6. Solutions for Proportions

    1. Let the number is x. Then, we have,
\[\frac{3}{4}=\frac{9}{x}~ \Rightarrow~x=12\]

Second and third part are left as exercise to the reader.

    1. Let mean proportional be x so we have,
\[x^2=ab~\Rightarrow~x=\sqrt{ab}\]

Second part is left as an exercise.

    1. Let third proportional be x. Then, we have,
\[9^2=7x~\Rightarrow~x=\frac{81}{7}\]

Second part is left as an exercise.

    1. Let us equate the given proportion’s ratios to k.
\[\frac{a}{b}=\frac{c}{d}~\Rightarrow~a=bk,~c=dk\]

Now let us evaluate L.H.S. by substituting above value for a and c.

\[\frac{a^c+ac^2}{b^d+bd^2}=\frac{b^2k^2dk+bkd^2k^2}{b^d+db^2}=k^3\]

Now let us evaluate R.H.S. by substituting above value for a and c.

\[\frac{(bk+dk)^3}{(b+d)^3}=k^3\]
\[\text{Hence,} L.H.S.= R.H.S.\]
  1. Proceeding the same way as previous problem we have L.H.S. as
\[\frac{pb^2k^2+qb^2}{p^2k^2-qb^2}=\frac{pk^2+q}{pk^2-q}\]

and R.H.S. as

\[\frac{pd^2k^2+qd^2}{pd^2k^2-qd^2}=\frac{pk^2+q}{pk^2-q}\]
\[\text{Hence,} L.H.S.= R.H.S.\]
  1. Processing similarly L.H.S. is
\[\frac{bk-dk}{b-d}=k\]

and R.H.S. is

\[\frac{\sqrt{b^2k^2+d^2k^2}}{\sqrt{b^2+d^2}} = k\]
\[\text{Hence,} L.H.S.= R.H.S.\]

7. We have computed R.H.S. to be k in last problem’s R.H.S. So let us try to calculate R.H.S.

\[\frac{\sqrt{bdk^2+\frac{d^3k^3}{bk}}}{\sqrt{bd+\frac{d^3}{b}}}=k\]
  1. Since a, b, c and d are in continued proportion we can write them as
\[\frac{a}{b}=\frac{b}{c}=\frac{c}{d}=k\]
\[\Rightarrow~a=dk^3, b=dk^2,~\text{and}~c=dk\]

First we do L.H.S.

\[\frac{a}{b+d}=\frac{dk^3}{dk^2+d}=\frac{k^3}{k^2+1}\]

now we do R.H.S.

\[\frac{c^3}{c^2d+d^3}=\frac{d^3k^3}{d^2k^2d+d^2}=\frac{k^3}{k^2+1}.\]

Problem no. 9 and 10 are left as exercises to the reader as they are similar.

11. Since b is mean proportional between a and b we can write \(b^2=ac\). Now let us evaluate the expression.

\[L.H.S.=\frac{a^2-ac+c^2}{\frac{1}{a^2}-\frac{1}{ac}+\frac{1}{c^2}}\]
\[\Rightarrow~\frac{(a^2-ac-c^2)a^2c^2}{c^2-ac+a^2}\]
\[\Rightarrow~a^2c^2 = b^4\]

12. Equating first set of ratio to \(k\) and second set to \(l\) we have following:

\[a=bk\text{ and }c=dk;~e=fl\text{ and }g=hl\]

Substituting these value for \(a, c, e \text{ and } g\) we get both left hand side and right hand side equal to

\[\frac{kl+1}{kl-1}\]

13. Proceeding similarly as previous problems we can say that \(a=bk\) and \(c=dk\). Now substituting for \(a\) and \(c\) we get L.H.S. as

\[L.H.S. = \{bk+b+dk+d\}\{bk-b-dk+d\} = \{(b+d)(k+1)\}\{(b-d)(k-1)\}\]
\[\Rightarrow~(b^2-d^2)(k^2-1)\]

Similarly for R.H.S. we have,

\[R.H.S. = \{bk-b+dk-d\}\{bk+b-dk-d\} = \{(b-d)(k+1)\}\{(b+d)(k+1)\}\]
\[\Rightarrow~(b^2-d^2)(k^2-1).\]

Remaining problems are left exercise to the reader.