« ΠροηγούμενηΣυνέχεια »
Example: I. A grocer would mix teas at 4s. per lb. 75. per lb. gs. per lb. and 105. per lb. in such proportion that the mixture may be worth 6s. per lb. ; what quantity of each must be taken?
In this example, I first ftate the work as before directed, placing the prices of the teas in a column over each other, with 6, the given price of the mixture, on the left hand.
Secondly, I conneet the prices with each other by curve lines; 4 the top figure, being lefs than the rate of the mix ture, I connect with 7, 9, and 10, becaufe they are all greater than 6, the rate of the mixture.
Thirdly, I find the difference between 6, the price of the mixture, and 4, that of the first fimple, which is 2 ; I therefore place 2 opposite the 7, 9, and 10, as the 4 is linked to all of them. Then I find the difference between the 6 and the next figure 7, which is 1, I therefore place I opposite the 4, being the figure to which the 7 is linked. Then the difference between the 6 and 9 is 3, which I place also opposite the 4 (the 9 being linked thereto), and the difference between 6 aud 10, which is 4, I also place opposite the 4 (as the to is also linked to it). These differences, so placed, 'contain the true proportion of each fort of tea at the price opposite to each, that thould be taken to form a mixture at the deGred rate.
But opposite the 4 there are three differences, viz. I, 3, and 4, which are to be added together, as seen in the last column. Thus, there mu£ be 8lb. of tea at 45. per Ib. 2lb. at 75. per Ib. 2lb. at gs. per ib. and 2/b.at jos, per lb.; and the whole quantity of the mixture is :41b. at 6s. per lb.
These questions are proved in the same manner as thofe in alligation medial, viz. by finding the total value of all the fimples in their separate ftate, and the total value of the mix. ture; and if these two vali:co de equal, the wet is right.
TI Proof of the foregoing Example.
d. 8lb. of tea, at 45. per lb. is i 12 141b. of the mix. 4
tùre at 6s. per lb. 32 Shillings
84 fhillings, or 21b. at 75. per lb. is
0 14 0
Qu. 2. ? A farmer mixed wheat at 45. the bushel, with rye at 35. the bushel, and barley at 18d. the bufhiel, how much must he mix of each to sell the whole mixture at 22d. the bushel ? - Answer, 40 bushels at 18d. per buchel, 4 bufhels * at 36. per bushel, and 4 bulhels at 41. per bufhel.
Qu. 3. A goldsmith has gold to melt of 24 carats fine, 21 carats fine, 19 carats fine, and 16 carats fine; how much of each muft he take to form an article of gold that shall make 27 carats fine? - Answer, 1 of 24, 1 of 21, I of 19, and- 13 of 16 carats fine.
When the whole niixture is limited to a certain quantity, after finding the quantity of each of the simples as before, * fay (by the rule of three), as the fun of the quantities is to the given quantity, so is the quantity of each fimple to the required quantity of each. :
Example 4. A vintner is desirous to mix 5 forts of wine together : viz. at 115. per gallon, jos. per gallon, gs. per gallon, 75. per gallon, and 6s. per gallon, in such proportion as to make 40 gallons of wine, worth 85. per gallon : how much of each fort must he tako?
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 say, Gall, Gall. Gall.
Gall. Pints. As it is to 2 so is 40 the quantity required to 7 241
3 511 40
3 54 5
18 14t The quantity required 40 o Question 5. A grocer has sugar at 10d. 8d. 6d. and 4d. per lb. of which he would make a mixture to consist of bolb. and worth 5d. per 1b.; how much of each fort muft he take infruer, 5lb. at rod. gib. at 8d. slb. at 6d. and 45lb. a: 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 uther simples. To find what proportion of the others is requisite, I say (by the rule of VOL. I.
three), as the quantity of that simple whose particular quantity is given is to the given quantity, so is the found quantity of any other simple to the quantity required.
Example 6. A grocer would mix raisins at ud, per lb, 10d. per lb. od. per lb. and 6d. per lb. with 120lb. at 7d. per lb.; how much of each fort must be take, that the whole. may be worth $d. per lb.?
31 6 Hence the several quantities requisite are as placed in the example; but to find the quantities which should be taken of each fort, I say (by the rule of three), as 5lb. (the quan. tity there foundy is to 120lb. (the quantity required to be taken), fo is 3lb. (the quantity at 11d. per lb.) to jalb. (the quantity that should be taken). Ib. Ib.
Quefiion-7A vintner mixes wine at 125. 10s. and bra per gallong with 20 gallons at 45. per gallon; how much of each fort must be take to make the mixture worth 8s. per gallon? - Answer, 20 gallons at 12s, jo gallons at 105, and 10 gallons at 6s,
From the foregoing examples it, is evident that there are feveral ways of working this rule, according to the method.
of stating the question; as may be partly feen in the fourth and fixth examples, which, though quite different questions from each other, yet confit of the very fame figures, and may be stated in the fame manner; but are here varied for the information of the learner, and admit of ftill greater variety, as the learner may prove at his leisure.
All the caution that is necessary, in linking the numbers cogether, is, that of every two numbers that are linked together, one must be greater and the orber less than the rate or price of the mixture. Therefore, the first example in this rule, having but one number less than the rate of the mixture, admits of no other method of Itating than that described.
OF VULGAR FRACTIONS.
ca. Reduction, Subtraction, Multiplication, Division, and
the Rule of Three.
A FRACTION is any part or parts of an integer or unit, and (in vulgar fractions) is represented by two numbers placed one above the other, with a line drawn between them.
The number below the line is called the denominator, and shews how many parts the integer is divided into; the number above the line is called the numerator, and shews how many of those parts the fraction represents. + Thus the fraction to represent three farthings is thus written Numer haror 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