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CHAPTER V.

RATIO AND PROPORTION.

Ratio.

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22. The student is referred to Chapter II. of the Arithmetic section of this work for definitions and observations. which need not be repeated here.

23. A ratio of greater inequality is diminished, and a ratio of less inequality is increased, by increasing the terms of the ratio by the same quantity. Let a:b or be the ratio, and let each of its terms be 6

a + m increased by m. It will then become

b + m a + m a Now, 3 귿 b + m

as (a + m) 6 = (b + m) a,

b' or, as ab + bm ab + am; or, as bm Zam, or as 6 z do Hence the ratio- is increased when bi a, that is, wheni

7 it is a ratio of less inequality; and is diminished when b < a, that is, when it is a ratio of greater inequality.

Cor. It may be shown in the same way that

A ratio of greater inequality is increased, and a ratio of less inequality is diminished, by diminishing the terms of the ratio by the same quantity.

24. When the difference between the antecedent and consequent is small compared with either, the ratio of the higher powers of the terms is found by doubling, trebling, &c., their difference. Let a t 30 : a or

be the ratio, where is small compared with a.

(a + c)2 aa + 2 ax + 2cm Then

1 * nearly = aa a + 2 x

nearly. a

a +

a

200

a

(a + x)

a + 3 a*x + 3 ax +

+ 2)

=l+

+39 nearly

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a

a + 330

nearly; and so on. Ex. (1002)2 : (1000) = 1004 : 1000 nearly.

(1002) : (1000)3 = 1006 : 1000 nearly.

or

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с

с

(1.) Since

a
or
с

с

с

or

a

Proportion. 25. Proportion, as has been already said, is the relation of equality expressed between ratios. Thus, the expression a : b = c:d,

or a:b::c:d,
a

с

6 đ
is called a proportion.
26. The following results are easily obtained :-

6
6

6
then Х

Х d' 7

d

d' .. a:0::6:d (alternando).

6 d (2.) 1:0 1 ;

d

c' .:: 6:a:: d:c (invertendo). Also, by Art. 64, page 214, we have (3.) a + 6:6:: c + d:d (componendo). (4.) a - 6:6::c-d:d (dividendo). (5.) a - b:a:: c-d:c (convertendo). (6.) a + b:a - 6:: c + d :c - d (componendo and

dividendo). 27. If a:b::c:d and e:f::g:h, we may compound the proportions. Thus we have

2. ......(2) b

cg (1) X (2), then,

bf

dh' or ae : bf :: cg : dh.

a

i ...... (1), and

ae

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a

a al

or Х с

с

.. a: C

:

And in the same way we may show that, if the corresponding terms of any number of proportions be multiplied together, the products will be proportional.

28. If three quantities are in continued proportion, the first has to the third the duplicate ratio of what it has to the second.

Let a, b, c be the given quantities in continued proportion; then

6 6 6

aa Hence,

Х 5

7 Ő

a> : 62. And, similarly, if a, b, c, d are four quantities in continued proportion, a :d :: a : 63, that is

The first has to the fourth the triplicate ratio of what it has to the second; and so on, for any number of quantities.

29. We shall now give one or two examples of problems in Proportion.

a® of c3 Ex. 1. If a:b::c:d, prove that

63 + a с Let

= ; ..a = bx, and c = da. 6 d

a + CO2 Hence,

73 + d3 (bx)3 + (dx)3

then, alter(a + c)3 (bac + d.)3 (6 + d)

a2 + 2 ( + nando, 63 + d3 b +

a + b Vac + Nod Ex. 2. If a :b :: c:d, prove that

b ac - Mod a Let

bx, and c =
6

do.
= N; .'. a =
ď
Ex + 6

vod.x + Vod
Hence,
6 bx 6

1

sod.x - Nod
sbx. dx + Vid Vac + dod
Nox . dx

sbd Jac Nod

a +

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a

с

a + b

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a

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ac

Or,

or

7

Х

b b à 7 Nod Hence, by Art. 26

a + 6 Jac + sod

a 6 Vac - Jod Ex. 3. If a :b :: c:d :: e:f, show that

ma" + nc" + pe" 6 mbi + nd" + pf"

1

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e

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Let

(1). b d f

a CT Di

fr Hence, a" b*oc", .. mam mb*ac"

nd och and .. by addition, en = fru", pe = pf"22"

ma" + nc" + pe" = (mb" + nd" + pf)a". mam + nom + pe"

maana t non + pe ::

= .. (2). mb" + nd" + poft mb" + nd* + pf) :'. Equating (1) and (2), we have mar + nc +

Q.E.D. mbt + nd" + pf")

1

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or

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1. Compare the ratios a + 6:a b, and a+ 6o: a 2. Which is the greater of the ratios a + b : 2 a, and

: a + b ? 3. What quantity must be subtracted from the consequent of the ratio a :b in order to make it equal to the ratio c:d?

2b:

and 1:30

4. Compound the ratios 1 x2 :1 +. Y, a xy2:1 + 2*,

xo. 5. There are two numbers in the ratio of 6 : 7, but if 10 be added to each they are in the ratio of 8 : 9. Find the numbers.

6 6. In what cases is x +

< 5 ?

-> or <

с

+

a

6 7. If

show that a t b + c = 0. C Y

C' 8. Find the value of x when the ratio x + 2 a : x + 2 6 is the duplicate ratio of 2 x + a + c:2 x + b + c.

9. Find x when the ratio x b : 2 + 2 a 6 is the triplicate ratio of a – a : x + a

b. 8C + Y

y + z 10. If

show that each of the fraca + 6

C + a' x + y + %

C y tions is equal to

and that a + b + c

6

la t mc + nc 11. If a

+ nfi hence, show that

6
2 % + 2 oC Y
2 x + 2 y

2 y + 2 z - a'

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a

* = 4, then each is equivalent to 16 + md +

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12. If a :b :: c:d, then

a + b:+ d :: Va? + ab + 62 : Vca i cd. + d. 13. Find a fourth proportional to the quantities

a + 1 ca + x + 1 2 + 1
l'aca
3 + 1 2 3

1° 14. Find c in terms of a and 6 when

(1.) a :a:: a 7:6
(2.) a : b:: a

6:0
(3.) a : c:: a :6

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b:

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