Question

The value of ‘x’ in \[x:5::2:10\]?

Answer 1

Hint: We have a proportion formula. We know that proportion validates if two ratios are equivalent to each other. We know the ratio formula, that is \[a:b \Rightarrow \dfrac{a}{b}\]. We also know the proportion formula, \[a:b::c:d \Rightarrow \dfrac{a}{b} = \dfrac{c}{d}\]. We use this to solve the given problem and we use cross multiplication to simplify it.

Complete step by step solution:
Given \[x:5::2:10\].
We need to find ‘x’.
Now we know the proportion formula
\[a:b::c:d \Rightarrow \dfrac{a}{b} = \dfrac{c}{d}\].
Then given problem becomes
\[x:5::2:10 \Rightarrow \dfrac{x}{5} = \dfrac{2}{{10}}\]
\[ \Rightarrow \dfrac{x}{5} = \dfrac{2}{{10}}\]
Now for cross multiplying, we multiply the numerator of the left hand side of the fraction with the denominator of the right hand side fraction and we multiply the denominator of the left hand side of the fraction with the numerator of the right hand side of the fraction.
\[ \Rightarrow x \times 10 = 2 \times 5\]
\[ \Rightarrow x \times 10 = 10\]
\[ \Rightarrow x = \dfrac{{10}}{{10}}\]
\[ \Rightarrow x = 1\].
Hence the value of ‘x’ is 1.
So, the correct answer is “x = 1”.

Note: We have a proportion \[a:b::c:d\]. Here the first term and the last term are called extremes and the second and the third term is called means. That is ‘a’ and ‘d’ are called extremes and b and c are called means. In proportion we know that the product of extremes is equal to the product of means. Using this concept we can solve the given problem.
  \[x:5::2:10\]
Since, product of extremes is equal to product of means, then
\[ \Rightarrow x \times 10 = 5 \times 2\]
\[ \Rightarrow x \times 10 = 10\]
\[ \Rightarrow x = \dfrac{{10}}{{10}}\]
\[ \Rightarrow x = 1\]
Note: We can verify the answer by substituting the ‘x’ value in the given proportions.
\[ \Rightarrow 1:5::2:10\]
\[ \Rightarrow \dfrac{1}{5} = \dfrac{2}{{10}}\]
\[ \Rightarrow \dfrac{1}{5} = \dfrac{1}{5}\]. Hence we have the correct answer. We also know that ratio is an expression and a proportion is an equation. If two ratios are equal, then their reciprocal must also be equal as long as they exist.