Question

How do you know if the pair \[4/3\] and \[16/12\] form a proportion?

Answer 1

Hint: We use the concept of proportion which means that two ratios are equal. We write the first ratio given to us and try to form the second ratio using the first ratio. Multiply the first ratio with such a number that gives the second ratio.
* Proportion is defined as two ratios being equal. If \[\dfrac{a}{b} = \dfrac{c}{d}\], then we can say the two ratios are in proportion.

Complete step-by-step solution:
We are given two ratios \[4/3\] and \[16/12\]
We first write the two ratios properly in fraction form.
First ratio is \[\dfrac{4}{3}\]..............… (1)
Second ratio is \[\dfrac{{16}}{{12}}\]...............… (2)
We see that the numerator of second ratio is a multiple of numerator of first ratio i.e. \[16 = 4 \times 4\]
Similarly, denominator of second ratio is a multiple of denominator of first ratio i.e. \[12 = 4 \times 3\]
We multiply the first ratio by 4 in both numerator and denominator, such that it transforms into the second ratio.
Since we know that when we multiply a fraction by same number in the numerator and in the denominator, the value of the fraction remains the same; so we multiply equation (1) by 4 in both numerator and denominator
\[ \Rightarrow \dfrac{4}{3} = \dfrac{4}{3} \times \dfrac{4}{4}\]
Multiply numerator by numerator and denominator by denominator in right hand side of the equation
\[ \Rightarrow \dfrac{4}{3} = \dfrac{{16}}{{12}}\]
We see that the right hand side is the ratio in equation (2)
So, we get the two ratios as equal.

\[\therefore \]The pair \[4/3\] and \[16/12\] form a proportion.

Note: Alternate method:
We can write the second ratio in simpler form by cancelling common factors between numerator and denominator.
First ratio is \[\dfrac{4}{3}\]............… (1)
Second ratio is \[\dfrac{{16}}{{12}}\].............… (2)
We see that the numerator of second ratio is a multiple of numerator of first ratio i.e. \[16 = 4 \times 4\]
Similarly, denominator of second ratio is a multiple of denominator of first ratio i.e. \[12 = 4 \times 3\]
Substitute the values of 12 and 16 in equation (2)
\[ \Rightarrow \dfrac{{16}}{{12}} = \dfrac{{4 \times 4}}{{4 \times 3}}\]
Cancel same factor from both numerator and denominator i.e. 4
\[ \Rightarrow \dfrac{{16}}{{12}} = \dfrac{4}{3}\]
So, we get the two ratios as equal.
\[\therefore \]The pair \[4/3\] and \[16/12\] form a proportion.