704. To find the Principal, the Interest, Rate, and Time being given.

EXAMPLE.-What principal, in 3 years and 2 months, at 6%, will gain $47.50

interest?

OPERATION.

$.18 int. of $1, at 6%, for 3 yr. .01 int. of $1, at 6%, for 2 mo.

$.19 int. of $1, at 6%, for 3 yr. 2 mo. $47.50 interest.19 $250, principal.

EXPLANATION.-Since $1 in 3 years, at 6 per cent., will gain $.18 interest, and in 2 months $.01 interest, it will in the given time gain $.19 interest; and if $1 will in the given time gain $.19 interest, the principal that will in the given time gain $47.50 interest must be as many times $1 as $.19

is contained times in $47.50, or $250; therefore $250 is the principal which will, in 3 yr. 2 mo., at 6%, gain $47.50 interest.

Rule. Divide the given interest by the interest of one dollar for the given time and rate.

REMARK.-Whenever the divisor contains a fraction not reducible to a decimal, as in case of some fractional or odd ratio per cent., it is better that the fractional form be retained. Before division in such cases, multiply both divisor and dividend by the denominator of the fractional divisor; the relative value of the terms will not be changed, and greater exactness will be secured in the result.

EXAMPLES FOR PRACTICE.

705. 1. What principal, at 7%, will gain $154 in 6 yr. 4 mo. 24 da. ?

2. What sum of money, loaned at 41%, for 7 yr. 11 mo. 15 da., will gain $1468.21 interest?

3. What sum of money, invested at 51%, will in 7 yr. 1 mo. 1 da. produce $131.50 interest ?

4. A money lender received $221.68 interest on a sum loaned at 8% July 17, 1885, and paid Oct. 11, 1888. What was the sum loaned?

5. A dealer who clears 121% annually on his investment, is forced by ill health to give up his business; he lends his money at 7%, by which his income is reduced $1512.50. How much had he invested in his business?

6.

How many dollars must I put at interest, at 9%, Jan. 27, 1889, so that on the 18th of Nov., 1895, $506.27 interest will be due?

706. To find the Principal, the Amount, Rate, and Time being given. EXAMPLE.-What principal, at 6%, will, in 4 yr. 6 mo. 15 da., amount to

$2372.25?

OPERATION.

$1.2725 amount of $1.00 for the time. $2372.251.2725 $1864.24, principal.

=

EXPLANATION. Since a principal of $1 will, in the given time, amount to $1.2725, it will require a principal of as many times $1 to amount to $2372.25 as $1.2725 is contained times in $2372.25, or $1864.24.

Rule.-Divide the amount by the amount of 1 dollar for the given

time and rate.

EXAMPLES FOR PRACTICE.

707. 1. What sum, put at interest at 7% for 5 yr. 11 mo. 3 da., will amount to $630.90?

2. A boy is now 15 years old. How much must be invested for him, at 71% simple interest, that he may have $15000 when he becomes of age?

3. What sum, put at interest June 1, 1888, at 7%, will amount to $687.50 July 1, 1890 ?

4, What sum of money, put at interest to-day at 5%, will amount to $1031.25 in 7 mo. 15 da.?

5. What principal will amount to $308.34 in 11 mo. 9 da., at 6% ?

6.

A man loaned a sum of money to a friend from June 13 to Dec. 1, at 7%, when he received $763.28 in full payment. How much was loaned ?

7. Owing a debt of $2146. 18, due in 1 yr. 7 mo. 18 da., I deposited in a bank, allowing me 6% interest, a sum sufficient to cancel my debt when due. Find the sum deposited.

708. To find the Rate Per Cent., the Principal, Interest, and Time being given. EXAMPLE.-At what rate per cent. must $750 be loaned, for 2 yr. 5 mo. 6 da., to gain $164.25 interest?

$18.25
$164.25

OPERATION.

int. of $750 for the time at 1%.
$18.25 9 or 9%.

EXPLANATION.-The principal will gain $18.25 interest in the given time at 1 per cent.; in order that it may in the given time gain $164.25, the rate must be as many times 1 per cent. as $18.25 is contained times in $164.25, or 9 per cent.

Rule.-Divide the given interest by the interest on the given principal for the given time, at 1 per cent.

REMARK.-When the amount, interest, and time are given, to find the rate per cent., subtract the interest from the amount, thus finding the principal, then proceed as by the above rule.

EXAMPLES FOR PRACTICE.

709. 1. If I pay $518.75 interest on $1250, for 5 yr. 6 mo. 12 da., what is the rate per cent.?

2. At what rate would $710, in 3 yr. 5 mo. 20 da., produce $172.56 interest ? 3. At what rate would $4187.50 amount to $4738.68, in 1 yr. 11 mo. 12 da.? 4. If $1200 amounts to $2135.80 in 12 yr. 11 mo. 29 da., what is the rate per cent.?

5. A lady deposited in a savings bank $3750, on which she received $93.75 interest semi-annually. What per cent. of interest did she receive on her money? 6. A debt of $480, with interest from August 24, 1886, to Dec. 18, 1888, amounted to $546.72. What was the rate per cent. of interest?

7. To satisfy a debt of $1216.80, that had been on interest for 4 yr. 4 mo. 21 da., I gave my check for $1751.18. What was the rate per cent. of interest?

710. To find the Time, the Principal, Interest, and Rate being given. EXAMPLE.-In what time will $540 gain $74.52 interest, at 6% ?

REMARK.-When by inspection it is apparent that the time is less than a year, divide the given interest by the interest on the principal for the highest apparent unit of time; the quotient will be in units of the order taken, which reduce as above.

Rule.-Divide the given interest by the interest on the principal for 1 year, at the given rate per cent.

REMARK.—When the amount, interest, and rate are given to find the time, subtract the interest from the amount, thus finding the principal, and proceed as above.

EXAMPLES FOR PRACTICE.

711. 1. How long will it take $360 to gain $53.64, at 6%.

2. How long should I keep $466.25, at 8%, to have it amount to $610.48? 3. A debt of $1650 was paid, with 51% interest, on Aug. 30, 1888, by delivering a check for $2316.85. At what date was the debt contracted?

4. How long must $612 be on interest, at 7%, to amount to $651.27 ? 5. On April 1, 1888, I loaned $1120, at 5%, and when the money was due I received $1202.60 in full payment. What was the date of the payment?

6. In what time will money, bearing 8% simple interest, double itself? EXPLANATION.—In order to double itself, the interest accumulated must be equal to the principal, or be 100 per cent. of the principal. And since the principal increases 8 per cent. in one year, it will require as many years to increase 100 per cent., or to double itself, as 8 per cent. is contained times in 100 per cent., or 121, equal to 12 yr. 6 mo.

SHORT METHODS FOR FINDING INTEREST.

712. To find Interest for Days, at 6 per cent., 360 day basis, or Common Interest. EXPLANATION.—A principal of $1 will, in 1 year, at 6 per cent., gain $.06 interest. A principal of $1 will, in year, or 2 months, or 60 days, at 6 per cent., gain .01 interest. Since $.01 equals of the principal, the interest on any sum of money for 60 days, at 6 per cent., can be found by pointing off two integral places from the right; and since 6 is of 60, the interest for 6 days can be found by pointing off three places; and since ten times 60 is 600, the interest for 600 days is ten times that for 60 days, and may be found by pointing off 1 place; and since 6000 is ten times 600, the interest for 6000 days can be found by multiplying the interest for 600 days by 10, or in other words, the interest for 6000 days will equal the principal; the principal thus being shown to double itself in that time at 6 per cent. This may further

be proved true from either of two illustrations:

1st. 6000 da. ÷ 360 (12 × 30) 2d. 100% 6%

=

=

163, or 16 yr. + 8 mo. 163, or 16 yr. + 8 mo.

Hence, assuming $2136 as a principal, we form the following

REMARKS.-1. Observe, as above stated, that the interest for 6000 days equals the principal, or that any sum of money will, at common interest, double itself in 6000 days.

2. Since interest is ordinarily computed on the basis of 360 days, or 12 periods of 30 days each, as illustrated above, all results will be required on that basis, unless otherwise specified. 713.-1. To find the interest of any sum of money, at 6%, for 6 days. RULE.-Cut off three integral places from the right of the principal.

2. To find the interest of any sum of money, at 6%, for 60 days. RULE.-Cut off two integral places from the right of the principal.

3. To find the interest of any sum of money, at 6%, for 600 days. RULE.—Cut off one integral place from the right of the principal.

4. To find the interest of any sum of money, at 6%, for 6000 days. RULE.— Write the interest as being equal to the principal.

=

REMARK.-Interest is a product of which the rate and time are factors. [Formula.-Interest Principal × Rate Time.] Since the rate, being a constant factor, may be ignored, it will be observed that it will make no difference if, for convenience, the term principal (in dollars), and that of time (in days), be interchanged. Illustration: The interest of 500 (dollars) for 93 (days), is the same as the interest of 93 (dollars) for 500 (days); and since 500 is of 6000, the interest required can be found by dividing 93 (dollars) by 12, which gives $7.75. Again, the interest of 150 (dollars) for 88 (days) equals the interest of 88 (dollars) for 150 (days); and since 150 is of 600, the required interest is obtained by pointing off one place from the right of 88 (dollars), as, $8.8, and dividing the result by 4, obtaining $2.2, or $2.20, as the interest.

714. To find Interest at Other Rates than 6 per cent., 360 Day Basis.

1. To find the interest on any sum of money for 12 days, at 6 per cent. RULE.-Point off 3 places and multiply by 2.

REMARKS.-1. For any number of days divisible by 6, proceed in like manner.

2. For other rates, add or subtract fractional parts of results, as in Art. 701.

3. For odd days, add fractional parts to the result.

2. To find the interest for 18 days, at 7%. RULE.-Point off 3 places, multiply by 3, and to the result add one-sixth of itself.

3. To find the interest for 24 days, at 5%. RULE.-Point off 3 places, multiply by 4, and from the result subtract one-sixth of itself.

4. To find the interest for 36 days, at 44%. RULE.-Point off 3 places, multiply by 6, and from the result subtract one-fourth of itself.

5. To find the interest for 78 days, at 8%. RULE.-Point off 3 places, multiply by 13, and to the result add one-third of itself.

6. To find the interest for 51 days, at 6%. RULE.-Point off 3 places, multiply by 8, and to the result add one-half of the first result.

REMARK.—In a similar way all changes of time and rate may be considered.

7. To find the interest for 10 days, at 6%. divide the result by 6.

RULE.-Point off 2 places, and

8. To find the interest for 20 days, at 7%. RULE.-Point off 2 places, divide the result by 3, and to the quotient add one-sixth of itself.

9. To find the interest for 30 days, at 71%. RULE.-Point off 2 places, divide the result by 2, and to the quotient add one-fourth of itself.

10. To find the interest for 40 days, at 9%. RULE.-Point off 2 places, subtract from the result one-third of itself, and to the remainder add one-half of itself. 11. To find the interest for 45 days, at 8%. RULE.-Point off 2 places, subtract from the result one-fourth of itself, and to the remainder add one-third of itself.

12. To find the interest for 54 days, at 6%. RULE.-Point off 2 places, and from the result subtract one-tenth of itself.

13. To find the interest for 240 days, at 6%. RULE.-Point off 2 places and multiply by 4.

REMARKS.-In a similar manner obtain interest for all terms of 60 days or parts thereof, and at any required rate.

14. To find the interest for 50 days, at 6%. RULE.-Point off 1 place and divide by 12.

15. To find the interest for 100 days, at 6%. RULE.-Point off 1 place and divide by 6.

16. To find the interest for 150 days, at 6%. RULE.-Point off 1 place and divide by 4.

REMARK.-Daily class drill for five or ten minutes, during the time given to the subject of interest and its varied applications, will impart to the class an astonishing degree of accuracy and rapidity in computing interest; and while odd rates are not in common use, valuable drill may be given by their occasional introduction, and the varied changes necessary to obtain interest for odd days will insure the very best results.