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13th, 1819, $416.08 ; May 10th, 1820, $152.00 : what was
due March 1st, 1821, the interest being 6 per cent?

LEGAL INTEREST,

258. Legal Interest is the interest which the law permits
a person to receive for money which he loans, and the laws
do not favor the taking of a higher rate. In most of the
States the rate is fixed at 6 per cent ; in New York, South
Carolina and Georgia, it is 7 ; and in some of the States the
rate is fixed as high as 10 per cent



250 PROBLEMS IN INTEREST.

PROBLEMS IN INTEREST.

259. In all questions of Interest there are four things con-
sidered, viz. :

1st, The principal ; 2d, The rate of interest ; 3d, The
time ; and &th, The amount of interest.

If three of these are known, the fourth can be found,

I. Knowing, the principal, rate, and time, to find the inter-
est. This case has already been considered.

II. Knowing the interest, time, and rate, to find the prin-
cipal.

Cast the interest on one dollar for the given time, and then
divide the given interest by it the quotient ivill be the princi-
pal.

III. Knowing the interest, the principal, and the time, to
find the rate.

Cast the interest on the principal for the given time at 1 per
cent and then divide the given interest by it the quotient will
be the rate of interest.

IV Knowing the principal, the interest, arid the rate, to
find the time.

Cast the interest on the given principal at the given rate
for 1 year and then divide the interest by it the quotient
will be the time in years and decimals of a year.

EXAMPLES

1. The interest of a certain sum for 4 years, at 7 per cent,
is $266 : what is the principal?

2. The interest of $3675, for 3 years, is $171.15 : what is
the rate?

3. The principal is $459, the interest $183.60, and the
rate 8 per cent : what is the time ?

4. The interest of a certain sum, for 3 years, at 6 per cent,
is $40.50 : what is the principal ?

5. The principal is $918, the interest $269.28, and the
rate 4 per cent : what is the time ?

258. What is legal interest ?

259. How many things are considered in every question of interest?
What arc they ? What is the rule for each ?



COMPOUND INTEKEST. 251



COMPOUND INTEREST.

260. Compound Interest is when the interest on a princi-
pal, computed to a given time, is added to the principal, and
the interest then computed on this amount, as on a new
principal. Hence,

Compute the interest to the time at which it becomes due ;
then add it to the principal and compute the interest on the
amount as on a new principal: add the interest again to
the principal and compute the interest as before ; do the
same for all the times at which payments of interest become
due ; from the last result subtract the principal, and the
remainder will be the compound interest.

EXAMPLES.

1. What will be the compound interest, at 7 per cent, of
$3150 for 2 years, the interest being added yearly?

* OPERATION.

$3750.000 principal for 1st year.

$3750 x. 07= 262.500 interest for 1st year

4012.500 principal for 2d "

$4012.50 x. 07= 280.875 interest for 2d "

4293.375 amount at 2 years.
1st principal 3750.000
Amount of interest $543.375.

2. If the interest be computed annually, what will be the
compound interest on $100 for 3 years, at 6 per cent?

3. What will be the compound interest on $295.37, at 6
per cent, for 2 years, the interest being added annually ?

4. What will be the compound interest, at 5 per cent, of
$1875, for 4 years?

5. What is the amount at compound interest of $250, for
2 years, at 8 per cent ?

6. What is the compound interest of $939.64, for 3 years,
at 7 per cent ?

7. What will $125.50 amount to in 10 years, at 4 per cent
compound interest ?

260. What of compound interest ? How do you compute it ?



252



COMPOUND INTEREST.



NOTE. The operation is rendered much shorter and easier, by
taking the amount of 1 dollar for any time and rate given in the
following table, and multiplying it by the given principal ; the
product will be the required amount, from which subtract the
given principal, and the result will be the compound interest.*

TABLE.

Which shows the amount of $1 or 1, compound interest, from 1 year
to 20, aud at the rate of 3, 4, 5, 6, and 7 per cent.



Years. jiSper cent.


4 per rent.io per cent.


ti per cent.


' per ci-Bt.


Vars.


1


1.03000


1.04000


1.05000


1.06000


1.07000


1


2


1.0(5090


1.08160


1.10250


1.12360


1.14490


2


3


1.09272


1.12486


1.15762


1.19101


1.22504 3




1.135501.109851.21550


1.26247


1.31079 4


5


1.15927


1.216'>5


1.27628


1. 33822


1.40255


5


6


1 19405


1.26531


1.34009


1.41851


1.50073


6


7


1.22987


1.31593 1. 40710


1.50363


1.60578


7


8 ft.26677
9 1.30477


1.36856! 1.47745
1.4233111.55132


1.59384
1.C8947


1.71818
1.83845


8
9


10 1.34391


1.480:38


1.62889


1.79084


1.96715


10


11 11.38433


1.53945


1.71033


1.89829


2.10485


11


12


1.4257(5


1.60103


1.79585


2.012192.25219


12


13


1.46853


1.66507


1.88564


2.13292240984


13


14


1.5! 258


1.73167


1.97993


2.260902,57853


14


15


1.55796


1.80094


2.07892


2.396552.75903


15


16


1.60470


1.8729812.18287


2.54035 2.95216


16


17


1. (55284


1.94790J2. 29201


2. 69277 ; 3. 15881


17


18


1.70243


2.02581


2.40661


2.854333.37993


18


19


1.75350


2.10684


2.52695


3.025593.61652


19


20


1.80611


2.19112


,2. (55329 3. 2071 3 i 3. 86968 1 20



NOTE. When there are months and days in the time, find the
amount for the years, and on this amount cast the interest for the
mcnths and days : this, added to the last amount, will be the re-
quired amount for the whole time.

8. What is the amount of $96.50 for 8 years and 6 months,
interest being compounded annually at 7 per cent ?

9. What is the compound interest of $300 for 5 years
8 months and 15 days, at 6 per cent ?

10. What is the compound interest of $1250 for 3 years
3 months and 24 days, at 7 per cent ?

11. What will $56.50 amount to in 20 years and 4 months,
at 5 per .cent compound interest ?

* The result may differ in the mills place from that obtained by the
other rule.



DISCOUNT. 253



DISCOUNT. x

261. DISCOUNT is an allowance made for the payment of
money before it is due.

THE FACE of a note is the amount named in the note.*

NOTE. DAYS OP GRACE are days allowed for the payment of
a note after the expiration of the time named on its face. By
mercantile usage a note does not legally fall due until 3 days
after the expiration of the time named on its face, unless the note
specifies without grace.

Days of grace, however, are generally confined to mercantile
paper and to notes discounted at banks.

262. The PRESENT VALUE of a note is such a sum as being
put at interest until the note becomes due, would increase to
an amount equal to the face of the note.

The discount on a note is the difference between the face
of the note and its present value.

1. I give my note to Mr. Wilson for $10 7, payable in
1 year : what is the present value of the note if the interest
is 7 per cent. ? what the discount ?

OPERATION.

ANALYSIS. Since 1 dollar in 1 year $107 -f- 1,07 $100.
at 7 per cent, will amount to $1.07, the PROOF

present value will be as many dollars y n 4. (frinn 1,1. <6 *r

as $1.07 is contained times in the face t, . \, ^ \ n A
of the note: viz., $100: and the dis- -Principal,
count will be $107- $100= $7: hence, Amount, $107

Discount, 7

Divide the face of the note by 1 dollar plus the interest of
1 dollar for the given time, and the quotient will be the pre-
sent value : take this sum from the face of the note and the
remainder will be the discount.



261. What is discount ? What is the face of a note ? What are days
of grace?

362. What is present value ? What is the discount ? How do you
find the present value of a note ?

* See Appendix, page 3l(X



254 DISCOUNT.

EXAMPLES.

1. What is the present value of a note for $1828,75, eke
in 1 year, and bearing an interest of 4 J per cent ?

2. A note of $1651.50 is due in 11 months, but the person
to whom it is payable sells it with the discount off at 6 per
cent : how much shall he receive ?

NOTE. When payments are to be made at different times, find
the present value of the sums separately, and tfieir sum will be the
present value of the note.

3 What is the present value of a note for $10500, on which
$900 are to be paid in 6 mouths ; $2700 in one year ; $3900
in eighteen months ; and the residue at the expiration of two
years, the rate of interest being 6 per cent per annum ?

4. What is the discount of <4500, one-half payable in six
months and the other half at the expiration of a year, at 7
per cent per annum ?

5. What is the present value of $5760, one-half payable in

3 months, one-third in 6 months, and the rest in 9 months,
at 6 per cent per annum ?

6. Mr. A gives his note to B for $720, one-half payable in

4 months and the other half in 8 months ; what is the present
value of said note, discount at 5 per cent per annum ?

7. What is the difference between the interest and discount
of $750, due nine months hence, at 7 per cent ?

8. What is the present value of $4000 payable in 9 months,
discount 4J per cent per ami am ?

9. Mr. Johnson has a note against Mr. Williams for
$2146.50, dated August 17th, 1838, which becomes due Jan.
llth, 1839 : if the note is discounted at 6 per cent, what
ready money must be paid for it September 25th, 1838 ?

10. C owes D $3456, to be paid October 27th, 1842 ; C
wishes to pay on the 24th of August, 1838, to which D con-
sents ; how much ought D to receive, interest at 6 per cent ?

11. What is the present value of a note of $4800, due 4
years hence, the interest being computed at 5 per cent per
annum ?

12. A man having a horse for sale, offered it for $225 cash
in hand, or $230 at 9 months ; the buyer chose the latter :
did the seller lose or make by his offer, supposing money to
be worth 7 per cent ?



BANK DISCOUNT, 255



BANK DISCOUNT.

263. BANK DISCOUNT is the charge made by a bank for the
payment of money on a note before it becomes due.

By the custom of banks, this discount is the interest on the
amount named in a note, calculated from the time the note
is discounted to the time when it falls due ; in which time
the three days of grace are always included.

The interest is always paid in advance.

RULE Add 3 days to the time which the note has to run,
and then calculate the interest for that time at the given rate.

EXAMPLES.

1. What is the Dank discount of a note for $350, payable
3 months after date, at 7 per cent interest ?

2. What is the bank discount of a note of $1000 payable
in 60 days, at 6 per cent interest ?

3. A merchant sold a cargo of cotton for $15720, for which
he receives a note at 6 months : how much money will he
receive at a bank for this note, discounting it at 6 per cent
interest ?

4. What is the bank discount on a note of $556. 2 1 paya-
ble in 60 days, discounted at 6 per cent interest?

5. A has a note against B for $3456, payable in three
months ; he gets it discounted at 7 per cent interest : how
much does he receive ?

6. What is the bank discount on a note of 367.47, having
1 year, 1 month, and 13 days to run, as shown by the face of
the note, discounted at 7 per cent ?

7- For value received, I promise to pay to John Jones, on
the 20th of November next, six thousand five hundred and
seventy-nine dollars and 15 cents. What will be the discount
on this, if discounted on the 1st of August, at 6 per cent per
annum ?

263. What is bank discount ? How is interest calculated by the
custom of banks ? How is the interest paid ? How do you find the
interest ?



256 BANK DISCOUNT.

8. A merchant bought 115 barrels of flour at $7.50 cents
a barrel, and sells it immediately for $9.75 a barrel, for
which he receives a good note, payable in 6 months. If he
should get this note discounted at a bank, at 6 per cent, what
will be his gain on the flour ?

264. To make a note due at a future lime, whose present
value shall be a given amount.

1. For what sum must a note be drawn at 3 months, so
that when discounted at a bank, at 6 per cent, the amount
received shall be $500 ?

ANALYSIS If we find the interest on 1 dollar for the given
time, and then subtract that interest from 1 dollar, the remainder
will be the present value of 1 dollar, due at the expiration of that
time. Then, the number of times which the present value of
the note contains the present value of 1 dollar, will be the num-
ber of dollars for which the note must be drawn : hence,

Divide the present value of the note by the present value of
1 dollar, reckoned for the same time and at the same rate of
interest , and the quotient will be the face of the note,

OPERATION.

Interest of $1 for the time, 3mo. and Ma. =$0.0155, which
taken from $1, gives present value of $1=0.9845; then, $500^-
0.9845= $507.872-1- =face of note.

PROOF.

Bank interest on $507.872 for 3 months, including 3 days of
grace, at 6 per cent =7.872, which being taken from the face of
the note, leaves $500 for its present value,

EXAMPLES,

1 . For what sum must a note be drawn, at 7 per cent,
payable on its face in 1 year 6 months and 1 5 days, so that
when discounted at bank it shall produce $307.27 ?

2. A note is to be drawn having on its face 8 months and
1 2 days to run, and to bear an interest of 7 per cent, so that
it will pay a debt of $5450 : what is the amount ?

364. How do you make a note payable at a future time, whose pre-
sent value shall be a given amount ?



EQUATION OF PAYMENTS. 257

3. What sum, 6 months and 9 days from July 18th, 1856,
drawing an interest of 6 per cent, will pay a debt of $674.89
at bank, on the 1st of August, 1856 ?

4. Mr Johnson has Mr. Squires' note for $814.57, having
4 months to run, from July 13th, without interest. On the
first of October he wishes to pay a debt at bank of $750.25,
and discounts the note at 5 'per cent in payment : how much
must he receive back from the bank ?

5. Mr. Jones, on the 1st of June, desires to pay a debt at
bank by a note dated May 1 6th, having 6 months to run and
drawing 7 per cent interest : for what amount must the note
be drawn, the debt being $1683.75 ?

6 Mr. Wilson is indebted at the bank in the sum of
$367.464, which he wishes to pay by a note at 4 months
with interest at 7 per cent : for what amount must the note
be drawn ?

EQUATION OF PAYMENTS,

265. EQUATION OF PAYMENTS is the operation of finding the
mean time of payment of several sums due at different times,
so that no interest shall be lost or gained.*

1. If I owe Mr. Wilson 2 dollars to be paid in 6 months,
3 dollars to be paid in 8 months, and 1 dollar to be paid in
12 months, what is the mean time of payment ?

OPERATION.

Int. of $2 for 6rao.=int. of $1 for 12mo. 2x 612

" of $3 for 8rao; int. of $1 for 24??io. 3x 8 = 24

" of $1 for 12wio.=mt. of $1 for 12 mo. I x 12^12

$6 48 48

ANALYSIS. The interest on all the sums, to the times of pay-
ment, is equal to the interest of $1 for 48 months. But 48 is
equal to the sum of all the products which arise from multiplying
each sum by the time at which it becomes duo: hence, the sum
of the products is equal to the time which would be necessary for
$1 to produce the game interest as would be produced by all the
principals.

* The mean time of payment is sometimes found by first finding the
jyrcsent value of each payment ; but the rule here given has the sanc-
tion of the best authorities in this country and England.
17



253 EQUATION OF PAYMENTS.

' $1 will produce a certain interest in 48 months, in what time
will $6 (or the sum of the payments) produce the same interest ?
The time is obviously found by dividing 48 (the sum of the pro-
ducts) by $6, (the sum of the payments.)
Hence, to find the mean time,

Multiply each payment by the time before it becomes due,
and divide the sum of the products by the sum of the pay-
ments : the quotient will be the mean time.

EXAMPLES.

1. B. owes A $600 ; $200 is to be paid in two months,
$200 in four months, and $200 in six months : what is the
mean time for the payment of the whole ?

OPERATION.
200x2-= 400

ANALYSIS. We here multiply each 200x4 800
sum by the time at which it becomes QHA f_ionn
due, and divide the sum of the products JUU
by the sum of the payments. 6|00 )24|00

Ans. 4 months.

2. A merchant owes $600, of which $100 is to be paid in
4 months, $200 in 10 months, and the remainder in 16
months : if he pays the whole at once, in what time must he
make the payment ?

3. A merchant owes $600 to be paid in 12 months, $800
to be paid in 6 months, and $900 to be paid in 9 months :
what is the equated time of payment ?

4. A owes B $600 ; one-third is to be paid in 6 months,
one-fourth in 8 months, and the remainder in 12 months :
what is the .mean time of payment ?

5. A merchant has due him $300 to be paid in 60 days,
$500 to be paid in 120 days, and $750 to be paid in 180
days : what is the equated time for the payment of the



6. A merchant has due him $1500 : one-sixth is to bo
paid in 2 months, one-third in 3 months, and the rest in 6
months : what is the equated time for the payment of the
whole ?

265. What is equation of payments ? How do you find the mean or
equated time ?



EQUATION OF PAYMENTS. 259

7. I owe $1000 to be paid on the first 'of January, $1500
on the 1st of February, $3000 on the 1st of March, and
$4000 on the 15th of April : reckoning from the 1st of Janu-
ary, and calling February 28 days, on what day must the
money be paid ?

NOTE. If one of the payments, as in the above example, is due
on the day from which the equated time is reckoned, its corres-
ponding product will be notliing, but the payment must still be
added in finding the sum of the payments,

8. I owe Mr Wilson $100 to be paid on the 15th of July,
$200 on the 15th of August, and 300 on the 9th of Septem-
ber : what is the mean time of payment ?

OPERATION

From 1st of July to 1st payment 14 days

" " " to 2d payment 45 days.

" to 3d payment 70 days.

100x14= 1400
200x45= 9000

Tlien by rule given above we 300 X 70 = 2 1 000
have,



600 6|00)314|00

fili

Hence, the equated time is 52^ days from the 1st of July ; that
is, on the 22d day of August.

But if we estimate the time from the 15th of July we shall have

From July 15th to 1st payment days.
" " to 2d payment 30 days.

" " to 3d payment 54 days.

Then, 100 x 0= 000

200x30= COOO
300x54 = 16200
600



Hence, the payment is due in 37 days from July 15th; or, on
the 22d of August the same as before.

Therefore : Any day may be taken as the one from, which
the mean time is reckoned.

NOTE. If one payment is due on the day from which the time is
reckoned, how do you treat it ? Can you compute the time from any
day?



260 ASSESSING TAXES.

9. Mr. Jones purchased of Mr. Wilson, on a credit of six
months, goods to the following amounts :

15th of January, a bill of $3t50,

10th of February, a bill of 3000,
6th of March, a bill of 2400,
8th of June, a bill of 2250.

He wishes, on the 1st of July, to give his note for the
amount : at what time must it be made payable ?

10. Mr Gilbert bought $4000 worth of goods ; he was to
pay $1600 in five months, $1200 in six months, and the re-
mainder in eight months : what will be the time of credit, if
he pays the whole amount at a single payment ?

11. A merchant bought several lots of goods, as follows :

A bill of $650, June 6th,
A bill of 890, July 8th,
A bill of 7940, August 1st.

Now, if the credit is 6 months, how many days from De-
cember 6th before the note becomes due ? At what time ?

ASSESSING TAXES.

26G. A tax is a certain sum required to be paid by the
inhabitants of a town, county, or state, for the support of
government or some public object. It is generally collected
from each individual, in proportion to the amount of his
property.

In some states, however, every white male citizen over the
age of twenty-one years is required to pay a certain tax.
This tax is called a poll-tax ; and each person so taxed is
called a poll.

267. In assessing taxes, the first thing to be done is to make
a complete inventory of all the property in the town on which
the tax is to be laid. If there is a poll-tax, make a full list
of the polls and multiply the number by the tax on each
poll, and subtract the product from the whole tax to be

266. What is a tax ? llow is it generally collected ? What is a
poll-tax ?



ASSESSING TAXES. 2C1

raised by the town : the remainder will be the amount to
be raised on the property Having done this, divide the
whole tax to be raised by the amount of taxable properly
and the quotient will be the tax on $1. Then multiply this
quotient by the inventory of each individual, and the product
will be the tax on his property

EXAMPLES.

1. A certain town is to be taxed $4280 ; the property on
which the tax is to be levied is valued at $1000000. Now
there are 200 polls, each taxed $1.40. The property of A
is valued at $2800, and he pays 4 polls.

B's at $2400, pays 4 polls. E's at $7242, pays 4 polls.
C's at $2530, pays 2 " F's at $1651, pays 6 "
D's at $2250, pays 6 " G's at $1600.80 pays 4 "

What will be the tax on 1 dollar, and what will be A's
tax, and also that of each on the list ?

First; $1.40 x 200 = $280 amount of poll-tax.
$4280 $280 4000 amount to be levied on property.
Then, $4000-i-$1000000=4 mills on $1.
Now, to find the tax of each, as A's, for example,

A's inventory $2800

_^004
TT200
4 polls at $1,40 each - - 5.60

A's whole tax - - - - - $16.800>
In the same manner the tax of each person in the town-
ship may be found.

Having found the per cent, or the amount to be raised on
each dollar, form a table showing the amount which certain
sums would produce at the same rate per cent. Thus, after
having found, as in the last example, that 4 mills are to be
raised on every dollar, we can, by multiplying in succession
by the numbers 1, 2, 3, 4. 5, 6, 7, 8, &c., form the following

267. What is the first thing to be done in assessing a tax ? If there
is a poll-tax, how do you find the amount ? How x then do you find the
per cent of tax to be levied on a dollar ? How do you then find the
amount to be levied on each individual ?



262



ASSESSING TAXES.



TABLE



$ $


$ $


$ $


1 gives 0.004


20 gives 080


300 gives 1.200


2 " 0.008


30 " OJ20


400 " 1.600


3 " 0-012


40 (< 0.160


500 " 2.000


4 " 0.016


50 " 0.200


600 " 2.400


5 " 0.020


60 " 0.240


700 " 2.800


6 " 0.024


70 " 0.280


800 " 3.200


7 " 0.028


80 " 0.320


900 " 3.600


8 " 0.032


90 " 0.360


1000 " 4.000


9 " 0.036


100 " 0.400


2000 " 8.000


10 " 0.040


200 " 0.800


3000 " 12.000



This table shows the amount to be raised on each sum in
the columns under $'s.

EXAMPLES.

1. Find the amount of B's tax from this table.

B's tax on $2000 - - is - $8.000

B's tax on 400 - - is, - $1.600

B's tax on 4 polls, at $1.40 - $5 600

B's total tax - is - $15.200

2. Find the amount of C's tax from the table.

C's tax on $2000 - - is - $8.000
C's tax on 500 - - is - $2.000
C's tax on 30 - - is - $0.120
C's tax on 2 polls - - is - $2.800
C's total tax - - is -"$12.920

In a similar manner, we might find the taxes to be paid
by D, E, &c.

3. If the people of a town vote to tax themselves $1500,
to build a public hall, and the property of the town is valued
at $300.000, what is D's tax, whose property is valued at
$2450?

4. In a school district a school is supported by a tax on
the property of the district valued at $121340. A teacher is
employed for 5 months at $40 a month, and contingent ex-
penses are $42,68 ; what will be a farmer's tax whose property
is valued at $3125?



COINS AND CURRENCY. 263



COINS AND CURRENCY.

268. Coins are pieces of metal, of gold, silver, or copper, of
fixed values, and impressed with a public stamp prescribed
by the country where they are made. These are called
specie, and are declared to be a legal tender in payment of
debts.

2(51). Currency is what passes for money. In our country
there are four kinds.

1st. The coins of the country :

M. Foreign coins, having "a fixed value established by
law :

3e?. Bank notes, redeemable in specie.

4th. Paper money declared a legal tender, by act of
Congress.

NOTE. The foreign coins most in use in this country are the
English shilling, valued at 22 cents 2 mills ; the English sove-


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Online LibraryCharles DaviesSchool arithmetic. Analytical and practical → online text (page 18 of 24)