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B.'s invoice was 76.50 marks, which must be reduced to sterling money on the basis of 21 marks = £1, or including the 5% commission, on the basis of 21 marks = £1.05, which gives 76.50 × £1.05 = £3.825 £3 16s. 6d., the amount of B.'s draft.
B. must give the Bank a cheque for £3.825 × $44 $18.70.
Question 5.-Multiply by contracted method and give the answer to three places of decimals.
64.4235 × 318.4851.
The question calls for three places of decimals, but to get the correct answer to three places it is better by the contracted method to multiply to four places on account of the "carrying figures." This is done by reversing the multiplier and placing it under the multiplicand with the units figure under the fourth decimal place; each of the other figures then falls naturally into its proper position. Multiplying by the 6 tens gives us four decimal places in our product. Next multiply by the four units, commencing at the
figure directly above it, which being a decimal of the fourth place gives a product of four decimal places; this product must be arranged under the previous one beginning at the extreme right. Now we multiply by 4, commencing at the 5 directly above it; this is really multiplying a decimal of 3 places by a decimal of 1 place, and gives, therefore, a decimal of four places for result, which is arranged as before at the extreme right. It will be noticed that all the results are in four decimal places, and when added give us an answer in four decimal places. The figures to the right of the one directly above that we are multiplying by are totally disregarded in the working.
Rule-Reverse the multiplier, placing the unit figure thereof directly under the decimal to which it is intended to extend the work (which should be one place further than an accurate answer is required to). Multiply as in ordinary multiplication, ignoring all figures in the multiplicand to the right of the figure we are multiplying by. Arrange the several products so that the figures on the extreme right are directly under each other. Add and point off the number of decimals to which work has been extended.
Question 6.-A Merchant holds a Mortgage of $6,250, Principal payable in four equal annual instalments of $1,562.50, Interest 5% payable annually. Being in need of money he offers to sell it to B. at a price that will give B. 8% on his investment. What amount did B. pay?
A. The annual Payment on Principal being $1,562.50, the Payment of interest for the several years is as follows:
1st year, 5% on $6,250
$312 50. Total payment
3rd year, 5% on $3, 125.00 = 156 25. Total payment = $1,562 50 + $156 25 = $1,718 75.
B. buys this mortgage at a price that will pay him 8% per annum on his money for the time it is invested. We must therefore find the present value of each of the above instalments.
Each $1 paid by B. for 1st instalment is worth $1.08 when 1st instalment is due: value of instalment No. 1 = $1875 $1736.11.
Each $1 paid for 2nd instalment is worth $1.08 × $1.08 = $1.1664 when 2nd instalment is due: value of instalment No. 2 $1796-88 1.1664
Each $1 paid for 3rd instalment is worth $1.08 × 1.08 × 1.08 = $1.2597 when 3rd instalment is due: value of instalment No. 3 = $1718.75 = $1364.40. 1.2597
Each $1 paid for 4th instalment is worth $1.08 × 1.08 × 1.08 × 1.08 $1.36049 when 4th instalment is due value of instalment = $1640.62 = $1205.90.
B. should pay for the mortgage $1736.11 + $1540.53 + $1364.40+ $1205.90 $5846.94.
Question 7-A Jeweller has 5 oz. Gold, 16 Carats fine, and melts it with 10 oz. Gold, 12 Carats fine. How much pure Gold must he add to his mixture to make it 20 Carats fine?
100 carats shortage ÷ 4 carats surplus
Pure gold needed to raise the mixture to required standard 25 oz.—Ans.
The fineness of Pure Gold is fixed at 24 Carats. Gold which is 20 Carats fine has only 20 parts out of 24 Gold, the rest of it is some other metal. In working this question each ounce is supposed to be divided into 24 parts. Each ounce of 16 Carat Gold contains only 16 parts Gold, which is 4 parts below the standard' called for in the mixture. The shortage of Gold in 5 ounces would therefore be 4 parts X5-20 parts. Similarly the shortage in each ounce of 12 Carat Gold is 8 parts, and in 10 ounces it would be 8 parts 10=80 parts. Both together show a total shortage of 20+80=100 parts. This is to be made up by putting in Pure Gold, each ounce of which gives us a surplus of 4 parts as the average required is only 20 and Pure Gold is 24. The question thus becomes a very simple one, viz.: "How many ounces, each showing an excess of 4 parts Gold, will be needed to make up a shortage of 100 parts?" Just 100 divided by 4, or 25 ounces.
Question 8.-A Farmer has a Mortgage on his Farm of $5,340, Principal payable in five years from 1st January, 1898. Interest payable half-yearly, 5%. He desires to deposit in a Savings Bank an annual deposit which compounded at 4% every six months will give him enough to pay off the Mortgage at the end of five years.
1st. What amount of Interest does he pay on the Mortgage every six months?
2nd. How much must he deposit on the 1st of January, 1898, and four following years, to have $5,340 at the end of five years.
A.-1. His half-yearly Interest on Mortgage is just 21% on $5,340 $133.50.
2. Each dollar deposited January 1, 1898, bears compound Interest at 2% half year for 10 half-years. Similarly each dollar deposited January 1, 1899, bears compound Interest at 2% per half-year for 8 half-years, etc.
$1 deposited Jan. 1/98 would at the end of the time be worth 1.0210
as often as $5.63964 is contained in $5340 there must be $1 in the
end of each year.
$5.42065, final value of $1 annuity for 5 years paid at
Add compound interest on $1 for time .21899 + $5.63964 value of $1, invested annually for 5 years, at the beginning of each year. $5340 $5.63964 $946,87.-Answer.
Annuities and Sinking Funds.
Before calculations in Annuities, Sinking Funds, Debentures, etc., can be understood, it is first necessary to have a thorough grasp of the principles underlying Compound Interest. For instance, what causes an investment of $1 to amount to $1.36857 if out at Compound Interest for 8 years? Simply an annuity of 4 cents added to it, which in its turn gathers interest at the same rate. If, instead of leaving the first year's Interest in the same Bank as the $1, we withdrew the 4 cents and deposited it at the same rate in another Bank, and do likewise with each successive year's 4 cents as it becomes due, we shall then have at the end of eight years just $1 in the first Bank and $.36857 in the second Bank. The amount in the second Bank is simply an annual deposit of 4 cents at the end of each year for eight successive years, with the Interest on these deposits at 4% compound added thereto.