Properties of proportions

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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,

=k^3/2

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