Question

If $a,b,c$ and $d$ are in proportion, which of the following are true?

• A. $b:a::d:c$
• B. $a:c::b:d$
• C. $(a+b):b::(c+d):d$
• D. $a:b::d:c$

Answer 1

Answer: (A), (B), (C)

The correct options are
A $b:a::d:c$
B $a:c::b:d$
C $(a+b):b::(c+d):d$

If $a,b,candd$ are in proportion, then we write it as $a:b::c:d$.
$⟹\frac{a}{b}=\frac{c}{d}$
$⟹\frac{b}{a}=\frac{d}{c}$
$⟹b:a::d:c$, and this property is known as Invertendo.
Now, if $a:b::c:d$, then by the rule of proportion $b\times c=a\times d$
$⟹c\times b=a\times d$
$⟹a:c::b:d$, and this property is known as Alternando.
Now, if $a:b::c:d$, then $\frac{a}{b}=\frac{c}{d}$
$⟹\frac{a}{b}+1=\frac{c}{d}+1$
$⟹\frac{a+b}{b}=\frac{c+d}{d}$, and this property is known as Componendo.
Note that $a:b::c:d$ need not imply $a:b::d:c$
For example, we have $2:3::4:6(\because 2\times 6=3\times 4)$
But, $2:3::6:4$ does not hold, as $2\times 4\ne 3\times 6$