Question

If \[A:B = 5:6\] and $B:C = 4:7$ , find $A:B:C$.

Answer 1

Hint:
In such types of questions, find $A$ and $C$ in terms of a common variable, i.e., $B$ here and then find$A:B:C$. Also, you must remember that $A:B$ means $\dfrac{A}{B}$.

Complete step by step solution:
 Given, \[A:B = 5:6\]
$ \Rightarrow \dfrac{A}{B} = \dfrac{5}{6}$
$ \Rightarrow A = \dfrac{{5B}}{6}$ …… (1)
Also we have, $B:C = 4:7$
$ \Rightarrow \dfrac{B}{C} = \dfrac{4}{7}$
\[ \Rightarrow \dfrac{C}{B} = \dfrac{7}{4}\]
\[ \Rightarrow C = \dfrac{{7B}}{4}\] ….. (2)
Now using (1) and (2), we get-
$A:B:C$$ = \dfrac{{5B}}{6}:B:\dfrac{{7B}}{4}$
$ \Rightarrow A:B:C = \dfrac{5}{6}:1:\dfrac{7}{4}$ (Since, $B$is a common factor; hence it eliminates from each term)
$ \Rightarrow A:B:C = \dfrac{5}{6} \times 12:1 \times 12:\dfrac{7}{4} \times 12$ (Multiply the terms by the LCM of $6$ and $4$, i.e., $12$)
$ \Rightarrow A:B:C = 5 \times 2:1 \times 12:7 \times 3$

$ \Rightarrow A:B:C = 10:12:21$

Note:
An another method to solve this question is described as follows:
Given, \[A:B = 5:6\]$ \Rightarrow \dfrac{A}{B} = \dfrac{5}{6}$ …. (1)
& $B:C = 4:7$$ \Rightarrow \dfrac{B}{C} = \dfrac{4}{7}$ …..(2)
To find $A:B:C$, we need to make the value of $B$ the same in both the ratio.
Therefore, multiply by $4$ in (1) and multiply by $6$ in (2).
Now, $\dfrac{A}{B} = \dfrac{5}{6} \times \dfrac{4}{4}$
$ \Rightarrow \dfrac{A}{B} = \dfrac{{20}}{{24}}$
$ \Rightarrow A:B = 20:24$ …. (3)
Similarly, $\dfrac{B}{C} = \dfrac{4}{7} \times \dfrac{6}{6}$
$ \Rightarrow \dfrac{B}{C} = \dfrac{{24}}{{42}}$
$ \Rightarrow B:C = 24:42$ …. (4)
Now using (3) and (4), we get-
$A:B:C = 20:24:42$
$ \Rightarrow A:B:C = 10:12:21$ ( Divide by $2$ in each term)