31.05 - The quotient and fractions

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 dividend and the bottom one the divisor. 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: $\frac{5}{12}=5/12$.

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 $\frac{1}{3}$.

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 dividend and divisor by this number. Take $\frac{30}{160}$ for example:

\begin{aligned} 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{aligned}

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

Equalizing fractions

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

One way of doing this is by multiplying the divisors 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 divisors $b$ 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:

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

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

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

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

next chapter

The next chapter will go over order of operations