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75; March 13, $971; and April 3, $763.37. What should be the date of a single note at 8 months, for the whole amount?

17. What must be the date of a note at 6 months, for the amount of the following charges, each payable in 8 months ? March 19, $59.63 ; April 3, $713.20; April 15, $2350; April 21, $1975.37; July 5, $163.25; and Sept. 19, $251.17.

18. What should be the date of a note at 9 months, for the amount of the following charges? May 23, $840 at 6mo.; May 30, $900 at 8mo. ; June 15, $325 at 4mo.; June 27, $563,50 at 3mo. ; July 13, $729 at 6mo.; and July 16, $421.75 at 9mo.

19. A jeweller mixed 31b. 602. silver, 90%. fine, 41b. of 11oz. fine, 2ib. 3oz. of 10žoz. fine, and 21b. of 10oz. fine. What was the fineness of the mixture ? [11oz. fine, denotes that 11oz. in the lb. are silver, and the rest alloy. Pure silver is 12oz. fine.]

20. If ilb. silver, 8oz. fine, 2lb. of 90%. fine, 3lb. of 10oz. fine, 4lb. of 11oz. fine, and 5lb. alloy, be melted together, what will be the fineness of the mixture ?

21. Marshall & Wood, to George W. Hall, Dr.

1844

4 mo. 16, To 143}yds. sheeting, at 12}cts.

8то. Ус 23, To 259 yds.

at 11 cts.

8то. 5 mo. 2, To 169yds. cambric, at 13cts.

6mo. 2311yds. bleached muslin, at 13 cts. 4mo. 5 mo. 9, To 258 yds. duck, at 114cts. 6mo.

What should be the date of a note at 6mo, for the whole amount?

22. A farmer mixed 18į bushels of wheat, at $1.00 per bushel; 16fbu. at $1.12; per bushel ; 133bu. of barley, at 62}cts. per bushel, and 10bu. of oats, at 37 cts. per bushel. What was the mixture worth per peck ?

23. What should be the date of a note at 60 days (or 63 days, including the days of grace) for $150, due April 7,-$200, due April 13,-$75, due April 17,-$325, due April 29,-$180, due April 30,-$400, due May 4,-and $439, due May 11?

CHAPTER XI. PROPORTION, OR THE RULE OF THREE. The Ratio of two numbers is the quotient resulting from the division of the first by the second. Thus, the ratio of 1 to 2, is į ; of 13 to 6, 122.

A ratio is usually expressed by two points written between the numbers, as 1 : 2; 13:6; which are read 1 is to 2 ; 13 is to 6.

A PROPORTION is the union of two equal ratios, by writing four points, or the sign of equality, between them. Thus, 2 : 3 ::4:6, or 2 : 3=4:6, is a Proportion, and may be read, 2 is to 3 as 4 is to 6. The numbers them. selves are called PROPORTIONALS.

The first term of every ratio is called the antecedent, and the second the consequent.

The first and fourth terms are called the extremes, the second and third terms the means, and in every proportion the product of the extremes is equal to the product of the means. Take for example the proportion 3 : 9:: 5:15. This may also be written g=is. Reducing these fractions to a common denomi. nator, we have 45 –

135

The numerators are now the same, but one is the product of the extremes, and the other the product of the means.

The antecedents and consequents may, therefore, change places in a variety of ways, the proportion always continuing so long as the product of the means is equal to the product of the extremes.

Then, whenever one of the extremes and the two means are given, to find the other extreme, Divide the product of the means by the given extreme.

If any number will divide either antecedent and its consequent, the division may be performed without destroying the proportion. Thus, the proportion 9 : 12 :: 15 : 20 is the same as 192=16. Dividing 9 and 12 by 3, or 15 and 20 by 5, the corresponding fraction will be reduced to its lowest terms, and the result, 3 : 4 :: 15 : 20, or 9:12 :: 3:4, is a proportion.

There are many other interesting properties of proportions, which belong more properly to the province of

135

Geometry. What we have already learned, is sufficient for all Arithmetical purposes.

EXAMPLES FOR THE BOARD.

men.

:

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If 8 men build 40 rods of wall in 3 days, how many men will build 100 rods in 12 days? To build 100 rods it will evidently take 1000

as many men as to build 40 rods, or 100 of 8, which may be expressed by the proportion rods. rods.

men. 40 100 8 To build the wall in 12 days, it will require but as many men as to build it in 3 days, which may be expressed by the proportion

days. days.

12 These two proportions may be written together, thus, 40 : 100

2:5 .::8:

::8: 3

men.

men.

:

3 ::

100 of 8

:

40

men.

men.

12:

Or by can-
celling the
like factors
in the ante-
cedents and
consequents,

4:15

men.
5

:

480 : 300 ::8:

8:5 ::8: Multiplying together the two antecedents and the two consequents of the first ratio, we obtain a single proportion, in which the two means and one extreme are given. Dividing the product of the means by the given extreme, we obtain 5 men for the answer, and our completed proportion is,

men. 8 : 5 :: 8

Ans. Hence we derive the RULE OF THREE.

Write that number which is of the same kind as the answer, for the third term of a proportion. Of the remaining quantities, compare any two of a kind, and consider whether the answer will be greater or less than the third term. If greater, write the greater number for the second term, and the less for the first. If less, write the less number for the second term, and the greater for the first. In the same way compare any other two of a kind, and so proceed until all the numbers are employed. Cancel the like factors in the antecedents and consequents, and for the fourth term, or answer, multiply the third term by the product of the second terms, and divide by the product of the first terms.

N.B. Before cancelling, all compound numbers must be reduced to the same denomination. The pupil should be required, after performing the ex.

To Messrs

. Noros & Chopin,}

amples in this chapter according to the rule, to give an analytical solution of each question.

1. State 274 x 16 in the form of a proportion, by making 1 the first term.

2. State 6517--19 in the form of a proportion, by making 1 either the second or third term. 19:11:611)

3. What is the value in dollars and cents of the following Bill of Exchange, - exchange at 5fr. 32c. per dollar ? $132

Exchange 10000fr. Second. Phil. 13th July, 1844. Sixty days after sight of this, our second of Exchange (first and third of same tenor and date unpaid), pay to the order of J. Villiers & Cie., ten thousand francs, for value received, as per advice, for the account of

THOMPSON & CLARK. Paris. 4. In 9450 milrees, 320 rees, how many dollars and cents ?

5. Reduce 487 roubles, 63 copecks, to Federal money ; exchange at 75 cents per rouble.

6. If 29 men can do a piece of work in 11 days, by working 10 hours a day, how long will it take 18 men, working 8 hours a day, to do the same work?

7. If 93 barrels of flour cost $418.50, what will 137 barrels cost?

8. When exchange on London is at par, and £9=$40, what is the value of £27 13s. 11d.?

9. How many men will reap 96 acres in 12 days, if 13 men reap 78 acres in 6 days ?

10. If I pay $14 for the freight of 3 tons, 45 miles, what must I pay for the freight of 11 tons, 17 miles ?

11. What is the value of 19 marcs 12 schillings Hamburg,-exchange at 35 cents per marc ?

It will readily be seen that all the terms of a proportion may be distinguished into multipliers and divisors. If, then, we write all the multipliers in one column, and all the divisors in another, and cancel the factors common to both columns, the answer may be obtained by dividing the product of the multipliers by the product of the divisors.

The terms of every proportion may also be distinguished into causes and effects, and the two products of each cause by its opposite effect are equal. For example, if 8 men build 4 rods of wall in a day, 4 men will build 2 rods in the same time. Stating the proportion, we have

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The product of the extremes, and the product of the means, give us the product of each cause by its opposite effect.

Times are causes, for 2 days will produce twice as much as one day.

In questions of freight, we may regard distances and bulk as causes, producing money for their effect.

A little practice will give great facility in making this distinction in all cases. If, then, we write each effect opposite to its cause, our multipliers and divisors will be obtained without difficulty.

EXAMPLE FOR THE BOARD.

causes.

effects.

If 18 men, in 6 days of 8 hours, build a wall 150 feet long, 2 feet wide and 4 feet high, in how many days of 12 hours will 24 men build a wall 200 feet long, 3 feet wide and 6 feet high?

men 18. | 24 men. Commencing our statements, days days.

we write 18 men,

days of 8 hours 8.1 19 hours. hours, as cause, and the effect, long Røø 1$ø long.

which is a wall 150ft. long, 2ft. wide 3 wide.

wide, and 4ft. high, on the ophigh 3 & 4 high.

posite side : opposite to each of

these terms, we write days, 12 9 days Ans.

hours, 24 men, as cause, and

200 long, 3 wide, 6 high, as effect. Cancelling the like factors, we have but 3x3 on the side of the multipliers, and 1 on that of the divisors. 3x3+1 is therefore the missing term, or number of days required. If either term were fractional, the denominator representing a divisor, should be transposed to the opposite side. By proceeding in this manner, a statement may be made as soon as the question can be proposed.

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