cent, has no more meaning than ten or one; all numbers signify possible measurements; they are empty of meaning till applied to measured quantity. It is not uncommon to find in published solutions of percentage problems different quantities used as defined by the same unit of measure because they are expressed in per cents. We have before us, for example, a solution in which the author takes it for granted that the difference between 110 per cent of one quantity and 90 per cent of a different quantity is 20 per cent. "Let 100 per cent equal the required quantity" is a very common presupposition in the solution of a percentage problem, and equally common to it to find the same 100 per cent "doing" duty for some other quantity which demands recognition in the same solution. So, in a recent English work of great pretensions, we have it posted, in all the emphasis of black letter-as a fundamental working principle-that "100 per cent is 1." One hundred per cent of any quantity-like 2-halves of it, or 3-thirds of it, or 4-fourths of it or n-nths of it is indeed the quantity taken once, or one time. But this loose way of making "100 per cent equal to 1," or to any quantity, is due to a total misconception of the nature of number as measurement of quantity, and of the function of the fraction as stating explicitly the process of measurement. It seems as if both teachers and pupils were often hypnotized by this subtle one hundred per cent.

... ,

SOME APPLICATIONS OF PERCENTAGE.

1. Profit and Loss.-We do not need either formal cases or formal rules, as "given the buying price and

the selling price to find the gain or loss per cent." A few examples will serve to illustrate the different 66 cases."

(1) Bought sugar for 6 cents per pound and sold it for 8 cents per pound; find the gain per cent.

The question simply stated is: 2 cents gain on 6 cents cost means how much gain on 100 cents of cost; that is, how many hundredths? Multiply both terms by 16 (= 100). Or,

6 cents outlay gains 2 cents;

(2) Cloth was bought at 60 cents a yard, and sold to gain 25 per cent; find the selling price.

Take the cost price as unit of comparison: Selling

On 100 cents of cost gain is 25 cents.

On 1 cent of cost gain is

25 66

100

On 60 cents of cost gain iscent × 60 = 15 cents. Hence 60 cents + 15 cents = 75 cents, the selling price.

(3) By selling cotton at 12 cents a yard there is a gain of 20 per cent; what was the cost price?

Take the cost price as unit of comparison: 20 per cent of cost is of cost; therefore, 1 cost = } cost = 12 cents. Therefore, cost 10 cents.

=

(4) By selling coffee at 30 cents a pound a grocer lost 25 per cent; what price would bring him a profit of 10 per cent?

=

of cost = 30 cents; therefore, New price of cost 11 of 40

cents 44 cents. Otherwise, the losing price,

(15)

of cost, must be increased to 1 (3) of cost-that is, must be increased in the ratio ; therefore, of 30

cents

44 cents, the price required.

(5) A merchant gains 30 per cent by selling goods at 39 cents a yard; at what selling price would he lose

40

per cent? Gaining price is

=

of cost. Losing price is of cost; therefore the latter is of the former of 39 = 18 (cents).

13

2. Stocks, Commission, etc.-A few examples will show that there is no new principle in these rules.

(1) How much cash will be realized by selling out $4,000 stock, Government 5's, at 951?

$100 stock brings $951 cash; $4,000 stock brings $951 X 40 $3,810 cash.

(2) What amount will be realized by selling out $4,400 six-per-cent stock at $1063, allowing brokerage ? Every 100 of stock brings $(1063)= $106; therefore, $4,400 of stock brings $106 X 44 = $4,675.

(3) What semi-annual income will be derived from investing $9,000 in bank stock selling at $120 and paying 4 per cent half yearly dividends?

$120 will buy $100 stock, which brings $4 incomethat is, the income is 4 ÷ 120 of the investment =

of $9,000 = $300.

(4) Which is the better investment, a stock paying 12 per cent at $140, or one paying 9 per cent at $120? What income from investing $1,400 in each?

In the first investment $140 brings $12 income; therefore, $1 brings $1 $35.

12 $140

In the second investment $120 brings $9; there

fore, $1 brings $130 = $2%; $3 is greater than $; therefore, the first is the better investment.

Income from the first =

3 of $1,400 = $120;

35

Income from the second of $1,400 $105.

=

=

(5) A commission merchant is instructed to invest $945 in certain goods, deducting his commission of 21 per cent on the price paid for the goods; find the agent's commission.

Since the agent receives $2 for every $100 he invests, $1023 must be sent for every $100 that is to be invested in goods; that is, for every $102 sent, the agent receives $21; therefore, he receives 2÷ 102

=

of the whole amount sent; therefore, amount of commission= $945 ÷ 41.

(6) For how much must a house worth $3,900 be insured at 2 per cent, so that the owner, in case of loss, may recover both the value of the house and the premium paid?

Since the premium is 24 per cent of the amount insured, the property must be 100 per cent - 23 per cent = 97 per cent of the amount insured; therefore, 28 38 of this amount = $3,900, and the amount is $4,000.

=

(7) What amount must a town be assessed so that after allowing the collector 2 per cent the net amount realized may be $24,500?

The collector gets 2 per cent = of total levy; therefore, town gets 48 of total levy; therefore, % of total levy = $24,500, and, therefore, total levy = $25,000.

INTEREST.

The pupil, having learned the meaning and the use of the term per cent, should find very little difficulty in the subject of interest. However in the problems of

interest and kindred commercial work pupils frequently fail; but if the cause of the failure is examined into, it will nearly always be found to be, not so much an inability to meet the mathematics of the problems, as a want of accurate knowledge of the terms used, and of acquaintance with the business forms and operations involved. On this account, in taking up the applications of arithmetic to commercial work, the teacher should be at great pains to ensure that every pupil understands well, and sees clearly through, all such forms and operations.

Simple Interest.—In accordance with what has been said, it is necessary first to explain to the class how men, when loaning money, require a certain payment for the use of the money, and how the amount to be paid for this use that is to say, the interest-depends on the time, twice, thrice, etc., the time (implying, as it does, twice, thrice, etc., the use), requiring twice, thrice, etc., the interest. The unit of time is generally taken as one year, and the rate for the year is given as a per cent. Accordingly, if we say that a man loans money for a year at 5 per cent per annum, we mean that at the end of one year he would receive as interest of the money loaned; if the money were loaned for half a year the interest would be of of the sum loaned, and if for fifty-three days, it would be 5 of 15 of the sum loaned. The pupil is now prepared to do any problem of calculation of simple interest, and after being trained in the formal working and stating of such problems—that is, after realizing the problem to the full-should be trained in making rapid calculations after the methods of men in business.