PROB. 3. To ascertain the weight of gold equivalent to any given sum, currency.

Rule 1. If the given sum be in pence, reverse Rule 1. Theorem 1. that is; As the numerator 8 is to the given sum in pence ; so is the denominator 3 to the weight required, in grains. What weight of gold is equal to 4s. ?

d.

As 8: 48: 3

3

8)144

Ans. 18 grains.

12

48

Rule 2. If the given sum be in pounds, shillings and pence.

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As is equal to ; therefore, divide the given sum by 8, and that quotient by 2; add the two quotients together, double the last denomination, and you will have the answer.

What quantity of gold is equivalent to £45 13s. 4d.

oz. pwt. gr.

Mark the pounds, shillings and

8)45 13 4

pence, as oz. pwt. and gr.

Oz. 8 11 6 Ans.

PROB. 4. To find the value of gold equivalent to any given sure

in Federal money.

Rule. As the numerator 8 is to the number of dollars; so is the denominator 9 to the answer in pennyweights: Or, divide the dollars by the numerator 8, and add the quotient to the dividend. Or, divide as before, and multiply the quotient by the denominator 9. In either case you will have the answer.

1. Required the weight of gold equal to 76 dollars. As 8: 76: 9

9

8)684

oz. pwt. gr.

Ans. 851pwt.4 5 12

Or thus, 8)76 Or, 91×9-851pwt.

9

Ans. 85 pwt.

2. Required the weight of gold equal $159 75c. As 8 159 75: 9: 179pwt. 171gr. Ans.

9

3)1437-75

179.71375

24

287500

143750

17.25 grains.

Or, 159.75-8 + 159.75 = 179pwt. 171gr. Ans.
Or, 159.75-8 × 9 = 179pwt. 171gr. as before.

RULE OF THREE DIRECT AND INVERSE.

Though Direct and Inverse Proportion, are properly only parts of the same rule, yet for the use of those who may desire it, the common distinctions will be made and the common rules given.

The Rule of Three Direct teaches, by having three numbers given, to find a fourth, which shall have the same ratio to the second, as the third has to the first.

The Role of Three Inverse teaches, by having three numbers given, to find a fourth, which shall have the same ratio to the second, as the first has to the third. It is also called reciprocal or indirect proportion.

If more require more, or less require less, the question belongs to the Rule of Three Direct. But if more require less, or less require more, the question belongs to the Rule of Three Inverse.

The principal difficulty, which will embarrass the learner, will be to distinguish when the proportion is direct, and when inverse. This must be done by an attentive consideration of the question proposed. For more requires more, when the third term is greater than the first, and the question requires the fourth term to be greater than the second; and less requires less, when the third term is less than the first, and the fourth is required to be less than the second.

More is said to require less, when the third term is greater than the first, and the question requires the fourth to be less than the second; and less requires more, when the third term is less than the first, and the fourth is required to be greater than the second.

RULE OF THREE DIRECT.

RULE.

*

1. State the question by making that number, which asks the question, the third term; that which is of the same name or quality as the demand, the first term; and that, which is of the same name or quality with the answer required, the second term.

2. Divide the product of the second and third terms by the first term and the quotient will be the answer.

Note. The directions under the General Rule, as well as the demonstration, apply to this rule.

EXAMPLES.

1. If 6lbs. of sugar cost 10s. what will 33lbs. cost at the same rate ?

lbs. S. lbs.

As 6 10 :: 33: the answer.

10

6)330

55s. £2 15s. Ans.

In this example 33lbs. asks the question, and is made the third term; 6lbs. being of the same quality, is made the first term; and 10s. being of the quality of the answer required, is placed for the second term.

To invert the question, say,

S. lbs. S. lbs.

As 10 6: 55: 33 the Ans.

2. If 100yds. of cloth cost $66 what will 1 yard cost?

Ans. 66c.

3. If my income be $1750 a year, and I spend 19s. 7d. a day, how much shall I have saved at the end of the year?

Ans. £167 12s. 1d.

RULE OF THREE INVERSE, OR RECIPROCAL PROPORTION.

RULE.†

State and reduce the terms as in the Rule of Three Direct; then, multiply the first and second terms together, and divide the product

* The term which asks or moves the question, has generally some words like these before it, viz. What will? what cost? How many? how long? how much? &c.

The reason of this rule may be explained from the principles of Compound Multiplication and Compound Division, in the same manner as the direct rule.

T

by the third; the quotient will be the answer in the same denomina tion as the middle term was reduced into.

If there be fractions in your question, they must be stated as before directed, and if they be vulgar, invert the third term: Then multiply the three terms continually together, and the product will be the answer.

EXAMPLES.

1. How much shalloon, that is a yard wide, will line 63 yards of cloth which is 14 yard wide?

yd. yds. qrs.

As 14: 63 :: 3

4 4

The same by Vulgar Fractions.

First. 1, 63=27, and 3qrs.-3. Then,

For example, if 4 men can do a piece of work in 12 days, in what time will 3 men do it?

As 4 men: 12 days :: 8 men:

4× 12
8

6 days, the Answer.

And here the product of the first and second terms, that is, 4 times 12, or 48, is evidently the time in which one man would perform the work. Therefore, & men will do it in one eighth part of the time, or 6 days.

3. Suppose I lend a friend £350 for 5 months, he promising the like kindness; but, when requested, can spare but £125, how long may I keep it to balance the favour? £ Mo. £ Mo. As 350 5: 125: 14 Ans.

4. Suppose 450 men are in a garrison, and their provisions are calculated to last but 5 months; how many must leave the garrison, that the same provisions may be sufficient for those who remain 9 months?

As 5450:9: 250, and 450-250-200 men, Ans. 5. If a man perform a journey in 15 days, when the day is 12 hours long, in how many days will he do it, when the day is but 10 hours? Ans. 18 days.

6. If a piece of land, 40 rods in length, and 4 in breadth, make an acre, how wide must it be, when it is but 19 rods long to make an acre? Ans. Brods 6ft. 117in.

7. If a piece of board be 30 inches in length, what breadth will make 11 square foot? Ans. 7.2 inches. 8. A wall, which was to be built 24 feet high, was raised 8 feet by 6 men, in 12 days: How many men must be employed to finish the wall in four days?

ft. m. ft.

m.

As 8; 6: 24-8: 12 to finish it in 12 days. And, d. m. d. m.

As 12 12 4: 36 to finish in 4 days.

9. There is a cistern having a pipe, which will empty it in 6 hours How many pipes of the same capacity will empty it in 20 minutes?

h. pi. mi. pi.

As 61: 20: 18 Ans.

Ans. 26 days.

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10. If a field will feed 6 cows 91 days, how long will it feed 21 Cows? 11. How much in length, that is 133 poles in breadth, will make a square acre? Ans. 11,5 poles. 12. If a suit of clothes can be made of 41 yards of cloth, 1 yard wide; how many yards of coating of a yard wide, will it require for the same person? Ans. 6yds. 1qr. 30.

ABBREVIATIONS.

To know whether a fraction, when abbreviated, be equivalent in all respects to the original fraction.

RULE.

As the numerator of the fraction, in its lowest terms, is to its denominator; so will the numerator of the original fraction be to its own denominator.

Or, as one numerator is to the other; so will one denominator be to the other, &c.

A owes B £75 13s. 6d. ; now £100 of A's money is equal to £140 of B's; what must A pay to satisfy the said debt?