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# If A: B = 6: 7 and B: C = 8: 9, then A: C isThis question has multiple correct options.(a) 21: 16(b) 16: 21(c) 48: 63(d) 63: 48

Hint: In order to solve this question, we convert the ratios into the fraction. Then we can get two-equation from two ratios. We are asked the ratio of A: C so we can eliminate B using these two equations and get the relation between A and C.

Complete step-by-step solution:
We are given two ratios.
The two ratios are A: B = 6: 7 and B: C = 8: 9,
We can write the ratios in form of a fraction.
Both of the ratios are given by $\dfrac{A}{B}=\dfrac{6}{7}$ and $\dfrac{B}{C}=\dfrac{8}{9}$ .
But need to find the ratio of A: C.
Therefore, our aim is to eliminate B.
We can cross multiply both the fractions and we can get two-equation.
Solving the first equation we get,
\begin{align} & \dfrac{A}{B}=\dfrac{6}{7} \\ & 7A=6B........................(i) \\ \end{align}
Solving the second equation we get,
\begin{align} & \dfrac{B}{C}=\dfrac{8}{9} \\ & 9B=8C.................(ii) \\ \end{align}
Let’s solve equation (i) to get the value of B in terms of A, we get,
\begin{align} & 7A=6B \\ & B=\dfrac{7A}{6}..................(iii) \\ \end{align}
Substituting the value of the equation (iii) in equation (ii) we get,
$9\left( \dfrac{7A}{6} \right)=8C$
Solving this equation, we get,
\begin{align} & 3\left( \dfrac{7A}{2} \right)=8C \\ & 21A=16C \\ \end{align}
As we are asked in the form of a ratio, we get,
$\dfrac{A}{C}=\dfrac{16}{21}$
Therefore, A : C = 16 : 21.
As we are given that there are multiple answers are correct, we need to try other options as well,
Let’s multiply both the sides by 3, we get,
$\dfrac{A}{C}=\dfrac{16\times 3}{21\times 3}=\dfrac{48}{63}$ .
Writing in the ratio form we get, A: C = 48: 63.
Hence, the correct options are (b) and (c).

Note: We can also solve this by first solving equation (ii) and then substituting in equation (i), the answer will stay the same. Also, we are told that there are multiple answers correct, so we need to check every option and where it simplifies to the option that we have arrived at.