§ LXXVIII. DAYS. To a pupil, the calculation of interest for odd days, as they are called, is usually perplexing. We hope to make it clear. In calculating interest, 30 days are reckoned to the month, and 12 months to the year. This is slightly erroneous, but custom has rendered it almost universal. It will be seen, that this mode of allowance makes the year to consist of only 360 days. For ordinary purposes, however, it is sufficiently accurate; and its convenience renders it general.

MENTAL EXERCISES.

1. If the rate for 2 months be 1 pr. ct., that is, if the rate for 60 days be 1 pr. ct., what is that for 6 days? NOTE. 6 days=60-10= of 60 days. A.

pr. ct.

2. If the rate for 6 days be pr. ct., what is that for 12 days? For 18 days? For 24 days? For 30 days? For 36 days? For 42 days? For 48 days? For 54 days. 3. What is the rate pr. ct. for 2 months and 6 days? NOTE. For 2 months, it is 1 pr. ct., and for 6 days pr. ct. Ans. 1% pr. ct. 4. What is the rate for 2 months, 12 days? For 2 mo. 18 d.? For 2 mo. 24 d. ? For 2 mo. 36 d.? 18 d.? For 4 mo. 24 d. ?

For 4 mo.

5. What is the rate for 6 mo. 6 d.? For 6 mo. 12 d. ? For 8 mo. 18 d.?

6. What is the rate for 8 mo. 6d. ? For 12 mo. 18 d. ? For 1 yr. 24 d. ?

It will be observed, that for once 6 days we have pr. ct.; for twice 6 days, we have pr. ct.; for three times 6 days, we have pr. ct. &c. Now pr. ct. is .001; (§ LXXIV.) is .002; is .003, and so on.

Hence, for every 6 days we have one thousandth in

the rate.

If there be fewer days than 6, as 3 days, we shall have a part of a thousandth, that is, in this case, 3=1 a thousandth =.0005. If the days be 2, we shall have = of a thousandth; =.0003; if 4, 4= of a thousandth = .0006; if 5, of a thousandth =.00083; if 1, of a thousandth=.00016. Hence, to find the rate decimally for any number of days.

DIVIDE THE DAYS BY 6, ANNEXING CYPHERS, IF NECESSARY, AND PUT THE FIRST FIGURE OF THE QUOTIENT IN THE THOUSANDTHS' PLACE, WHEN

THE DAYS ARE 6 OR MORE, IN THE TEN THOUSANDTHS' WHEN THEY ARE UNDER 6.

Thus, for the rate for 23 days; 23÷6=383. As the days are over 6, put the first figure 3 in the thousandths' place, thus, .00383. The rate for two days is found, thus; 2÷6=3. As the days here are under 6, the quotient must begin in the ten thousandths' place, thus .0003. In these cases the cyphers, though not written, were supposed to be annexed.

EXAMPLES FOR PRACTICE.

1. What is the interest of $100.00 for 2 mo. 6d. ? A. $1.10. Rate for 2 mo.=.01 For 6 d. .001. Rate for the whole time =.011.

2. Find the interest on $600.00 for 4 mo. 18d. A. $13.80. Rate for 4 mo.=.02. For 18 d.=.003. Rate for whole time =.023.

3. Find the interest of $800.00 for 6 mo. 21 d. A. $26.80. Rate for 6 mo.=.03.

=.0335.

For 21 d.=.0035. Rate for whole time

4. Find the interest on $9,000.00 for 8 mo. 4 d. A. $366.00. Rate for 8 mo.-.04. For 4 d.=.0006. For whole time=.0406. 5. Find the interest on $127.47 for 2 mo. 12 d. A. $1.53— 6. Find the interest on $115.42 for 7 mo. 15 d. A. $4.328+ 7. Find the interest on $143.18 for 1 yr. 7 mo. 14 d. A$13.936Rate for 1 yr.=.06. For 7 mo.=.035. For 14 d.=.0023. For whole time=.0973.

NOTE. We would here mention, that when the rate repeats, or is a long decimal, the vulgar Fraction may be retained. This will often be most convenient, and convenience should decide the question. Thus, instead of using the decimal .0973, we would recommend the use of .0973. This is likewise perfectly accurate, whereas, when the repetend is used, perfect accuracy cannot be obtained, by the common mode of multiplying.

8. Find the interest on $625.00, for 3 yrs. 2 mo. 7 d.

A. 119.479. Rate for 3 yrs. =.18. For 2 mo. =.01. For 7 d. =.001. For whole time =.191.

9. Find the interest on $930.00 for 6 yrs.

mo. 11 d.

Ans. $350.455. 10. Find the interest on $865.25 for 9 yrs. 4 mo. 15 d.

NOTE. When the months are even, and the days are an aliquot, or an even part of 2 months or 60 days, it is well to make a vulgar Fraction for the days instead of a decimal. For example, if, as in the last question, there are 15 days, this will be of 60. Now the rate for 60 d. or 2 mo. is .01. Therefore the rate for 15 d. is of .01.04. This will often save figures. Then, in the last question, the rate for 9 yrs.=.54, for 4 mo.-.02, for 15 d.=.04, for the whole .564. Ans. $486.703.

11. Find the interest of $750.00 for 12 yrs. 3 mo.

NOTE. The learner will readily see that an odd month, being of 2 months, may have .0 as its rate. Then, in the last example, rate for 12 yrs.=.72, for 3 mo.01, whole rate .73. Ans. $551.25.

These abbreviations are only suggested, and if they do not seem clear, the pupil is recommended to follow the general mode. Other vulgar Fractions may sometimes be conveniently used, which we leave the pupil to discover. Hence, the general rule for calculating interest seems to be

FIND THE DECIMAL RATES FOR YEARS, MONTHS AND DAYS SEPARATE-
LY, ADD THEM TOGETHER, AND MULTIPLY THE PRINCIPAL BY THEIR SUM.
12. Find the interest on $725.34 for 16 yrs, 6 mo. 12 d.
NOTE. 12 d. is of 60 d. Hence the rate is .99.

13. Find the interest on $348.31 for 11 yrs. 8 mo. 10 d.
14. Find the interest on $795.333 for 13 yrs. 9 mo.

15. Find the interest on $5,863.63 for 7 yrs. 11 mo.

16. Find the interest on $325,965.813 for 15 yrs. 8 mo. 20 d. 17. A note was given with interest, for $2,635.00, on the 17th of February, 1827, and paid on the 12th of March, 1830. What was the interest due, and what was the amount of the note ?

A. Int. $485.2791. Am. $3,120.2791.

NOTE. In this example we are obliged to find the time. In business, we are almost always under the necessity of doing this. It may be done by Subtraction of Compound numbers, ($XXXVII.) The greater number consists of 1830 years, 2 months, (Jan. and Feb.) and 12 days, (in March.) The less, consists of 1827 Yrs. mo. d. years, 1 month, (Jan.) and 17 days, (in Feb.) 1830; 2; 12 1827; 1; 17

Set down the numbers, thus, In subtracting, disregard entirely the inequality of the months, and call them all 30 days, allowing 12 months to the year. If interest be required for days exactly, the inequality of months must be taken into account, (see § LXXIX.)

3; 0; 25

18. A note was given with interest, for $1,143.16 on the 9th of August, 1826, and paid on the 16th of April, 1830. What was the amount of the note, when paid? A. $1,395.989

19. A note was given, with interest, for $6,325.13 on the 8th of October, 1823, and paid on the 11th of August, 1829. What was then due ?

20. A note was given, with interest, for $2,647.53, on the 27th of December, 1821, and paid on the 15th of November, 1828. What was then due ?

21. A note was given for $15,833.75, on the 8th of July, 1819, and paid, with interest, on the 11th of June, 1827. What was then due?

22. A note was given on the 23d of March, 1813, and paid, with interest, on the 17th of September, 1830. The note was for $12,981 .375. What was due at the time of payment?

23. A note continued accumulating interest for 50 years, 9 mo. 3 d. It was given for $25,864.29. What was the interest? What was the amount ?

24. A note of $30,329.18, lay on interest from July 19th, 1763, to August 17th, 1829. What did it amount to?

§ LXXIX. In BANKS, notes are usually written for a certain number of days, as 30 days, 60 days, 95 days, &c. Sometimes, also, but not in Banks, interest is calculated for years and days, without regard to months,

A

of

The interest accurately for 1 day is, of course, the interest for 1 year. For 2 days, then, it is; for 3 days,; for 47 days, 7 of the interest for 1 year, and so on.

Hence, to find the interest accurately for any number of days: FIND WHAT FRACTIONAL PART OF 365 THE DAYS ARE, AND MULTIPLY THE INTEREST FOR 1 YEAR BY IT.

NOTE. Reduce the Fraction to its lowest terms, before multiplying. This method is so tedious that it is little used. The common mode is usually as accurate as is necessary.

1. Find the interest on $200.00 for 73 days.

73 365

200.06=$12.00. $12.00×3=$2.40 Ans. Ans. By com. meth. $2.433.

2. Find the interest on 600.00 for 55 days. NOTE..

A. $5.4244.

Ans. By com. meth. $5.50.

3. A note lay at interest from the 15th Jan. 1824, to the 19th April, 1827. It was given for $1,025.67. What was its interest at the time of settlement?

A. $200.126623.

Ans. By com. meth. $200.68943.

NOTE. The error of the common mode was here greater on account of the intervention of February.

4. Find by both modes, the amount of $943.61 for 5 years, 47 d. 5. Find the interest, by both modes, on $1,864.00 for 6 yrs. 125 d. 6. What was the amount of a note for $7,684.33, which had been lying on interest from Dec. 19th, 1811, to May 27th, 1825, by both modes ?

§ LXXX. We mentioned (§LXXVII.) that it was easier to find interest at pr. ct. than at any other rate. The pupil is now prepared to understand the reason of this; which is, that 6 is just half the number of months in the year. To find interest at any other rate, for months and days, let it be remembered, that

1 pr. ct. is of 6 pr. ct.

2 pr. ct. is 21 of 6.

3

pr. ct. is 3=1 of 6.

4 pr. ct. is 4 of 6.

5 pr. ct. is of 6.

7 pr. ct. is 14 times 6. 8 pr. ct. is &=4=1}

66 6.

ct. is 2=3=11

ct. is ==13

11 pr. ct. is=15

In this manner, may be found what part any rate is of 6 pr. ct. Hence, we have a universal rule for interest at any rate pr. ct.

có có

1. FIND THE RATE FOR THE WHOLE TIME AT 6 PR. CT., TAKE SUCH PART OF THIS AS THE GIVEN RATE IS OF 6 PR. CT., AND MULTIPLY THE GIVEN PRINCIPAL BY IT. Or the following:

II. FIND THE INTEREST At 6 pr. ct., and take sucH PART OF IT as THE GIVEN RATE IS OF 6 PR. CT.

We leave abbreviations to the ingenuity of the pupil.

EXAMPLES FOR PRACTICE.

1. What is the interest on $29.25 for 4 months, at 3 pr. ct.? NOTE. 3 pr. ct. is of 6 pr. ct. Therefore, find the interest at 6 pr. ct. and multiply it by . A. $0.292.

2. Find the interest for 3 months on $72.05, at 2 pr. ct.

A. $0.360.

3. Find the interest on $600.00, at 4 pr. ct., for 4 months and 18 days? A. $9.20. 4. Find the interest on $9,000.00, at 5 pr. ct., for 8 mo. 4 d. ?

A. $305.00.

5. Find the interest on $800.00 for 6 mo. 21 d., at 7 pr. ct.?

A. $31.266. yrs. 6 mo. 9 d. A. $281.00.

6. Find the interest on $1,000.00, at 8 pr. ct., for 3

7. Find the interest on $1,200.00, for 1 yr. 3 mo. 15 d. at 9 pr. ct.

A. $139.50.

8. Find the interest on $625.00, at 10 pr. ct., for 3 yrs. 2 mo. 7 d. A. $199.131.

9. Find the interest on $930.00, for 6 yrs. 3 mo. 11 d., at 11 pr. ct. A. $642.500.

10. Find the interest on $780.00, for 9 yrs. 7 mo. 16 d. at 12 pr. ct. 11. Find the interest on $1,825.25, for 6 yrs. 8 mo. 13 d., at 13 pr. ct.

ct.

12. Find the interest on $3,976.18, for 2 yrs. 4 mo. 8 d., at 8 pr.

13. Find the interest on $1,964.43, for 9 yrs. 5 mo. 13 d. at 7 pr. ct. 14. Find the interest on $2,675.33, for 7 yrs. 7 mo. 7 d. at 5 pr. ct. 15. Find the interest on $3,865.49, for 12 yrs. 3 mo. 5 d. at 7 pr. ct. Besides the modes given above, there is another, which is often convenient. We have seen how to compute the interest for even years at any rate. (§ LXXVI.) A rate for the whole time is found, by multiplying the rate pr. an. by the number of years. Now if we reduce months and days to decimals of a year, we can treat them in the same way. Thus,

16. Find the interest on $100.00, for 2 yrs. 6 mo., at 7 pr. ct. A. $17.50. 2 yrs. 6 mo. 2 yrs.=2.5 yrs. Then, 2.5X.07.175 rate for whole time.

17. Find the interest on $700.00, for 3 yrs. 2 mo. 12 d., at 5 pr. ct. A. $112. The time is 3.2 yrs. Therefore, the rate for whole time is 3.2X .05.16.

Hence, the rule is

III. MULTIPLY THE RATE PER CENT. PER ANNUM BY THE TIME IN YEARS AND DECIMALS OF A YEAR; THE PRODUCT WILL BE THE RATE FOR THE WHOLE TIME, WITH WHICH PROCEED AS USUAL. Or,