I can solve fractions abstractly, for example, $\frac{5}{2}$ divided by $\frac{3}{2}$, you can flip $\frac{3}{2}$ so that $\frac{5}{2}$ multiplied by $\frac{2}{3}$. Specifically, math makes sense abstractly, but concretely it just won't make sense, like in word problems. I understand the concept of complex fractions I know how to solve them, but by applying it on practical use such as a shape it does not make sense. How to make sense of fractions concretely?? or perhaps there is a book that you can advice me that help solve this problem
4 Answers
So what you would like to know is why $$\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{a\cdot d}{b\cdot c}$$
to understand this you first need to understand $\frac{1}{x}\cdot x=1$. Knowng this we can see $\frac{c}{d}\cdot\frac{1}{\frac{c}{d}}=1\implies \frac{1}{\frac{c}{d}}=\frac{d}{c}$ (just assume $\frac{c}{d}$ is $x$).
Therefore
$$\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{a}{b}\cdot\frac{1}{\frac{c}{d}}=\frac{a}{b}\frac{d}{c}=\frac{ad}{bc}$$
as desired.
-
$\begingroup$ Again as I told I understand the abstract way of dividing a/b by c/d. But if I put it in concrete way like in terms of a shape or practical means it does not make sense. So what I want is to make sense of it concretely. Again I understand the concept of complex fractions $\endgroup$ Apr 20, 2014 at 2:47
-
$\begingroup$ If you really understand it abstractly then you should understand it concretely also. $\endgroup$– AsinomásApr 20, 2014 at 2:50
In general, $\dfrac{a}b$ denotes the quantity $x$, which when multiplied by $b$ gives $a$. In fact, this is the correct way to interpret $\dfrac{a}b$.
In your case, $\dfrac{5/2}{3/2}$ denotes the quantity $x$, which when multiplied by $\dfrac32$ gives us $\dfrac52$, i.e., $$\dfrac32x = \dfrac52 \implies x = \dfrac53$$
Taking your example: $5/2$ divided by $3/2$ is the number of bottles of $3/2$ liters that you need to complete $5/2$ liters. What is the same that how many bottles of $3$ liters are needed to contain $5$ liters. That is, $5/3$ bottles.
-
$\begingroup$ Alternatively, if $5/2$L represents $3/2$ containers, then how big is one container? "Partitive" and "Quotative " are good words to Google. $\endgroup$– John JoyMar 5, 2017 at 20:04
This is a playlist of videos especially targeted to understand fractions concretely, hope you enjoy.