Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

a

=

a a

or Х с

C

[ocr errors]

a a

Х

;

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
b

a
Hence,
7

7 72 ... a:0:: a : 62. And, similarly, if a, b, c, d are four quantities in continued proportion, a :d :: 28: , that isThe first has to the fourth the triplicate ratio of what it

to the second; and so on, for any number of quantities. 29. We shall now give one or two examples of problems in Proportion.

as + c3 Ex. 1. If a:6::c:d, prove that

73 + a C Let Ő

= X; .'. a = bw, and c = doc. d

a3 + 2 (6x)3 + (doc) 63 + d Hence,

then, alter-
(a + c) (bx + dx) (6 + d)8'
a3 + c3 a + C

C
73 + d 6 + d

Jac + Nod Ex. 2. If a :b :: c:d,

prove
that
6 Vac

Nod

3

[ocr errors]

nando,

=

[ocr errors]

a + 6

a

[merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors][ocr errors][merged small][merged small][merged small][ocr errors]
[ocr errors]

nearly =

(a + c)* as + 3 a-x + 3 ax + 23

1+
as
a>

Q. a + 32

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

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

a

с

or

a

[ocr errors]

с

a

Х

[ocr errors]

a or

[ocr errors]

a

C

[ocr errors]

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 : 6:: c:d,

a 5

d is called a proportion. 26. The following results are easily obtained :

6 (1.) Since õ

then à

Х

6 с à
.. a:0::6:d (alternando).

6 d
(2.) 1: 1
7
d

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

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

9
ū

f
h

..(2)
(1) x (2), then,

cg bf

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

:

[ocr errors]
[ocr errors][ocr errors][merged small][merged small][ocr errors][merged small][merged small]

ae

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

a

6

a

- or

[ocr errors]
[ocr errors]

:

[ocr errors]

a

6 a a a? Hence,

Х

Х
7 с 6 ū 72

.. a:0:: a : 62. And, similarly, if a, b, c, d are four quantities in continued proportion, a :d ::a3: 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.

a3 c3 Ex. 1. If a : 6::c:d, prove that

73 d3 a C Let 7 à

=X; .. a = ba, and c = doc.

a + ඒ Hence,

(6x)3 + (dx)8 73 + da (a + c) (63C + dx)3

as + c3 nando, 73 +

Vac + sod Ex. 2. If a : 6 :: 0 :d, prove that

6 Nac – Nod a Let

X; .. a = bx, and c = dx.
d
bx + b

Nid. x + sid
b bx 6

1 sod. x Nod
Nox. dx + Nod Vac + sod
sbx. dx Nod

Sac - Vod

=

(6 + dja, then, alter

a + c

[ocr errors]

16 td

a + 6

a

C =

.

Hence,

a + 6

[ocr errors]
[ocr errors]

a

[blocks in formation]

Ja Ja

Ja y or a

Jac

[ocr errors]

Х

a + 6

a

[ocr errors]

e

=

= X....

Or,

b
7

Jod
Hence, by Art. 26—

Jac + sod

6 sac sod Ex. 3. If a :b :: c:d :: e:f, show that

a ma" + naa + pe"

Ő Imb+ nd" + pof" a Let

..(1). b d

a" CT et

d" fr Hence, a b*ac", :; may mbaca

CH d'u", . nca ndaac" and ... by addition, en = fix", :: pe"

pfra" ma" + nca + pe*

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

ma" + nc + pe

= 2... (2). mb" + nd" + pfr mb+ nd" + pf .. Equating (1) and (2), we have— ma" + no + pe)

Q.E.D. 5 mb" + nd" + pf

[ocr errors]
[ocr errors]

=

=

[ocr errors]

or

a

= (y2

[merged small][ocr errors]

1. Compare the ratios a + b:a b, and as + 62:a 62. 2. Which is the greater of the ratios a + b 2

an

and 26:

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?

: a + 6 ?

4. Compound the ratios 1 - 2c* : 1 + y, 2 xy2 :1 + ac*, and 1:3

2cm. 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 a +

> or < 5 ?

[ocr errors]
[ocr errors]

y + z

[ocr errors]

=

b + c

6 7. If

show that a + b + c = 0. y 8. Find the value of ac when the ratio x + 2 a : x + 2 5 is the duplicate ratio of 2 x + a + c:2 x + b + C. 9. Find x when the ratio & 6

: x + 2 a 6 is the triplicate ratio of a a:X + a b.

8C + Y 10. If

show that each of the fraca + b

C + a' 2 + y + %

Y tions is equal to

and that
a + b + c

a
7

la + mc + nc 11. If

then each is equivalent to 7 d fi

16 + md + nfi hence, show that

6
2 2 + 2 x


2 x + 2 y

2 y + 2x

ac

Y when 2 a + 2 6 2 b + 2 c a 2 c + 2 a

7

[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]

e

a

[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]

C

12. If a : 6 :: c:d, then

a + 6:0 + d :: Na? + ab + 62: Vo+ cd. + d2. 13. Find a fourth proportional to the quantities

X + 1 + x + 1 2 + 1
I x2
2C + 1' 23

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

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

7:6
(3.) a :c :: a 6:6

[merged small][ocr errors][ocr errors]
« ΠροηγούμενηΣυνέχεια »