Addition and Subtraction of Fractions
To add % to % we must first multiply numerator and
denominator of each fraction by the denominator of the
96 ARITHMETIC
other ; this gives % and %, and the sum of these is % ;
the numerators are added because they tel
is the first quantity to be added to the originaldividend.
To get the next number we cut oflEfrom 8726 its two right
hand numbers^ 26 and add them to what is left of the
number; we have thus: 87 + 26=113. This is the
second number to be added to our origi
and are one, two or three figure decimals, .5, .25, .125,
.375, .625 and .875.
This leaves the numbers 7 and 9 to be considered.
The remainders given by the number 7 are repetends,
identical in all cases, except that the decimals begin with
diflFerent num
13 inches ?
5. What is the content of a stick of timber 27 feet.
What is the rule for finding the area of a board ? What is the rule
for finding the content of four-sided timber ?
262 OAUomo.
4 inches long, the average breadth 18 inches, and the
average t
35 rods. What is the circumference of a piece 2i
times as large ?
385. The areas of all similar surfaces are to each
other as the squares of their like dimensions,
386. All solid bodies are to each other as the cube
of their diameters or similar sides,
6.
the diameter of the larger base being 20 inches, of the
smaller base 8 inches, and the slant height 12 feet ?
383. To find the solidity of a frustum of a mpiydr,aor
a cone, "
Rule. Find the area of the two ends, and add the
What is the rule for finding th
Rule. Multiply the cube of the diameter by .5236.
4. What is the solidity of a sphere whose diameter
is 16 inches ?
163x. 5236=2144.6656.
EXAMPLES.
5. What is the solidity of a sphere whose axis is
4 feet ?
6. What is the solidity of the moon, supposing h
of the largest square.
EXAMPLES.
24. What is the side of a square inscribed in a cilerwhose
diameter is 16 feet ?
25. What is the side of a square inscribed in a cilerwhose
circumference is 156 feet ?
What is tiie rule for finding the side of a square equ
12 inches wide at one end and 16 inches at the
other ?
367. A circle is a figure bounded
by a curve, every part of which is
equally distant from a point called
the centre.
368. The curve line bounding a circle is called
the circumference. A straight line
8. What is the area of a triangle,whose base is 75
rods and whose perpendicular is 96 rods ?
9. What is the area of a triangle,whose base is 60
yards and whose perpendicular is 80 yards ?
What is the rule for finding the area of a parallelogram ? What ii
solved by analysis.
1. If the exchange of London on Hamburg be 1
pound sterling equal to 13 marcs and 14 schillings,
and that of Hamburg on Paris be 100 marcs equal to
186 francs and 4 cents, how many francs in Paris,
"
" " " I " ^ " " " 1^
What is arbitr
it contains.
356* A square is a figure of
four equal sides, which are penr-dicular
to each other.
357. A rectangle is a figure
of four sides, which are dpeicrpuelan-r
to each other, but the
opposite sides of which only are
equal.
358* To find the area of
To Messrs. Stevens " Williams,
Bankers, Paris.
Form of a Domestic Bill of Exchange.
$5000. New York, September 20, 1850.
Seven days after sight, pay to Darius Alden, or
order, the sum of five thousand dollars, for value cre-ived,
with or without further a
348. To find the value of a pound sterling at any
per cent, above par, "
Rule. Multiply the mimher of pounds hy the value
What is the rule for finding the value of a pound at any pei^ cent
above par ?
EXCHANGE. 235
of 1 'pound at the given per cent., as s
hands it passes.
344. The process of exchange is as follows : A,
in Boston, owes B, in London, and C, in London, owes
D, in Boston. A, instead of remitting the money to
B, pays the money to D for a bill of exchange, or
order, on C, requesting him to pay t
What is the rule for reducing federal money to any state currency!
What IS the rule for reducing any state currency to federal money?
232 EXCHANGE.
20. Reduce "100, 10 s. sterling to federal money.
100.5 X 4.8#=$486.866.
EXAMPLES.
21. Reduce "240, 12 s. 6
Delaware, and Maryland.
$1=8 s., or "f, in New York, Ohio, Michigan, and
North Carolina.
$l=4s. 8d.,.or ";fe,in South Carolina and Georgia
$l=5s., or "j, in Canada and Nova Scotia.
Louisiana, Florida, Arkansas, and Alabama have no
currency but federal mon
which inrrcase or decrease bv a common ratio
Thus, 2, 4, 8, 16, "c., is au increasing series, in which
the ratio is 2; and 81, 27, 9, 3, 1, ^, i, "c., is a cder-easing
series, in which the ratio is 3.
317. The numbers forming the series are called the
,
Z
16. What is the present worth of an annuity of
$500 at 5 J per cent., to commence in 10 years, and to
continue 15 years afterwards ?
*
331. To find an annuity, the present worth being
given, "
Rule. Divide the present worth by the worth of
an annuity o/*
not paid for 5 years, what will the rent amount to, at
6 per cent, compound interest ?
328. To find the present worth ot an annuity, "
Rule. Subtract the present worth o/ $1 for the
givefi time from 1, and divide the difference by the
interest q/* $1 for
the number of terms will be one more than the number of payments.
14. What is the amount of $100 for 4 years, at 6
per cent, compound interest ?
100=first term; 1.06=:ratio ; 4-f-l=:5ziznumber
of terms; 1.06^ Xl00=126.2476=last term, or amount.
Obs. The a
He agreed to pay 4 cents for the first yard, 8 cents for
the second, and doubling the price for each succeeding
yard. What did he gi\re for the last yard ?
7. If a farmer plant ^ grain of wheat which yields
the first year 20 grains, and if each grain be p
314. To find the num.ber of terms, when the etrx-emes
and common diflference are known, "
Rule. Divide the difference of the extremes by the
common difference,and the quotient, increased by 1,
tpill be the number of terms.
This rule may be represented by
day 5 miles^ and increases his journey every day the
same number of miles ; on the twelfth day he travels
49 miles. What was the daily increase ?
313- To find the sum of all the terms, when the
two extremes and the number of terms are known, "
Rule. Multi
of the power, beginning with units.
Find the first figure of the root, and subtract its
power from, the left hand period, and to the dremrainannex
the first figure of the next period, for a
dividend.
' Involve the root to the power next inferior to thp\
d
16. What is the cube root of fj ?
17. What is the cube root of ^iJ ?
18. What is thc'Cube root of -^l
19. What is the cube root of 91i?
20. What is the cube root of ^?
21. What is the cube root of ^?
306. To find two mean proportionals between two
given n
three times the square of any number P
CUBE BOOT. 211
Obs. The priAciple of the rule may be escpressed and demonstrate
"d by the following formida. Let a = the tens, b =s the units.
(a + 6)3 = a3 4-3 a" 6 + 3 a ft^+ ft'ssany cube.
3 a'=the first trial div
the square of the last figure, and annex two ciphers
for the succeeding trial divisor.
Obs. 1. If there be a remainder, periods of three ciphers each may
be axmexed. If there are decimals, they must be separated into
periods of three figures each, beginni
77. The wall of a castle, 45 ft. high, is surrounded
with a ditch ; and a ladder 75 ft. long will reach from
the outside of the ditch to the top of the wall. What
is the breadth of the ditch ?
78. Two ships sail from the same port; one of
them sails due s