Question

If $A:B=3:4$ and $B:C=6:7$ , then $A:B:C$ is $9:12:14$ state true or false.
A.True
B.False

Answer 1

Hint: Use equations $\dfrac{A}{B}=\dfrac{3}{4}$ and $\dfrac{B}{C}=\dfrac{6}{7}$ to get the value of B in terms of a and c individually. Now, get an equation by equating values of B and hence, equate the whole equation to k and get the values of A, B and C in terms of k. Now, find A:B:C with the help of the relation A,B,C in terms of k individually.

Complete step-by-step answer:
Here, ratio of A to B and B to C are given as:
$A:B=3:4$ ………………….. (1)
$B:C=6:7$ …………………… (2)
And hence, we need to verify the statement that value of $A:B:C=9:12:14$ . So, let us find the value of $A:B:C$ with the help of two equations.
So, we can write equation (1) as:
$\dfrac{A}{B}=\dfrac{3}{4}$
On cross-multiplying the above equation, we get:
$4A=3B$ ……………………….. (3)
Similarly, we can write equation (2) as:
$\dfrac{B}{C}=\dfrac{6}{7}$
On cross-multiplying the above equation, we get-
$7B=6C$ ……………………. (4)
Now, we can get value of B from equation (3) as:
$4A=3B$ or
$B=\dfrac{4A}{3}$ ……………………………. (5)
Similarly, we get value of B from equation (4) as:
$B=\dfrac{6C}{7}$ ……………………………. (6)
Now, we can equate equation (5) and (60 as both are representing the values of B. So, we get:
$B=\dfrac{4A}{3}=\dfrac{6C}{7}$ or
$\dfrac{4A}{3}=B=\dfrac{6C}{7}$ …………………………… (7)
Now, on equating the above equation to a real number ‘k’, we can rewrite the equation (7) as:
$\dfrac{4A}{3}=B=\dfrac{6C}{7}=k$ …………………………… (8)
Now, we can get value of A from first and last term of the above equation as:
$A=\dfrac{3k}{4}$
Similarly, value of B, can be given as:
$B=k$
And hence, value of C is also given by the same approach as:
$C=\dfrac{7k}{6}$
Therefore, A:B:C can be given as:
$\dfrac{3k}{4}:\dfrac{k}{1}:\dfrac{7k}{6}$
On dividing the whole equation by ‘k’ and multiplying with the L.C.M of 4, 1, 6 i.e. 12, we get the above relation as:
$\begin{align}
  & \dfrac{3k}{4}\times \dfrac{12}{k}:\dfrac{k}{1}\times \dfrac{12}{k}:\dfrac{7k}{6}\times \dfrac{12}{k} \\
 & 9:12:14 \\
\end{align}$
So, we get the value of A:B:C as:
\[A:B:C=9:12:14\]
So, the given statement is true as \[A:B:C=9:12:14\] is also given in the problem as well.
Hence, (A) is the correct answer.

Note: Another approach to solve the question would be that:
For, $A:B=3:4$
Suppose $A=3x$ and $B=4x$
For, $B:C=6:7$
Suppose $B=6y$ and $c=7y$
Now, equate 4x and 6y to get relation in x and y and hence, write the values of A, B, C either in x or y to get the value of A: B: C.
We cannot calculate exact values of A, B and C from the given equation. So, don’t confuse yourself with it as we cannot calculate values of three variables using only two equations. So, one may go wrong, If he/she goes for calculating values of A, B, C. Hence, we need to take care of it with the problem and for future reference as well.