31.05 - The quotient, fractions, ratios and percentages

The quotient of two numbers

Dividing two numbers by one another is called the quotient. The quotient is usually mathematically denoted as an obelus: $÷$

You can also write divisions as fractions, where the left number goes on the top, and the right number on the bottom

The left number is called the dividend (the number being divided)

• the dividend is called the numerator (top) in a fraction The right number is called the divisor (the number you are dividing by)
• The divisor is called the denominator (bottom) in a fraction Dividing means you will calculate how many times the divisor(right) fits in the dividend(left).

For example:

• The quotient of 6 and 2 is: $6÷2$
  • 6 is the dividend, and 2 is the divisor
  • Their fraction is: $\frac{6}{2}$
• The quotient of 12 and 3 is: $12÷3$
  • 12 is the dividend, and 3 is the divisor
  • Their fraction is: $\frac{12}{3}$
• The quotient of 15 and 4 is: $15÷4$
  • 15 is the dividend, and 4 is the divisor
  • Their fraction is: $\frac{15}{4}$

Quotients and negatives

You can apply the rules of products for quotients as well, lets try it with $20÷-5$:

1. Take both the dividend and divider in the equation: $20$ and $-5$.
   1. If either number is positive, write down a $+$ in front of it: $20$ → $+20$
2. remove the numbers: $+$ and $-$
3. • $++$ will be yes yes, which is the same as saying yes ($+$)
   • $+-$ will be yes not, which is the same as saying not ($-$)
   • $-+$ will be not yes, which is the same as saying not ($-$)
   • $--$ will be not not, which is the same as saying yes ($+$)
4. The outcome of step 3 will decide whether the result is positive or negative: $20÷-5=-4$

Keep in mind that the second operator isnt actually an operator

It is an indicator that the number is negative. The dual operator trick i showed is just to make adding/subtracting with negative numbers easier. [!important] Tip: If you have a hard time knowing the outcome at all

Since whether the outcome is positive or negative has been found out, you can remove all minuses, calculate the outcome, and add the minuses back.
$-20÷5$ → $20÷5=4$ → $-20÷5=-20$

Tip: If you have a hard time knowing the outcome at all

Since whether the outcome is positive or negative has been solved, you can remove all minuses, calculate the outcome, and add the minuses back.
$-5\times 3$ → $5\times 3=15$ → $-5\times 3=-15$

in case the equation has more than 2 numbers

If for example you have $5\times -3\times -4$, you apply the order of operations and do the equation in groups of 2 (in this case left to right).
OR, you could count the number of minuses and, if it’s an even amount, the solution is positive and vice versa.

Integer part division

Usually, the quotient is defined as the greatest whole number of times a divisor can be subtracted from a dividend before the remainder is below $1$. If this is the case, the quotient of two numbers is always an integer itself, or you could say that you divide and remove the decimal numbers.

For example:

• The quotient of 15 and 4 is: $15÷4$
  • 15 is the dividend, and 4 is the divisor
  • Their fraction is: $\frac{15}{4}$
  • The integer part quotient would be $3$ because $15÷4=3.75$
    • $4$ can be subtracted from $15$ $3$ times before the remainder is below 1

Fractions

If you take a pizza and cut it into 12 equal pieces…you’d have 12 pieces.

But then if i took 5 of those slices and put strawberries on them, 5 out of 12 slices would have strawberries on them. If you want to note this down as a fraction, it would be $\frac{5}{12}$. The top number of a fraction is called the numerator and the bottom one the denominator. A fraction and division are the same in the sense that you could write a fraction top-to-bottom to left-to-right and get the same equation: $\frac{5}{12}=5/12$.

Simplifying fractions

$\frac{1024}{2048}$ Seems like a really scary fraction, but its actually just $\frac{1}{2}$.
A way of simplifying factors is by calculating the greatest common divisor and dividing both the numerator and denominator by this number. Take $\frac{30}{160}$ for example:

$$\begin{array}{rl}{2}^{\boxed{1}}\times 3^1\times 5^{\boxed{1}}& =30\\ 2^5\times 3^{\boxed{0}}\times 5^1& =160\\ 2^1\times 3^0\times 5^1& =\boxed{10}\end{array}$$

So now we divide 30 and 160 by ten to get 3 and 16.

Another way is just to try and figure it out in your head :P

Equal fractions

If you took a pizza and cut it into 6 equal pieces and took 2 of those, you’d get $\frac{2}{6}$ of a took pizza.
If you took a pizza and cut it into 3 equal pieces and took 1 of those, you’d get $\frac{1}{3}$ of a the whole pizza.

$\frac{2}{6}$ and $\frac{1}{3}$ are equal to one another because $\frac{2}{6}$ divided by 2 is equal to $\frac{1}{3}$.

Equalizing fractions

You can make two fractions equal by equalizing the denominator, of course the numerator has to stay in proportion with its denominator, so whatever we do to the denominator has to be done to the numerator as well.

One way of doing this is by multiplying the denominators with one another since you know that number is a common multiple:

$$\frac{a}{b}+\frac{c}{d}=\frac{da}{db}+\frac{bc}{bd}$$

The right hand side of the equation is written this way because, to equalize the denominators $b$ and $d$, you can multiply both to get a common multiple.
But because $b$ gets multiplied by $d$, $a$ has to also get multiplied by $d$ to in proportion to $b$ (and vice versa, so $b\times d$ or $bd$ and $b\times c$ or $bc$).

Now lets simplify the equation. Because $db$ and $bd$ is the same ($5\times 3=3\times 5$), the bottom can be combined, because $db/bd=0$:

$$\frac{a}{b}+\frac{c}{d}=\frac{da}{db}+\frac{bc}{bd}=\frac{da+bc}{db}$$

Let’s now fill this in with numbers:

$$\frac{2}{3}+\frac{4}{5}=\frac{5\times 2}{5\times 3}+\frac{3\times 4}{3\times 5}=\frac{5\times 2+3\times 4}{3\times 5}=\frac{10+12}{15}=\frac{22}{15}$$

equal denominators

If the fractions have equal denominator, you can jump to the 3rd step immediately: $\frac{2}{5}+\frac{4}{5}=\frac{2+4}{5}=\frac{6}{5}$ which is a lot simpler

After this we can simplify the fractions by calculating the greatest common denominator and dividing both the numerator and denominator by this number.

$$gcd(22,15)=1$$

As you can see, the gcd is 1, so the numbers will stay the same

Adding and subtracting multiple fractions together

If you just have the sum of two fractions, use Equal fractions. Take for example:

$$\frac{4}{9}+\frac{1}{4}-\frac{1}{2}$$

First we calculate the denominator by calculating the least common multiple (or if you’re lazy just multiply all the numbers together, which is usually the same).

$$9\times 4\times 2=42$$

This is your denominator. Now the numerator is a bit trickier.

You multiply ones numerator by the others denominator (since you also did that with the same fraction’s denominator).

$$\begin{array}{rl}4\times 4\times 2& =32\\ 1\times 9\times 2& =18\\ 1\times 9\times 4& =36\end{array}$$

And now you replace these numbers for their respective fractions:

$$32+18-36=14$$

And finally you add back the denominator (and simplify if possible):

$$\frac{14}{72}=\frac{7}{36}$$

This is your final answer.

$$\frac{4}{9}+\frac{1}{4}-\frac{1}{2}=\frac{14}{72}=\frac{7}{36}$$

Multiplying fractions

Multiplying fractions is shockingly easy. You calculate the product of every numerator and every denominator and place them back into your fraction.

$$\frac{5}{10}\times \frac{6}{12}\times \frac{7}{13}=\frac{5\times 6\times 7}{10\times 12\times 13}=\frac{210}{1560}=\frac{7}{52}$$

The opposite of a fraction

If we flip the numerator and denominator of $\frac{2}{3}$, we get $\frac{3}{2}$.
If we multiply these together we get $\frac{2}{3}\times \frac{3}{2}=\frac{6}{6}=1$.

This is kinda similar to the opposite of an integer, but in this case, the product of two opposite fractions is 1 instead of 0

Dividing with fractions

If something is being divided by a fraction, it’s the same as multiplying it with its opposite.

$$\frac{a^1}{b^1}÷\frac{a^2}{b^2}=\frac{a^1}{b^1}\times \frac{b^2}{a^2}=\frac{a^1{b}^2}{b^1{a}^2}$$

Or an example with numbers would be:

$$\frac{3}{6}÷\frac{2}{5}=\frac{3}{6}\times \frac{5}{2}=\frac{15}{12}$$

Another way to notate $\frac{a^1}{b^1}÷\frac{a^2}{b^2}$ is this:

$$\frac{a^1}{b^1}÷\frac{a^2}{b^2}=\frac{\frac{a^1}{b^1}}{\frac{a^2}{b^2}}$$

No idea why you’d want to but here you go i guess

For dividing with negatives, just look at negative fractions.

Finding a missing number in 2 equal fractions

Let’s say you have the fraction: $\frac{18}{63}=\frac{x}{7}$ and you want to calculate $x$. Since $x$ is equal to $18$ and $7$ is equal to $63$, you can calculate the ratio of 63 to 7 and apply that to 18.

$$\begin{array}{rl}63÷7& =9\\ 18÷9& =2\\ \frac{18}{63}=\frac{2}{7}& \end{array}$$

Negative fractions

If there’s a negative number inside a fraction, you apply the following rule.

Take for example: $\frac{-5}{15}$

1. Take both numbers in the fraction: $-5$ and $15$.
   1. If either number is positive, write down a $+$ in front of it: $15$ → $+15$
2. remove the numbers: $+$ and $-$
3. • $++$ will be yes yes, which is the same as saying yes ($+$)
   • $+-$ will be yes not, which is the same as saying not ($-$)
   • $-+$ will be not yes, which is the same as saying not ($-$)
   • $--$ will be not not, which is the same as saying yes ($+$)
4. The outcome of step 3 will decide whether the result is positive or negative: $\frac{-5}{15}=-\frac{5}{15}$ which is easier to read.

Important

A fraction with a negative can also be written as -$\frac{1}{-15}$; it doesn’t matter where the minus is put Just keep in mind that a fraction is negative if there’s 1 or 3 minuses(uneven)

Fractions and powers

more info on powers

For more info on powers, click here

positive powers

Fractions with positive powers are done as followed:

$${\left(\frac{3}{5}\right)}^4=\frac{3}{5}\times \frac{3}{5}\times \frac{3}{5}\times \frac{3}{5}=\frac{3\times 3\times 3\times 3}{5\times 5\times 5\times 5}=\frac{3^4}{5^4}=\frac{81}{625}$$

negative powers

Fractions with negative powers are a bit silly to solve. As mentioned in 31.05 - The quotient and fractions: the opposite of a fraction is the same as turning multiplication into division and vice versa: this also applies to fractions since negatives do the opposite of multiplication.

Following this logic, we can deduce that if we flip the fraction, the exponent must also flip. To prove it, i will first solve the fraction without flipping it:

$$\begin{array}{rl}{\left(\frac{3}{5}\right)}^{-4}& =\frac{3^{-4}}{5^{-4}}=3^{-4}÷5^{-4}\\ {\left(\frac{3}{5}\right)}^{-4}=\frac{\frac{1}{81}}{\frac{1}{625}}=\frac{1}{81}\times \frac{625}{1}=\frac{1\times 625}{81\times 1}& =\frac{625}{81}\end{array}$$

This seems really drawn out and time consuming right?
Now let’s do it by flipping the exponent and fraction:

$${\left(\frac{3}{5}\right)}^{-4}={\left(\frac{5}{3}\right)}^4=\frac{5}{3}\times \frac{5}{3}\times \frac{5}{3}\times \frac{5}{3}=\frac{5\times 5\times 5\times 5}{3\times 3\times 3\times 3}=\frac{5^4}{3^4}=\frac{625}{81}$$

That’s a lot easier :3

Fractions and roots

more info on roots

For more info on roots, click here

Calculating the (square) root of a fraction is the exact same as multiplying it: you calculate the (square) root of the numerator and the denominator.

$$\sqrt{\frac{4}{9}}=\frac{\sqrt{4}}{\sqrt{9}}=\frac{2}{3}$$

Percentages

Percent is French for per cent, which means per 100. A percent is noted using the percent sign($\%$).

50 percent/50% means 50 per 100, or in other words a half of something($\frac{50}{100}=\frac{1}{2}$). For instance, $50\%$ of $20$ is 10.

Info

A good point of knowledge to know is that $x\%$ of $y$ is also $y\%$ of $x$. $50\%$ of $20$ is $10$. $20\%$ of $50$ is $10$

Using the percentage

When you have a number and a percentage, you can multiply the number by the percentage divided by 100:

$$\begin{array}{r}n2=n1\times (percentage÷100)\\ 20\times (50÷100)=20\times 0.5=10\end{array}$$

Finding(calculating) the percentage

When you have two numbers, you can calculate the percentage of one from other using the following steps:

We have two numbers: $n1$ and $n2$. We want to calculate how much percent $n2$ of $n1$ is.

Since $n1$ is the full number, $n1$ is $100\%$.

This lets us write down the numbers like this:

$n1$	?	$n2$
100%	?	?
The next step is to turn $n1$ into $n2$ and do the same to the percentage, because it needs to stay the same [[#ratios	ratio]].

The best way to turn $n1$ into $n2$ is to divide $n1$ by itself, turning it into one, and multiply it by $n2$. (Example with 90 and 54: $90÷90\times 54=54$).

Doing the same with the percentage will give us the percentage of $n2$ in relation to $n1$.

Let’s do it with some real numbers now: $n1=40$ and $n2=15$
Divide $100\%$ by $40$: $100÷40=2.5$
Multiply it by $15$: $2.5\times 15=37.5$
$100÷40\times 15=37.5$

\40	*15
40	1	15
100%	2.5	37.5
\40	*15

So to recap: to calculate how many percent a number of another number is, we do $percentage=100÷n1\times n2$

Ratios

Rations are kind of like percentages in the sense that they show a difference in size. A ratio shows how much bigger one number is opposed to another.

A ratio is also just another way to write a division or a fraction:
$1:20=1÷20=\frac{1}{20}=0.05=5\%$

When you fill a glass with 100ml of water and 20ml of lemonade, you have an 100:20, or 5:1 ratio of water to lemonade. Ratios have to written as an integer, so 1:0.2 is not a valid ratio.

Some other ratios are:

• 4:3
• 19:8
• 5:3
• etc…

Simplifying ratios

Simplifying a ratio(finding the smallest valid one) is done in the same way as fractions: you find the greatest common divisor and divide both numbers by it.

Example:
18:24 → 6 is the gcd → $18÷6=3$ and $24÷6=4$ → 3:4

Using ratios

When we make water with lemonade at a ratio of 10:1 and a glass has 200ml of water in it, how much lemonade is added?

To calculate this, we divide by the left number and multiply it by the right (same method as used in percentages).

$200÷10\times 1=20$ml of lemonade added.

If we had 220ml of prepared drinks and it was a water to lemonade ratio of 10:1, we can calculate how much water is added by dividing 100 by left + right number, and multiplying it by the right number:

$220÷11\times 10=200$ml of water inside the cup.

next chapter

The next chapter will go over order of operations