

8)13824

1728 shillings or fourth term.

Here remember that when the fourth term is in a low denomination, it must be reduced to its proper quantity; as, pints to gallons; inches to feet; pounds to hundreds; and, as in this case, shillings to pounds. Therefore divide the sum 1728 shillings, by 20, the answer will be £86 8s.

2|0)17218

£86 88.

Lesson 5.-Suppose the aforesaid pieces of chints contain 432 yards in the whole, and cost £86 8; how Ans. 4 shillings.

much is that a yard?

R*

Here the third term being one, division only is ne

cessary.

£86 8s. 1728s.

yds.

s. yd.

432 : 1728 :: 1

Read thus," as 432 yards are to 1728 shillings: so is 1 yard to a fourth number."

Operation.

432)1728(4s. the Answer,

1728

KEY TO CARD No. 31.

Lesson 6. If 6 horses eat 9 bushels of oats, how much will 24 horses eat in the same time?

Answer, 36 bushels.

But, some of those particulars before mentioned, must come under consideration: In the Rule of Three, there are two sorts of proportion; direct, and

inverse.

That part which we have already scrutinized, is direct proportion; yet we know not why it is direct. In every problem there are certain signs by which we can know direct from inverse proportion.

1st. When more requires more, the proportion is di

rect.

If the third term be greater than the first, and the nature of the question shows that the answer must be greater than the second term, then more requires

more.

EXAMPLE.

If one bushel of salt cost 8 shillings, what will 2bushels cost?

Here the third term is greater than the first, and we readily perceive that the answer must be greater than the second term. Again,

2nd. When less requires less, the proportion is direct.

* Problem, a question, a mathematical question.