Question

How do you determine whether the ratios are equivalent: \[\dfrac{5}{6}\] ; \[\dfrac{{15}}{{18}}\] ?

Answer 1

Hint: Here in this question, we have to check if the given ratios are equal or not. For solving this first we have written the two given ratios are equal to each other, next cross multiplying the ratios, on multiplication if we get the same value in both LHS and RHS then the ratios are equivalent otherwise not equivalent.

Complete step by step solution:
Ratio is a comparison of two or more numbers that indicates their sizes in relation to each other. A ratio compares two quantities by division, with the dividend or number being divided termed the antecedent and the divisor or number that is dividing termed the consequent.
Consider the given two ratios:
 \[\dfrac{5}{6}\] and \[\dfrac{{15}}{{18}}\]
Here, we have to check these two ratios are equivalent or not
When calculating equivalent ratios you must multiply or divide both numbers in the ratio. This keeps both numbers in direct relation to each other. i.e.,
 \[\dfrac{a}{b} = \dfrac{c}{d} \Leftrightarrow ad = cb\]
Where, \[b \ne 0\] and \[d \ne 0\]
In the given question
 \[a = 5\] , \[b = 6\] , \[c = 15\] , and \[d = 18\] .
Then the ratios are written as
 \[ \Rightarrow \dfrac{5}{6} = \dfrac{{15}}{{18}}\]
On cross multiplying the ratios:
 \[ \Rightarrow 5 \times 18 = 15 \times 6\]
On multiplication, we get
 \[ \Rightarrow 90 = 90\]
 \[\therefore \] LHS = RHS
Hence, the two ratios \[\dfrac{5}{6}\] ; \[\dfrac{{15}}{{18}}\] are equivalent.

Note: The ratios are equivalent means it is equal. The ratios are equivalent means it should satisfies the condition and it is given as \[\dfrac{a}{b} = \dfrac{c}{d} \Leftrightarrow ad = cb\] , where the a, b, c and d are the real values and hence we can simplify, if it is simplifying. Hence we obtain the solution for the question.