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**Hint:**The given question will be solved with the concept of ratio - proportion. The question asks us to equate the two given fractions. For solving the question we should know to cross multiply the fractions. This will then lead us to the values of v.

**Complete step by step solution:**

Let us consider the fraction $\dfrac{p}{q}=\dfrac{r}{s}$ , then after cross- multiplication this could be written as $p\times s=r\times q$ or $p=\dfrac{q\times r}{s}$ .

The question given to us is

$\dfrac{v}{0.23}=\dfrac{7}{1.61}$

On seeing the problem we can infer the two fractions need to be equated.

So we will have to cross multiply the fraction.

$\Rightarrow \dfrac{v}{0.23}\nearrow \dfrac{7}{1.61}$

$\Rightarrow v=\dfrac{0.23\times 7}{1.61}$

To make the solving simpler the decimal point can be removed by converting it into fraction which is dividing it by 100 in this case. So here the decimal point from 0.23 and 1.61 both are divided by 100.

$\Rightarrow v=\left( \dfrac{23}{100} \right)\left( 7 \right)\left( \dfrac{100}{161} \right)$

On multiplying the numerators and denominators respectively, we get

$\begin{align}

& \Rightarrow v=\dfrac{161}{161} \\

& \Rightarrow v=1 \\

\end{align}$

$\therefore $ The value of $v$ for the expression is 1 .

**Note:**We should have a knowledge of cross-multiplication. In cross-multiplication we multiply the numerator of the first fraction to the denominator of the second fraction, and vice versa. It is an important concept for solving questions on ratio and proportion.

We can check whether the value of v is correct or not. For this we put the value of v as $1$ and check whether the first fraction(LHS) is same as the second one.

$\begin{align}

& \dfrac{v}{0.23} \\

& \Rightarrow \dfrac{1}{0.23} \\

\end{align}$

Second fraction (RHS) is $\dfrac{7}{1.61}$ which on solving turns to $\dfrac{1}{0.23}$ .

$\therefore $ The both the fractions are equal. So you check whether our answer is correct or not .

To change decimal into fraction we need to remove the point from the given number and write number as the fraction’s numerator without the decimal and denominator becomes ${{10}^{n}}$ where n is the number of digits after the decimal in the decimal number given .

Example: 4.55 in fraction is $=\dfrac{455}{100}$

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