Download Ratio, proportion and variation (Terms are considered to be positive

Ratio, proportion and variation
(Terms are considered to be positive, and can be extended to negative in some of the statements below.)
a c
=
b d
⇔
1.
a:b=c:d
2.
Cross multiplication:
3.
⎧a
⎪b > 1⇒
⎨a
⎪ <1⇒
⎩b
4.
a c
a b
= ⇔ =
b d
c d
(Alterando)
,
a c
b d
= ⇔ =
b d
a c
(Invertendo)
a c
a+b c+d
= ⇔
=
b d
b
d
(Compoendo)
,
a c
a −b c−d
= ⇔
=
b d
b
d
(Dividendo)
a c
a+b c+d
= ⇔
=
b d
a −b c−d
(Compoendo et Dividendo)
5.
a a+x
>
b b+x
a a+x
<
b b+x
a a−x
<
b b−x
a a−x
>
b b−x
, a > x > 0,
b > x > 0.
Equal Ratios Theorems
a
a1 a 2
=
= ... = n = k
b1 b 2
bn
⇒
k=
a1 + a 2 + ... + a n
,
b1 + b 2 + ... + b n
a
a1 a 2
=
= ... = n = k
b1 b 2
bn
⇒
k=
m1a 1 + m 2 a 2 + ... + m n a n
,
m1b1 + m 2 b 2 + ... + m n b n
⇒
⎛ m a p + m 2 a 2 p + ... + m n a n p
k = ⎜⎜ 1 1p
p
p
⎝ m1b1 + m 2 b 2 + ... + m n b n
b1 + b 2 + ... + b n ≠ 0
where
where
⎞
⎟
⎟
⎠
m1b1 + m 2 b 2 + ... + m n b n ≠ 0
1/ p
,
where m1b1 + m 2 b 2 + ... + m n b n ≠ 0
p
p
p
Unequal Ratios Theorem
a1 a 2
a
<
< ... < n
b1 b 2
bn
7.
a 1 : b 1 : c1 = a 2 : b 2 : c 2
8.
Continued proportion
9.
ad = bc
⎧a
⎪b > 1⇒
⎨a
⎪ <1⇒
⎩b
, a, b, x > 0.
a
a1 a 2
=
= ... = n = k
b1 b 2
bn
6.
⇔
a:b=c:d
⇒
a 1 a 1 + a 2 + ... + a n a n
,
<
<
b1 b1 + b 2 + ... + b n b n
⇔
a 1 b1 c1
=
=
a 2 b2 c2
where
a1, a2, ..., an ; b1, b2, …, bn > 0
⎧a 1 = ka 2
⎪
⎨b1 = kb 2
⎪ c = kc
2
⎩ 1
⇔
a:b=b:c
⇒
a : c = a2 : b2 = b2 ; c2
a:b=b:c=c:d
⇒
a : d = a3 : b3 = b3 : c3 = c3 : d3
Rule of Cross Multiplication
x
⎧ a 1 x + b 1 y + c1 z = 0
⎨
⎩a 2 x + b 2 y + c 2 z = 0
⇒
b1
b2
y
c1
c2
z
a1
a2
b1
b2
x
y
z
=
=
b 1 c 2 − b 2 c1 c1 a 2 − c 2 a 1 a 1 b 2 − a 2 b 1
1
10.
Direct variation
x ∝ y ⇔ x = ky ⇒
k>0
x 1 y1
=
x 2 y2
-2
11.
Inverse variation
x∝
k<0
y
2
y
2
1
1
O
-1
1
2
-2
x
O
-1
-1
-1
-2
-2
2
5
10
x
k<0
k>0
1
k
⇔x=
⇒ x 1 y1 = x 2 y 2
y
y
y
y
5
15
1
-10
-5
O
5
5
10
x
15
-10
-5
-5
x ∝ yz ⇔ x = kyz
12.
Joint variation
13.
x ∝ y, when z is kept constant and x ∝ z
14.
Partial variation
⇒
when
O
x
-5
x 1 y1 z 1
=
x 2 y2z2
⇒ x ∝ yz
y is kept constant
x is partly constant and partly varies directly as y
⇒
x is partly varies directly as
y and partly varies directly as z
⇒
x = k1y + k2z
x is partly varies directly as
y and partly varies inversely as z
⇒
x = k1y +
x = k1 + k2y
k2
z
x ∝ y and y ∝ z ⇒ x ∝ z
15.
Transitive law:
16.
x ∝ yz ⇔
17.
x ∝ z and y ∝ z ⇒
(x ± y) ∝ z
18.
x ∝ z and y ∝ z ⇒
xy ∝ z2
19.
a : b = ka : kb =
20.
x ∝ y and z ∝ w
⇒
xz ∝ yw
21.
x ∝ y and z ∝ w
⇒
x
y
∝
z w
22.
x∝y ⇒
x
∝z
y
a b
:
, k ≠ 0.
k k
(ax + by) ∝ (cx + dy)
,
(a, b) , (c , d) ≠ (0, 0)
2