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The money of the association is loaned to the stockholder who offers the highest premium for its use. The premium in some instances is deducted at once, in others it is paid in monthly installments.
Interest is sometimes charged on the amount loaned, in other cases on the amount loaned plus the premium.
When the value of each share amounts to $200, the stockholders receive that sum per share, either in cash, or by the return of their own securities to that amount.
TABLE. The amount of $1 compounded monthly, at 6% per annum, also the amount of a monthly payment of $1, compounded in the same manner.
$1.19668 $39.336 1.27049 54.098 1.34885 69.771 1.43205 86.410 1.52038 104.076 1.56656 113.313 1.61415 122.831 1.66319 132.638 1.71371 142.743 1.76577 153.155 1.81941 163.883 1.87468 174.937 1.93163 186.327 1.99031 198.063 2.05073/ 210.146
To estimate the probable duration of a series of
Stock, the value of a share being $200. 1. A building association has loaned its money on 500 shares of stock, at an average premium of 25%, thus paying $150 on each share, and receiving interest monthly at 6% per annum, along with the monthly installments. If there are 1000 shares of stock in all, how long will the association run before its shares are worth $200 each?
ANALYSIS.—The borrowing stockholders have received their money, and the owners of the remaining 500 shares are entitled to all the proceeds of the association. When the stock, therefore, is worth $175 a share, there will be enough to pay 500 shares at the rate of $200 each, as there will then be a margin of $25 on each share of the 500 shares that have already been paid, at $150 each. In the above table, $174.937 corresponds to 126 months; the shares will then be worth $200 each, and the society will therefore probably close in 126 months, or 104 years.
RULE. Make the number of shares on which money has been loaned the numerator of a fraction, and the total number of shares, the denominator. Multiply this fraction by twice the number which denotes the average premium, and subtract the product from the final value of a share ($200). Find the time in which a monthly payment of $1 will amount to the sum thus obtained, and the number of months most nearly corresponding will be the probable duration of the series.
Note.-In the practical working of building associations, so many contingencies arise, that the value of the stock can only be determined hy balancing the books.
2. What is the probable duration of an association in which the money is loaned at an average premium of 20% on of 1200 shares; the average amount loaned on a single share being $160?
3. A borrows $1800 for 10 years at 6% simple interest. B borrows $2000 of a building association at 10% premium. How much more does A's loan cost than B's, the association closing in 10 years?
ANALYSIS.—$1800 at 6% for 10 years amounts to $2880. B pays $20 monthly for 10 years=$2400. $2880 – $2400=$480, additional cost of A's loan.
4. The money of a building association has been loaned at an average premium of 33} % on the whole number of shares. In what number of months should its shares reach their final value of $200 each?
5. What is the probable duration of an association, whose money has been loaned at an average premium of 40% on 1100 shares, the whole number of shares being 1200?
6. If a building association closes in 8 years, having disposed of all its shares at a premium, what was the average rate obtained for the money?
CIRCULATING DECIMALS. A Circulating Decimal is a decimal in which certain figures are continually repeated in the same order. Thus, .111+, which is the result obtained in changing 1 to a decimal, is a circulating decimal; for if the division be continued, the figures will be found to repeat in the same order continually.
Strictly speaking, these fractions are not decimals, but nonary fractions, as they have unexpressed denominators whose value is always nine, or some multiple of nine.
A consideration of the manner in which circulating decimals originate will lead to the proper method of
treating them. Let it be required to change 1 to an
equivalent decimal. Here we observe 9)1.00.11} that the figure 1 will be repeated con
tinually, with a remainder, 1, which is evidently 1 of the value of a decimal unit in the column to which it is annexed. The 1 which takes the place of this , in case
the division is carried further, is į of the preceding decimal unit, and the entire expression might properly be written jil = .
A Repetend is the figure or series of figures repeated in a circulating decimal.
The repetend is written once, and a dot (-) is placed over a single figure when there is but one, and over the first figure and the last figure of a series. Thus, .1 and .142578 are repetends.
A Pure Circulating Decimal contains no decimal figures but those of the repetend; as .3, .66.
A Mixed Circulating Decimal contains one or more decimal figures which do not form part of the repetend; as .6333, .06.
The figures which precede the repetend are called the finite part.
A pure circulating decimal is equivalent to a fraction whose numerator is the repetend, and whose denominator is as many places of nines as there are repeating figures. For we have seen that .i = 1; hence .2= ,9=%, etc. .
A mixed circulating decimal is equivalent to a decimal and a fraction. Thus, .06 = 10=4 =10= fx bo = t. REDUCTION. 1. Change .333 to an equivalent common fraction.
ANALYSIS.—We write the repetend 333
without the decimal point, as the numeramoce 999 3 tor, and for the denominator we take as
many nines as there are places of figures
333. in the repetend, obtaining 990* which we reduce to its lowest terms .
2. Change .003 to an equivalent common fraction.
ANALYSIS.–We write the mixed circulating decimal as a mixed decimal .003; then, omitting the decimal point, and writing the denominator 100, we have an expression of division, g = 100, which we find to be equal to do.
RULE. If the circulating decimal is pure, omit the decimal point, make the repetend the numerator, and take as many nines for the denominator as there are figures in the repetend.
If the circulating decimal is mixed, change it to an equivalent mixed decimal, and simplify the resulting fraction.
3. Change .09 to an equivalent common fraction. 4. What common fraction is equivalent to .076923? 5. Reduce .133 to a common fraction. 6. Change .428571 to a common fraction. 7. What common fraction is equivalent to .4666? 8. Reduce .353 to a common fraction. 9. Reduce .222 to a common fraction. 10. What common fraction is equivalent to .42857142?