Question

If \[6{\text{ }}:{\text{ }}8{\text{ }} = {\text{ }}9{\text{ }}:{\text{ }}x\]; then \[x\] is
A. \[18\]
B. \[12\]
C. \[8\]
D. \[6\]

Answer 1

Hint: In this given question, the terms are given in ratio. So, we can solve this question by using the factorization method. Ratio is nothing but a relationship between two numbers which shows how much one of the quantities is greater than the other. For example, we have 6 toffees and \[4\] chocolates in a box, then the ratio of toffees to chocolates is \[6:{\text{ }}4\] whereas the ratio of chocolates to toffees is \[4:6\].

Complete step by step answer:
We are given that
\[6:8{\text{ }} = {\text{ }}9:x............(i)\]
We have to find \[x\]
Now ratio is written as the how much one thing is completed to the other
\[6:{\text{ }}8\] is written as \[\dfrac{6}{8}\] and \[9:x\] is written as \[\dfrac{9}{x}\]
Equation (i) can be written as \[\dfrac{6}{8}\]\[ = \dfrac{9}{x}\]
Cross multiplication both sides, we get
$6 \times x = 9 \times 8$
\[6x{\text{ }} = {\text{ }}72\]
6 is written in denominator of other by cross-multiplication
$x = \dfrac{{72}}{6}$\[\]
Now, here we divide \[72\] by \[6\] and we get \[12\]
\[x\; = 12\]
So, according to given question
Option \[\left( B \right)\] is correct
Also If we substitute the value of \[x{\text{ }}in{\text{ }}1\] equation, we get
\[6{\text{ }}:{\text{ }}8{\text{ }} = {\text{ }}9{\text{ }}:{\text{ }}12\]
Which can be written as $\dfrac{6}{8} = \dfrac{9}{{12}}$
\[a:b\] can be written as $\dfrac{a}{b}$
Here, we make factors of the numerator \[9\] and denominator \[12\] on the right hand side.
$\dfrac{6}{8} = \dfrac{{3 \times 3}}{{4 \times 12}}$
Here, we make factors of the numerator \[6\] and denominator \[8\] on the left hand side.
 $\dfrac{{2 \times 3}}{{2 \times 4}} = \dfrac{{3 \times 3}}{{4 \times 3}}$
Cancel \[2\] on L.H.S & \[3\] on R.H.S., We get,
$\dfrac{3}{4} = \dfrac{3}{4}$
So, At this step Left hand side is equal to the right hand side.

Hence, answer of the question is option (B)

Note: 1) Ratio of \[2\] quantities is dimensionless. For Example: Ratio of rupees is unit less and similarly, the ratio of weight to weight is dimensionless.
2) Ratio \[a:{\text{ }}b\] is not equal to b: a. For example, \[\dfrac{3}{4}\] is not equal to $\dfrac{4}{3}$