ALLIGATION.

214. Under this head come questions which relate to the mixing of different kinds of the same commodity-tea, sugar, wine, &c.; the object of such mixing being (a) to improve the quality of an inferior article, or (b) to make a superior article cheaper and more saleable.

The following examples explain themselves :

EXAMPLE I.—If 20 lbs. of tea worth 2s. 6d. a pound, 16 lbs. worth 3s. a pound, and 12 lbs. worth 38. 4d. a pound were mixed together, how much a pound would the mixture be worth?

EXAMPLE II.-In what proportions does a grocer mix sugar worth 6d. a pound and sugar worth 4d. a pound, when he wants the mixture to be worth 54d. a pound?

On every pound of 6-penny sugar in the mixture, the grocer loses (6d.-54d.=) 3 farthings; whilst on every pound of 4-penny sugar he gains (54d.-4d.=) 5 farthings. Consequently, as the total loss and the total gain must exactly counterbalance each other, the grocer, instead of mixing the two sugars in equal proportions, makes the 6-penny sugar as much larger in quantity than the 4-penny, as 5 farthings are larger in amount than 3 farthings. So that 5 lbs. of the dearer, and 3 lbs. of the cheaper sugar would form the required mixture; 5 times 3 farthings being the loss on the former, and 3 times 5 farthings the gain on the latter :

In practice, the work would assume the following form-the numbers indicating the gain and the loss per pound being taken cross-wise for the required proportions:

NOTE. If a particular quantity of the mixture were required, the number denoting the quantity should be divided into parts proportional to 5 and 3. Thus, in 4 lbs. of the mixture there would be 2 lbs. of the 6-penny and 1 lbs. of the 4-penny sugar; in 24 lbs. of the mixture, 15 lbs. of the 6-penny and 9 lbs. of the 4-penny sugar; &c.—

d. d.

15 @ 6 = 90

9" 4 = 36

24 51=126

EXAMPLE III.-In what proportions should a vintner mix four different kinds of sherry-worth, respectively, 158., 208., 30s., and 32s. a gallon-in order to form a mixture worth 24s. a gallon?

On every gallon of the

It is evident, therefore, after what has already been explained, that the gains would counterbalance the losses if (a) the mixture were made to contain 6 gallons of the first sherry for every 9 of the third, and 8 gallons of the second for every 4 of the fourth; or if (b) 8 gallons of the first were taken for every 9 of the fourth, and 6 gallons of the second for every 4 of the third.

According to the first arrangement, the gains and losses

According to the second arrangement, the gains and losses would be

So that the wines could be mixed in either of two ways— corresponding to the two different ways in which the gains and losses could be made to counterbalance one another :

EXAMPLE IV.-There are three different kinds of rum-worth 168., 158., and 8s. a gallon, respectively; what quantity of each would be required to form 120 gallons of a mixture worth 118. a gallon?

We first find the proportions in which the three kinds should be mixed. On every gallon of the

In order, therefore, that the loss on the first rum may be counterbalanced by the gain on the third, the mixture should contain 5 gallons of the latter for every 3 of the former; and that the loss on the second rum may be counterbalanced by the gain on the third, there should be 4 gallons of the 8s. for every 3 of the 158. kind:

15", 11=165

Dividing 120 into three parts proportional to the numbers 3, 3, and 9, we thus find the required quantities to be 24 gallons of the 168., 24 of the 158., and 72 of the 8s. rum:

galls.

8.

7 @ 16 = 112 13, 15 = 195

NOTE. The "proportions" found in the last example are by no means the only ones which would answer. Thus, if 7 gallons (say) of the 16s. and 13 gallons of the 158. rum were taken, there would be a loss on the former, of (7 X 5=) 358.; on the latter, of (13 X 4=) 528.; altogether, of (35+52) 878. Dividing 878., therefore, by 38.-the gain on a gallon of the 8s. rum, we see that the necessary quantity of this rum would be (873) 29 gallons. So that if 120 were divided into parts proportional to 7, 13, and 29, those parts would fulfil the required conditions.

29,,

49,,

8=

232

11=539

POWERS AND ROOTS.

215. A number which is the product of two or more equal factors is said to be a POWER of one of those factors; and a number from whose repetition, as factor, a certain product can be obtained is called a ROOT of that product.

We say that 49 is a power of 7, or that 7 is a root of 49; that 125 is a power of 5, or that 5 is a root of 125; that 1,296 is a power of 6, or that 6 is a root of 1,296; and so on:

49=7×7; 125=5×5×5; 1,296=6×6×6×6; &c.

216. The number denoting how many times the root is contained, as factor, in the power is termed the INDEX.

We say that 49 is the second power of 7, or that 7 is the second root of 49; that 125 is the third power of 5, or that 5 is the third root of 125; that 1,296 is the fourth power of 6, or that 6 is the fourth root of 1,296; and so on. Here the indices are 2, 3, 4, &c., and are written in either of the following two ways:—

It will be observed that 72, 53, and 64 are short expressions for 7×7, 5×5×5, and 6×6×6×6, respectively. It will also be observed that "49=72" and "7=49 are two different statements of the one fact: 49=72" meaning that 49 is the second power of 7; and "7=49,' that 7 is the second root of 49. Again, "125=53" and "5=V125" are two different statements of the one fact, and the remark is equally applicable to the expressions" 1,296=64” and “6=√1,296”: because,