Arithmetical Formulæ and Rules

for the Use of Candidates for Responsions

G. C. M.:—
(α) two numbers	divide greater by less, and divisor by remainder, and so on till there is no remainder. The last divisor is the G. C. M.
(β) three or more numbers	find G. C. M. of first two; then G. C. M. of answer and third number; and so on.
L. C. M.:—
(α) two numbers	product divided by G. C. M.
(β) three or more numbers	divide out all primes which are factors of 2 or more of them; then multiply together remaining numbers and primes so divided out.
Vulgar Fractions:—
(α) to multiply by integer	multiply numerator.
(β) to divide by integer	multiply denominator.
(γ) to multiply by fraction	multiply numerators for new numerator, and denominators for new denominator.
(δ) to divide by fraction	invert divisor, and proceed as in multiplication.
Decimal Fractions:—
(α) to multiply together:—
(1) rule for multiplying	multiply as in whole numbers.
(2) rule for pointing answer	mark off in answer so many decimal places as there are in all the factors together.
(β) to divide one by another:—
(1) rule for dividing	divide as in whole numbers, annexing ciphers to decimal part of dividend if necessary.
(2) rule for pointing answer:—
(a) if dividend has more decimal places than divisor	mark off in answer so many places as difference denotes.
(b) if as many	answer is integer.
(c) if less	answer is integer; annex so many ciphers as difference denotes.
Circulating Decimals, to reduce to vulgar Fractions	for numerator, take to end of first circulating period, subtracting from it non-circulating period; for denominator, take so many nines, as there are figures that circulate, and so many ciphers as there are figures that do not.

Tables of Weights, &c.

Avoirdupois Weight:—

		= 1 Dram (dr.)
16	Drams	= 1 Ounce (oz.)
16	Ounces	= 1 Pound (lb.)
28	Pounds	= 1 Quarter (qr.)
4	Quarters	= 1 Hundredweight (cwt.)
20	Hundredweights	= 1 Ton.

Troy Weight:—

		= 1 Grain (gr.)
24	Grains	= 1 Pennyweight (dwt.)
20	Pennyweights	= 1 Ounce (oz.)
12	Ounces	= 1 Pound (lb.)

Apothecaries’ Weight:—

		= 1 Grain (gr.)
20	Grains	= 1 Scruple (scr.) = 1 _
3	Scruples	= 1 Dram (dr.) = 1 _
8	Drams	= 1 Ounce (oz.) = 1 _
12	Ounces	= 1 Pound (lb.)

Miscellaneous Weight:—

14	Pounds (Avoird.)	= 1 Stone (st.)
7000	Grains (Troy)	= 1 Pound (Avoird.)
gr., oz., lb.	are same in Troy and Apoth. Weight.

Length:—

		= 1 Barleycorn.
3	Barleycorns	= 1 Inch (in.)
12	Inches	= 1 Foot (ft.)
3	Feet	= 1 Yard (yd.)
220	Yards	= 1 Furlong.
8	Furlongs, or	}= 1 Mile (m.)
1760	Yards	}= 1 Mile (m.)
3	Miles	= 1 League.

Surface:—

		= 1 Square Inch.
$12 \times 12$	Square Inches	= 1 Square Foot.
$3 \times 3$	Square Feet	= 1 Square Yard.
4840	Square Yards	= 1 Acre.
640	Acres, or	}= 1 Square Mile
$1760 \times 1760$	Square Yards	}= 1 Square Mile

Solid Contents:—

		= 1 Cubic Inch.
$12 \times 12 \times 12$	Cubic Inches	= 1 Cubic Foot.
$3 \times 3 \times 3$	Cubic Feet	= 1 Cubic Yard.

Miscellaneous Lengths:—

4	Inches	= 1 Hand.
2	Yards, or	}= 1 Fathom.
6	Feet	}= 1 Fathom.
5½	Yards	= 1 Rod, Pole, or Perch.
40	Poles	= 1 Furlong.

Proportion:—

if $a : b :: c : d$ ,
values of `a`, `b`, `c`, `d`, each in terms of the other three, are	$\frac{b c}{d}$ , $\frac{a d}{c}$ , $\frac{a d}{b}$ , $\frac{b c}{a}$ .
Proportional parts: to divide a given number into parts which shall be proportional to certain other given numbers	divide by sum of given numbers and multiply by each separate.

Simple Interest, and Discount:—

let $P =$ Principal, or Present value of future debt,
let $T =$ Time (in years),
let $R =$ Rate per cent,
let $D =$ Interest on principal, or Discount on future debt,
let $F =$ Future amount of Principal, or Future debt;
then,
(α) Interest on £100=	$T \times R$ .
(β) Amount of £100=	$£100 + T \times R$ .
(γ) formula connecting `P`, `D`, `F`	$P + D = F$ .
(δ) formula connecting `P`, `T`, `R`, `D`	as $£100 : P :: T \times R : D$ .
(ε) formula connecting `P`, `T`, `R`, `F`	as $£100 : P :: £100 + T \times R : F$ .

Compound Interest:—

given Principal, Time and Rate:
(α) to find Amount	find Amount of £1 in one year: multiply Principal by this so many times as there are years.
(β) Interest =	Amount minus Principal.

Stocks:—

let $S =$ amount of Stock,
let $R =$ Rate per cent.,
let $Y =$ Yearly income,
let $M =$ Market price of £100 Stock,
let $C =$ Cash value of amount of Stock,
formula connecting these	as $£100 : S :: R : Y :: M : C$