Division misconception

Question

I'm having trouble understanding division when the divisor is greater than the dividend, for ex 1/4.

I think of division as "how many times can the divisor fit into the dividend evenly".

Intuitively, when I see 1/4 in the context of slices of pizza, I think of it as 1 "out of" 4, but I can't seem to grasp it in terms of "how many times does 4 fit into 1" if that makes any sense.

In other words my question could be why do we use division to represent "one out of 4"?.

If it helps you guys understand, this came about as I was trying to find the percentage representation of two populations, as in "there are 1253 A's and 747 B's, what is the proportion of each in % ?".

Conceptually I understand that I need to add up those two populations and then find the proportion they represent of that total. However, when I got to that second part, I couldn't reason through whether I needed to divide the total by a population, or a population by the total in order to find the desired result.

Obviously I eventually found the right way to do it, but it still doesn't make sense to me.

Sorry if it's very vague, this is really bothering me; I can't seem to reconcile those two ways of thinking about division.

Answer 1

Suppose you have 1 bread and you want to distribute it among 4 people equally. The only solution is to cut the single bread into 4 equal pieces.

For the population suppose that you have collected 8 from A and B such that their donations are 5 and 3 respectively. So if someone asks you that what is the contribution of A or B in the donation in terms of the percentage. Then what should you do?

Answer 2

$X$ equals $\,\dfrac{1}4\,$ of the total $\,T$ means $\,X\, =\, \dfrac{1}4 T\, =\, \dfrac{T}4\ \,$ or $\ \color{#c00}{\dfrac{X}T = \dfrac{1}4}\,\ $ or $\,\ 4X = T$

$X$ equals $\dfrac{A}B$ of the total $\,T$ means $\,X = \dfrac{A}B T = \dfrac{AT}B\ $ or $\ \color{#c00}{\dfrac{X}T = \dfrac{A}B}\ $ or $\ BX = AT$

$X$ equal $A$% of the total $\,T$ is the special case $\,B=100\,$ of the above, i.e. $A$% denotes the fractional proportion $\,A/100\,$ (here $A$ can be any real number). For example, $25$% denotes $25/100 = 1/4$.

To convert a fraction to percent form $\ \dfrac{X}{100} = \dfrac{A}B\ \Rightarrow\ X = \dfrac{100A}B\, $ (usually written in decimal).

Said equivalently, write $\,A/B\,$ in decimal form, then shift the point two places rightward.

Hence the meaning of "percent" boils down to the meaning of fractional proportions, which is simply the ratio of the part $X$ to the whole $T$, expressed as a fraction (see the red equations above). Hopefully this makes it clearer which number goes on the top vs. bottom of the fraction.

Answer 3

Division is often represented algebraically as the fraction $\frac{a}{b}$ where "$a$ is divided by $b$". $a$ is the numerator (dividend) and $b$ the denominator (divisor). The numerator represents the number of equal parts and the denominator indicates how many of those parts make up the whole. Take for example $\frac{3}{4}$; the numerator indicates that the fraction represents $3$ equal parts while the denominator indicates that $4$ parts make up the whole.

You can still think of division as "how many times can the divisor fit into the dividend evenly" in instances where the divisor is larger than the dividend; but you should realize that $\frac{1}{4}$ is just one way of expressing division. What might help you see this is considering writing the fraction as its decimal equivalent: $$\frac{1}{4} = 4 \div 1 = 0.25$$

And so, $1$ is divided by $4$ evenly $0.25$ times.

---

Following the definition of a fraction: a representation of a part of a whole, it should now be easier for you to see which (either the total of $A$/$B$'s population or the total population) goes on top and which goes on bottom. Using above, the numerator represents the number of equal parts, in this case $A=1253$ and $B=747$ and the denominator would be how many parts make up the whole, i.e. $A+B=1253+747=2000$.

Hence the numerator would be either $1253$ (for $A$'s percentage) or $747$ (for $B$'s percentage) and the denominator for both would the be total, $2000$.

Answer 4

I'm not sure of your level, but it might not be helpful to think of $\frac{a}{b}$ as "the number of times" $b$ fits into $a$ simply because that is not so intuitive when it comes to non-integer number of times. Instead, $\frac{a}{b}$ (where $b \ne 0$) is the amount such that $\frac{a}{b} \times b = a$. In other words, dividing by $b$ is the inverse (opposite) of multiplying by $b$. This also means that $\frac{a \times b}{b} = a$. For example, $\frac{2}{5}$ is the amount such that $5$ times of it is $2$, and $\frac{2 \times 5}{5}$ is the amount such that 5 times of it is $2 \times 5$. This definition remains valid for irrational amounts (and later complex numbers and more generally mathematical structures called fields). For example, a square with length $2r$ has greater area than a circle with radius $r$, and the ratio of the square's area to the circle's area is $\frac{4r^2}{\pi r^2} = \frac{4}{\pi} \approx 1.27324$. Intuitively, this means that if you divide up both shapes into tiny squares of equal size using a grid (and ignore the pieces that aren't square), the ratio of the number of pieces will get closer to $\frac{4}{\pi}$ as the size of the grid square decreases.