Question 4.2.26

1. What decimal represents $\frac{3}{4}$? What about $\frac{6}{8}$? What about $\frac{9}{12}$? Show answer

   \begin{align*}\shortdiv{\ms{3}\ms{.}\ms{0}\ms{0}}{4}{\ms{0}\ms{.}\ms{7}\ms{5}}{\ms{}\ms{}\ms{^2}}\end{align*} \begin{align*}\shortdiv{\ms{6}\ms{.}\ms{0}\ms{0}}{8}{\ms{0}\ms{.}\ms{7}\ms{5}}{\ms{}\ms{}\ms{^4}}\end{align*} \begin{align*}\shortdiv{\ms{9}\ms{.}\ms{0}\ms{0}}{12}{\ms{0}\ms{.}\ms{7}\ms{5}}{\ms{}\ms{}\ms{^6}}\end{align*} The answer in all three cases is $0.75$. All three fractions are actually the same number.

   

2. Why did we get this answer? What is it about these three fractions? Show hint

   Look at the factors of each number.

   Show answer

   There are a few ways to look at this. You might have noticed that the numerators are all multiples of $3$, so you decided to look at the fractions like this: \begin{align*} \frac{6}{8} &= \frac{3\times2}{4\times2}\\ \frac{9}{12} &= \frac{3\times3}{4\times3}. \end{align*} and realised that if we multiply the top and bottom of the fraction by the same number, then the numerator will still be $\frac{3}{4}$ of the denominator, and the fraction will always be equivalent to $\frac{3}{4}$.

   Or you might have pictured the fractions a bit like this: and noticed that $\frac{6}{8}$ is the same as $\frac{3}{4}$ but with every part split into $2$, and similarly for $\frac{9}{12}$ split into $3$, but the amount shaded remains the same.