Question

A fraction which bears same ratio to \[\dfrac{1}{{27}}\] that \[\dfrac{3}{{11}}\] does to \[\dfrac{5}{9}\] , is equal to
A. \[\dfrac{1}{{55}}\]
B. \[\dfrac{1}{{11}}\]
C. \[\dfrac{3}{{11}}\]
D. \[55\]

Answer 1

Hint: In the above given question, we are given three fractions as \[\dfrac{1}{{27}}\] , \[\dfrac{3}{{11}}\] , and \[\dfrac{5}{9}\]. We have to find an another fraction such that that its ratio to \[\dfrac{1}{{27}}\] becomes equal to the ratio of the fractions \[\dfrac{3}{{11}}\] and \[\dfrac{5}{9}\]. In order to approach the solution, first we have to assume the required fraction as a variable \[x\]. Then we can find the ratio of the other two fractions \[\dfrac{3}{{11}}\] and \[\dfrac{5}{9}\] to substitute the value of their ratio equal to the ratio of \[x\] and \[\dfrac{1}{{27}}\]. In that way, we can obtain the value of the required fraction \[x\].

Complete step by step answer:
Given that, the three fractions as \[\dfrac{1}{{27}}\] , \[\dfrac{3}{{11}}\] , and \[\dfrac{5}{9}\].We have to find an another fraction such that its ratio to \[\dfrac{1}{{27}}\] is equal to the ratio of \[\dfrac{3}{{11}}\] and \[\dfrac{5}{9}\]. First, let us find the ratio of the last two fractions \[\dfrac{3}{{11}}\] and \[\dfrac{5}{9}\]. Their ratio will be given by the expression \[\dfrac{3}{{11}}:\dfrac{5}{9}\] as,
\[ \Rightarrow \dfrac{3}{{11}}:\dfrac{5}{9} = \dfrac{3}{{11}} \div \dfrac{5}{9}\]
That gives us,
\[ \Rightarrow \dfrac{3}{{11}}:\dfrac{5}{9} = \dfrac{3}{{11}} \times \dfrac{9}{5}\]
Now, let the first required fraction be \[x\]. Therefore, the ratio of the fractions \[x\] and \[\dfrac{1}{{27}}\] is given by the expression \[x:\dfrac{1}{{27}}\].

Now, it is given that the two ratios are equal .
That gives us the following equation,
\[ \Rightarrow x:\dfrac{1}{{27}} = \dfrac{3}{{11}}:\dfrac{5}{9}\]
Substituting the value of the first ratio, we get
\[ \Rightarrow x:\dfrac{1}{{27}} = \dfrac{3}{{11}} \times \dfrac{9}{5}\]
We can also write is as,
\[ \Rightarrow x \div \dfrac{1}{{27}} = \dfrac{3}{{11}} \times \dfrac{9}{5}\]
\[ \Rightarrow x \times 27 = \dfrac{3}{{11}} \times \dfrac{9}{5}\]
Dividing both sides by \[27\] we get the equation,
\[ \Rightarrow x = \dfrac{3}{{11}} \times \dfrac{9}{5} \times \dfrac{1}{{27}}\]
Evaluating above equation gives us the value,
\[ \Rightarrow x = \dfrac{1}{{55}}\]
That is the required fraction. Therefore, the fraction which bears same ratio to \[\dfrac{1}{{27}}\] that \[\dfrac{3}{{11}}\] does to \[\dfrac{5}{9}\] , is equal to \[\dfrac{1}{{55}}\].

Hence, the correct option is A.

Note: A ratio is the comparison between two quantities. It is the relation between two numbers which shows how much bigger one quantity is than the other. In simple words, the ratio is the number which can be used to express one quantity as a fraction of the other ones. The two numbers in a ratio can only be compared when they have the same unit.A ratio can be written using a colon as \[a:b\] or as a fraction as \[\dfrac{a}{b}\] .