Question

How do you determine whether the two ratios form a proportion: $\dfrac{6}{9},\dfrac{10}{15}$?

Answer 1

Hint: We first try to describe the relation between the denominator and the numerator to find the simplified form. We use the G.C.D of the denominator and the numerator to divide both of them. We get the simplified form when the G.C.D is 1. Then either we will have the same fraction or the cross-multiplication of the fractions will give equal value.

Complete step by step answer:
We need to find the simplified form of the proper fractions $\dfrac{6}{9},\dfrac{10}{15}$.
Simplified form is achieved when the G.C.D of the denominator and the numerator is 1.
This means we can’t eliminate any more common root from them other than 1.
For any fraction $\dfrac{p}{q}$, we first find the G.C.D of the denominator and the numerator. If it’s 1 then it’s already in its simplified form and if the G.C.D of the denominator and the numerator is any other number d then we need to divide the denominator and the numerator with d and get the simplified fraction form as $\dfrac{{}^{p}/{}_{d}}{{}^{q}/{}_{d}}$.
For our given fraction $\dfrac{6}{9}$, the G.C.D of the denominator and the numerator is 3.
Now we divide both the denominator and the numerator with 6 and get $\dfrac{{}^{6}/{}_{3}}{{}^{9}/{}_{3}}=\dfrac{2}{3}$.
For our given fraction $\dfrac{10}{15}$, the G.C.D of the denominator and the numerator is 5.
Now we divide both the denominator and the numerator with 6 and get $\dfrac{{}^{10}/{}_{5}}{{}^{15}/{}_{5}}=\dfrac{2}{3}$.
Therefore, the simplified form for both $\dfrac{6}{9},\dfrac{10}{15}$ is $\dfrac{2}{3}$.
The fractions are equal. So, they are proportional.

Note:
The cross-multiplication of the fractions $\dfrac{6}{9},\dfrac{10}{15}$ will give $6\times 15=9\times 10=90$.
The equal value of the cross-multiplication of the fractions proves that they are in proportion.