Why does $\frac{3}{4} \times 1\frac{1}{3}$ equal $1$?

Question

Can someone please explain how this works and how to apply it to other fractions to get 100%?

I thought it might have something to do with the Dividend in the first Part (In this Example the 3 in $\frac34$) being the divisor in the second part (The $3$ in $1\frac13$) since $\frac56 \times 1\frac{1}{5}$ also equals $1$ but it doesn't really work out with other Fractions like $\frac46$.

Maybe those two examples have by pure coincidence so nicely even and fitting numbers. Any help is appreciated.

Answer 1

We have $$\begin{align} 1\frac{1}{a} &= 1 + \frac{1}{a} \\ &= \frac{a}{a} + \frac{1}{a} \\ &= \frac{a+1}{a}. \end{align}$$

Therefore,

$$\begin{align} \frac{a}{a+1} \times 1\frac{1}{a} &= \frac{a}{a+1} \times \frac{a+1}{a} \\ &= 1, \end{align}$$

provided $a\neq 0$ and $a \ne -1$

Answer 2

Since $1\,\frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3}$, you get $$\frac{3}{4}\times 1\,\frac{1}{3} = \frac{3}{4}\cdot\frac{4}{3} = 1.$$ For your other example, $$\frac{4}{6}\times 1\,\frac{\color{red}{2}}{4} = \frac{4}{6}\cdot\frac{6}{4} = 1.$$

Answer 3

Here

$$\begin{align} 1\frac{1}{3}&=\frac{3}{3}+\frac{1}{3}\\ &=\frac{3+1}{3}\\ &=\frac{4}{3}; \end{align}$$

therefore, we have

$$\begin{align} \frac{3}{4}\times 1\frac{1}{3}&=\frac{3}{4}\times \frac{4}{3}\\ &=\frac{3\times 4}{4\times 3}\\ &=\frac{12}{12}\\ &=1. \end{align}$$

Answer 4

Note that $1\frac{1}{3}$ is the same as $1+\frac{1}{3}$. Therefore, we can rewrite the product as $$ \frac{3}{4}\times\left(1+\frac{1}{3}\right). $$ Now, we can use the distributive property to rewrite this as $$ \frac{3}{4}\times 1+\frac{3}{4}\times\frac{1}{3}. $$ The first product is $\frac{3}{4}$ since we have multiplication by $1$. The second product simplifies to $\frac{1}{4}$ since there is a $3$ in the numerator and a $3$ in the denominator, which cancel. Putting this together, this simplifies to $$ \frac{3}{4}+\frac{1}{4}=\frac{3+1}{4}=1. $$

Answer 5

The intuition of multiplying with a fraction is:

Take that fraction out of the other factor!

E.g. multiplying $x$ by $1/2$ means taking a half of $x$. Multiplying $x$ by $3/5$ means taking three fifths of $x$. And so on.

Now, you are multiplying $3/4$ by $1\frac{1}{3}$, which may mean one of two things, which will both boil down to the same:

• Either you are taking three quarters of $1\frac{1}{3}=\frac{4}{3}$. In this case, one quarter of it is $1/3$ and three quarters are $3/3=1$,
• Or you are taking one-and-a-third of $3/4$. A third of $3/4$ is $1/4$ so one-and-a-third of $3/4$ is $3/4+1/4=4/4=1$.

In both cases, the result is $1$.