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Analysis. The interest of $75 at 1 per cent. for 1 year, is $.75, and for 4 years it is $.75×4=$3. (Art. 238.) Now since $3 is 1 per cent. interest on the given principal for the given time, $24 must be 24 of 1 per cent., which is equal to 8 per cent. (Art. 121.)

Or, we may reason thus: If $3 is 1 per cent. on the principal for the given time, $24 must be as many per cent. as $3 is contained times in $24; and $24-$38. Ans. 8 per cent.

PROOF.-$75.08 $6.00, the interest for 1 year at 8 per cent., and $6×4=$24, the interest of $75 for 4 years at 8 per cent. Hence,

253. To find the rate per cent. when the principal, interest, and time, are given.

Make the given interest the numerator, and the interest of the given principal at 1 per cent. for the given time, the denominator of a common fraction, which reduced to a whole number will give the per cent. required. (Art. 121.)

Or, simply divide the given interest by the interest of the given principal at 1 per cent. for the given time, and the quotient will be the per cent.

2. If I borrow $300 for 2 years, and pay $42 interest, what rate per cent. do I pay?

Operation. $6)$42

7 Ans. 7 per ct.

PROOF.

The interest of $300 for 2y., at 1 per ct., is $6. (Art. 238.)

$300.07×2=$42.

3. If I borrow $460 for 3 years, and pay $82.80 interest, what is the rate per cent.?

4. A man loaned $500 for 8 months, and received $40 interest: what was the rate per cent.?

5. At what rate per cent. must $450 be loaned, to give $56.50 interest in 1 year and 6 months?

QUEST.-253. When the principal, interest, and time, are given, how is the rate per cent. found?

6. At what per cent. must $750 be loaned, to give

$225 in 4 years?

7. A man has $8000 which he wishes to loan for $600. per annum for his support: at what per cent. must he loan it?

8. A gentleman deposited $1250 in a savings bank, for which he received $31.25 every 6 months: what per cent. interest did he receive on his money?

9. A capitalist invested $9260 in Railroad stock, and drew a semi-annual dividend of $416.70: what rate per cent. interest did he receive on his money?

10. A man built a Hotel at an expense of $175000, and rented it for $8750 per annum: what per cent. interest did his money yield him?

PROBLEM III.

11. What sum must be put at interest, at 6 per cent., to gain $30 in two years?

Note. In this example it will be observed that the interest, the rate per cent., and the time, are given, to find the principal.

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Analysis. The interest of $1 for 2 years at 6 per cent., (the given time and rate,) is 12 cents. Now 12 cents is of its principal $1. Consequently $30, the given interest, must be 12 of the principal required. The question therefore resolves itself into this: $30 is 12 of what number of dollars? If $30 is 12 is $24; and 100 =$2×100, which is $250, the principal required.

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100, 100

Or, we may reason thus: Since 12 cents is the interest of 1 dollar for the given time and rate, 30 dollars must be the interest of as many dollars for the same time and rate, as 12 cents is contained times in 30 dollars. And $30÷.12=250. Ans. $250.

PROOF.-$250.06=$15.00, the interest for 1 year at the given per cent., and $15×2=$30, the given interest. Hence,

254. To find the principal, when the interest, rate per cent., and time, are given.

Make the interest of $1 for the given time and rate, the numerator, and 100 the denominator of a common fraction; then divide the given interest by this fraction; (Art. 141;) and the quotient will be the principal required.

Or, simply divide the given interest by the interest of $1 for the given time and rate expressed in decimals; and the quotient will be the principal.

12. What sum put at interest will produce $13.30 in 6 months, at 7 per cent. ?

Operation. $.035) $13.300

380. Ans. $380.

The int. of $1 for 6 mo. at 7 per cent. is $.035. (Art. 239.)

PROOF.-$380×.035 $13.30. (Art. 247.)

13. A father bequeaths his son $500 a year: what sum must be invested, at 5 per cent., to produce it?

14. What sum must be put at 6 per cent. interest, to give $350 interest semi-annually?

15. A gentleman retiring from business, loaned his money at 7 per cent., and received an income of $1200 a year how much was he worth?

PROBLEM IV.

16. A man loaned $80 at 5 per cent., and received $10 interest: how long was it loaned?

Note. In this example it will be observed that the principal, the interest, and the rate per cent., are given, to find the time.

Analysis. The interest of $80 at 5 per cent. for 1 year is $4. (Art. 237.) Now, since $4 interest requires the given principal 1 year at the given per cent., $10 inter

QUEST.-254. When the interest, rate per cent., and time, are given, how is the principal found?

est will require the same principal 10 of 1 year, which is equal to 21⁄2 years. (Art. 121.)

Or, we may reason thus: If $4 interest requires the use of the given principal 1 year, $10 interest will require the same principal as many years as $4 is contained times in $10. And $10÷$4=2.5. Ans. 2.5 years.

PROOF.-$80.05 $4, the interest for 1 year;

$4×23=$10, the given interest. Hence, 255. To find the time, when the principal, interest, and rate per cent., are given.

Make the given interest the numerator, and the interest of the given principal for 1 year at the given rate the denominator of a common fraction, which being reduced to a whole or mixed number, will give the time required.

Or, simply divide the given interest by the interest of the given principal at the given rate for 1 year, and the quotient will be the time.

OBS. If the quotient contains a decimal of a year, it should be reduced to months and days. (Art. 166.)

17. How long will it take $100, at 5 per cent., to double itself; that is, to gain $100 interest?

Operation. The interest of $100 for 1 year, at 5 per $5)$100 cent., is $5. (Art. 237.)

20 Ans. 20 years.

PROOF.-$100 ×.05×20=$100. (Art. 238.)

18. In what time will $500, at 6 per cent., produce $100 interest?

19. How long will it take $100, at 6 per cent., to double itself?

20. How long will it take $100, at 7 per cent., to double itself?

QUEST.-255. When the principal, interest, and rate per cent., are given, how is the time found? Obs. When the quotient contains a decimal of a year, what should be done with it?

TABLE,

Showing in what time any given principal will double itself at any rate, from 1 to 20 per cent. Simple Interest.

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256. Compound Interest is the interest arising not only from the principal, but also from the interest itself, after it becomes due.

OBS. 1. Compound Interest is often called interest upon interest. 2. When the interest is paid on the principal only, it is called Simple Interest.

Ex. 1. What is the compound interest of $500 for 3 years, at 6 per cent. ?

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$530 X.06

$561.80 x .06=

530 Amt. for 1 year.

31.80 Int. for 2d year.

561.80 Amt. for 2 years.

33.70 Int. for 3d year.

$595.50 Amt. for 3 years.

500.00 Prin. deducted.

Ans. $95.50 compound Int. for 3 years.

QUEST. 256. From what does compound interest arise? Obs. What is compound interest often called? What is Simple Interest?

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