These questions are proved in the fame manner as thofe in alligation medial, viz. by finding the total value of all the fimples in their separate state, and the total value of the mixture; and if these two values be equal, the work is right.

Qu. 2. A farmer mixed wheat at 45. the bufhel, with rye at 35. the bufhel, and barley at 18d. the bufhel, how much must he mix of each to fell the whole mixture at 22d. the bufhel?-Anfwer, 40 bushels at 18d. per bufhel, 4 bushels at 3s. per bufhel, and 4 bushels at 45. per bufhel. Qu. 3. A goldfmith has gold to melt of 24 carats fine, 21 carats fine, 19 carats fine, and 16 carats fine; how much of each must he take to form an article of gold that shall make 17 carats fine-Answer, 1 of 24, 1 of 21, 1 of 19, and 13 of 16 carats fine.

When the whole mixture is limited to a certain quantity, after finding the quantity of each of the fimples as before, Lay (by the rule of three), as the fun of the quantities is to the given quantity, fo is the quantity of each fimple to the required quantity of each.

Example

Example 4. A vintner is defirous to mix forts of wine 5 together: viz. at 11s. per gallon, 10s. per gallon, 9s. per gallon, 75. per gallon, and 6s. per gallon, in fuch proportion as to make 40 gallons of wine, worth 85. per gallon: họw much of each fort must he take?

In this example, after linking the prices together, as before directed, I have the quantity of each wine to form a mixture of 8s. per gallon; but the whole quantity of the mixture thus found is only 11 gallons, whereas it should be 40 gallons, therefore I fay,

Gall.

Gall. Gall.

Gall. Pints.

As 11 is to 2 fo is 40 the quantity required to 7 2

Queftion 5. A grocer has fugar at 10d. 8d. 6d. and 4d. per lb. of which he would make a mixture to consist of 60lb. and worth 5d. per lb.; how much of each fort must he take? Anfwer, 5lb. at 10d. 5lb. at 8d. 5lb. at 6d. and 45lb. at 4d. per lb.

Sometimes it is required to take a certain quantity of any one fimple to mix with the others, and which generally alters the quantities of the other fimples. To find what proportion of the others is requifite, I fay (by the rule of VOL. I. Cc three),

three), as the quantity of that fimple whofe particular quantity is given is to the given quantity, fo is the found quantity of any other fimple to the quantity required.

Example 6. A grocer would mix raifins at 11d. per lb. 10d. per lb. 9d. per lb. and 6d. per lb. with 120lb. at 7d. per lb.; how much of each fort must he take, that the whole may be worth 8d. per lb. ?

Hence the feveral quantities requifite are as placed in the example; but to find the quantities which fhould be taken of each fort, I fay (by the rule of three), as 5lb. (the quantity there found) is to 120lb. (the quantity required to be taken), fo is 3lb. (the quantity at 11d. per lb.) to 72lb. (the quantity that fhould be taken).

As 5 is to 120 fo is 3 to the quantity required

Ib.

72

Question 7. A vintner mixes wine at 125. 10s. and 6s. per gallon, with 20 gallons at 45. per gallon; how much of each fort must he take to make the mixture worth 8s. per gallon?-Anfwer, 20 gallons at 125. 10 gallons at 10s. and 10 gallons at 65.

From the foregoing examples it is evident that there are feveral ways of working this rule, according to the method

of

of ftating the question; as may be partly feen in the fourth and fixth examples, which, though quite different questions from each other, yet confift of the very fame figures, and may be stated in the fame manner; but are here varied for the information of the learnier, and admit of still greater variety, as the learner may prove at his leisure. '

All the caution that is neceffary in linking the numbers together, is, that of every two numbers that are linked together, one must be greater and the other lefs than the rate or price of the mixture. Therefore, the, firft example in this rule, having but one number lefs than the rate of the mixture, admits of no other method of ftating than that defcribed.

SECT. XIII.

OF VULGAR FRACTIONS.

Reduction, Subtraction, Multiplication, Divifion, and the Rule of Three.

A FRACTION is any part or parts of an integer or unit, and (in vulgar fractions) is reprefented by two numbers placed one above the other, with a line drawn between them.

The number below the line is called the denominator, and fhews how many parts the integer is divided into; the number above the line is called the numerator, and thews how many of thofe parts the fraction reprefents.

Denominator

Thus the fraction to reprefent three farthings is thus written Numerator and is properly called three-fourths of a penny; a penny being the integer, or unit, of which the fraction is a part.

Cc 2

Vulgar

Vulgar fractions are either proper or improper, fingle or compound.

A proper fraction has its numerator lefs than its denominator: as, two fixths, three fourths, three ninths; and always ftands for less than the integer it is taken from.

An improper fraction has the numerator equal to or greater than the denominator, as,,, and always reprefents as much or more than the integer.

A fingle fraction is only a single expreffion of any affigned parts of an integer.

A compound fraction confifts of more than one fraction, and is a fraction of a fraction, and they have the particle of placed between them, as of, of 3, &c.

There are also mixed numbers, which are whole numbers united with fractions, as 8, 121%, &c.

The common measure of two or more numbers, is that number which will divide each of them without a remainder; thus 4 is the common measure of 12 and 16.

A number that can be exactly measured by two or more numbers is called their common multiple; and if it be the leaft number that can be measured, it is called their leaft common multiple thus, 45 and 60 are the common multiples of both and 5, but their least common multiple is 15.

3

:

Before the learner can proceed to reduction of fractions, it is neceffary that he be able to folve the two following Problems.

To find the greatest common Measure of two or more Numbers.

:

Rule. If there be only two numbers, divide the greater by the lefs and if there be no remainder, the divifor is the greatest common measure; but if there be a remainder, the divifor is to be divided by fuch remainder, and if there still

be