Ratio and Proportion I


   
 
Properties of proportions
Invertendo
 
If a:b=c:d then b:a = d:c
 
 
a:b=c:d
 
 
 
Dividing 1 by each of these ratios,
 
 
 
Alternendo
 
If a:b=c:d, then a:c=b:d.
 
 
 
 
a:c=b:d
 
Componendo
 
If a:b=c:d, then a+b:b=c+d:d
 
 
Adding 1 to both sides, we get
 
 
 
 
Dividendo
 
If a:b=c:d then a-b:b=c-d:d
 
 
Subtracting 1 from both sides, we get
 
 
 
 
Componendo and Dividendo
 
If a:b=c:d then a+b:a-b::c+d:c-d.
 
a:b=c:d
 
a+b:b=c+d:d by componendo
 
a-b:b=c-d:d by dividendo
 
Dividing,
 
 
 
 
 
Illustrative examples
 
1) If a:b::c:d, show that 2a+3b:2a-3b::2c+3d:2c-3d.
 
Suggested answer:
 
a:b::c:d
 
 
Multiplying both sides by 2/3, we get
 
 
Applying componendo and dividendo,
 
 
2a+3b:2a-3b=2c+3d:2c-3d
 
 
Suggested answer:
 
 
By alternendo,
 
 
Applying Componendo/Dividendo, we get
 
 
 
 
 
we get
 
 
Dividing both sides by 2x, we get
 
 
 
By componendo and dividendo, we get
 
 
 
 
 
Adding (1) and (2),
 
 
 
 
 
 
 
 
=2
 
4) If a, b, c and d are in continued proportion then prove that
 
 
Suggested answer:
 
a, b, c, d are in continued proportion.
 
 
then a=bk, b=ck, c=dk
 
Consider,
 
 
 
 
 
Consider,
 
 
 
 
 
Suggested answer:
 
 
then x=ak, y=bk, z=ck
 
Consider,
 
 
 
 
 
 
=k3/2
 
 
 
 
Suggested answer:
 
 
By componendo and dividendo, we get
 
 
 
Squaring both sides,
 
 
By componendo and dividendo, we get
 
 
 
x(b2+1)=2ab
 
 
 
Suggested answer:
 
(a2+b2) (m2+n2)=(am+bn)2 given
 
a2m2+b2m2+a2n2+b2n2=a2m2+b2n2+2ambn
 
b2m2+a2n2-2ambn=0
 
(bm-an)2=0
 
 
bm=an
 
 
 
     
   
Get FREE Live Tutoring
Get FREE Live Tutoring
(No credit card required)

Customer Care

Click to get customer service, technical support and subscription help.

Customer Care Chat


Refer-A-Friend

Get One Month Free!
When you refer a friend