Question

How do you know if the pair \[\dfrac{6}{9}\] and $\dfrac{2}{3}$ form a proportion?

Answer 1

Hint: In this question, we are given two fractions and we have to find if they are proportional or not. Proportional means that one fraction is a multiple of the other fraction, that is, if we multiply one fraction by some number and we get the other fraction, then we can say that the two fractions form a pair of proportions. For example, if one fraction is $\dfrac{a}{b}$ and the other fraction is $\dfrac{{2a}}{{2b}}$ , then these two fractions form a proportion as on simplifying the second fraction, we get the first fraction as the answer. This way we can solve the given question.

Complete step-by-step solution:
We are given two fractions \[\dfrac{6}{9}\] and $\dfrac{2}{3}$ , $\dfrac{6}{9}$ can be written as $\dfrac{6}{9} = \dfrac{{2 \times 3}}{{3 \times 3}}$
As 3 is common in both the numerator and the denominator, so we cancel it out and get –
\[\dfrac{6}{9} = \dfrac{2}{3}\]
Hence, the pair \[\dfrac{6}{9}\] and $\dfrac{2}{3}$ forms a proportion.

Note: A fraction is defined as an expression in which terms are present that are separated by a horizontal line, the term on the upper side of the horizontal line is called the numerator and the term on the lower side is called the denominator. For simplifying a fraction, we write the numerator and the denominator as a product of its prime factors and cancel out the common factors. This question can also be solved by equating the given two fractions and then cross multiplying them –
$
  \dfrac{6}{9} = \dfrac{2}{3} \\
   \Rightarrow 6 \times 3 = 2 \times 9 \\
   \Rightarrow 18 = 18 \\
 $
As 18 is equal to 18, so the fractions do show a proportion.