Question

If \[A:B=3:4\] and \[B:C=5:6\] then \[A:C\] is

Answer 1

Hint:Ratio is the comparison of quantities of the same kind of unit. Now multiply each term of the ratio by then unit ratio assigned to B and as given in the formula below:
\[{{B}_{2}}\left( A:{{B}_{1}} \right)::{{B}_{1}}\left( {{B}_{2}}:C \right)\]
where \[{{B}_{2}}\] is the value of the second ratio and \[{{B}_{1}}\] is the value of the first ratio. By cross multiplying the ratios we will get a ratio in terms of \[{{A}_{n}}:{{B}_{n}}:{{C}_{n}}\] , here
\[{{A}_{n}}:{{B}_{n}}:{{C}_{n}}\] are the new number ratio.

Complete step by step solution:
To find the ratio of \[A:C\], first let us find the value of the ratio \[{{A}_{n}}:{{B}_{n}}:{{C}_{n}}\](are the new number ratio.) And to find the value of the three ratios we need to cross multiply the ratio value of \[A:B=3:4\] and \[B:C=5:6\].

Now to cross multiplication we need to transfer the value of B of the first ratio and multiply it with the second ratio as:

\[{{B}_{1}}\left( {{B}_{2}}:C \right)\]

Similarly, we need to transfer the value of B of the first ratio and multiply it with the second ratio as:
\[{{B}_{2}}\left( A:{{B}_{2}} \right)\]

Now placing the values of the ratios in the given cross multiplication we have the new ratios as:
\[{{B}_{1}}\left( {{B}_{2}}:C \right)=5\left( 3:1 \right)\]
\[{{B}_{2}}\left( A:{{B}_{2}} \right)=3\left( 2:5 \right)\]

Calculating the ratios we get the value as:
\[\Rightarrow {{A}_{n}}:{{B}_{n}}:{{C}_{n}}=3\left( 2:5 \right):5\left( 3:1 \right)\]
\[\Rightarrow {{A}_{n}}:{{B}_{n}}:{{C}_{n}}=6:15:15:5\]
\[\Rightarrow {{A}_{n}}:{{B}_{n}}:{{C}_{n}}=6:15:5\]

Hence, as we have found the ratio of the three valued number ratio of \[{{A}_{n}}:{{B}_{n}}:{{C}_{n}}\],

we can find the ratio of new \[A:C\] as \[6:5\].

Note:Another method to solve for the ratio is by fraction method. Fractionalize the ratios of \[A:B=3:4\] and
\[B:C=5:6\] as:
\[\dfrac{A}{B}=\dfrac{2}{5}\ and \\dfrac{B}{C}=\dfrac{3}{1}\]

Multiplying the fractions side by side as:
\[\Rightarrow \dfrac{A}{C}=\dfrac{A}{B}.\dfrac{B}{C}\]
\[\Rightarrow \dfrac{A}{C}=\dfrac{2}{5}.\dfrac{3}{1}\]
\[\Rightarrow \dfrac{A}{C}=\dfrac{6}{5}\]

Hence, the ratio is \[A:C=6:5\].
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