Question

What are three equivalent ratios for 12 to 9?

Answer 1

Hint: First write the ratio 12 to 9 in the fractional form with 12 as the numerator and 9 as the denominator. Now, cancel the common factors (if present) and get the simplest form of the fraction. This simplest form obtained will be an equivalent fraction. To find more equivalent fractions multiply the numerator and the denominator of the simplest form with the same numerical value. Choose a different numerical value each time and get the answer. Write them in the ratio form to get the answer.

Complete step by step answer:
Here we are provided with the ratio 12 to 9 and we are asked to write its three equivalent ratios. First let us find its equivalent fractions which can be converted into the ratios easily.
The ratio 12 to 9 can be written in fractional form by considering 12 as the numerator and 9 as the denominator as ${\displaystyle \frac{12}{9}}$. Let us see if it has common factors, so we get,
$\begin{array}{rl}& \Rightarrow {\displaystyle \frac{12}{9}}={\displaystyle \frac{3\times 4}{3\times 3}}\\ & \Rightarrow {\displaystyle \frac{12}{9}}={\displaystyle \frac{4}{3}}\end{array}$
Therefore, the simplest form of the fraction ${\displaystyle \frac{12}{9}}$ is ${\displaystyle \frac{4}{3}}$.
Now, two or more fractions are said to be equivalent if their simplest forms are equal. So, ${\displaystyle \frac{4}{3}}$ is an equivalent fraction of ${\displaystyle \frac{12}{9}}$. To find equivalent fractions we multiply the numerator and the denominator with the same numerical value other than 0. So let us find two more equivalent fractions.
$\begin{array}{rl}& \left(i\right){\displaystyle \frac{4}{3}}={\displaystyle \frac{4\times 5}{3\times 5}}={\displaystyle \frac{20}{15}}\\ & \left(ii\right){\displaystyle \frac{4}{3}}={\displaystyle \frac{4\times 6}{3\times 6}}={\displaystyle \frac{24}{18}}\end{array}$
Therefore, ${\displaystyle \frac{4}{3}}$,${\displaystyle \frac{20}{15}}$ and ${\displaystyle \frac{24}{18}}$ are the three equivalent fractions of ${\displaystyle \frac{12}{9}}$. We know that a fraction of the form ${\displaystyle \frac{a}{b}}$ is written in the ratio form as $a:b$. Hence, we have $4:3,20:15$ and $24:18$ as the three equivalent ratios of $12:9$.

Note: You must remember the definition of equivalent fraction to solve the above question. Do not multiply the numerator and denominator with 0 otherwise you will get the fraction undefined as the denominator of a fraction cannot be 0. So other than 0 you may choose any real number whether positive or negative, decimal or natural. There can be infinite equivalent ratios of a given ratio.