We will now combine everything we have learned about positive and negative (directed) numbers and the order of operations with our knowledge of fractions and decimals.
Comparing and ordering rational positive and negative numbers
The symbol $<$< represents the phrase is less than. For example, $-\frac{3}{2}$−32 is less than $\frac{3}{4}$34 can be represented by $-\frac{3}{2}<\frac{3}{4}$−32<34.
The symbol $>$> represents the phrase is greater than. For example, $\frac{4}{3}$43 is greater than $-\frac{2}{3}$−23 can be represented by $\frac{4}{3}>-\frac{2}{3}$43>−23.
On the number line below, each tick is labelled with a multiple of the fraction $\frac{1}{5}$15. We can see that the point further to the left is plotted at the fraction $-\frac{3}{5}$−35, and the point further to the right is plotted at the fraction $\frac{6}{5}$65.
Each tick is labelled with a fraction.
This means $\frac{6}{5}$65 is greater than $-\frac{3}{5}$−35. It is the numbers' positions on the number line that helps us decide which number is greater (not their magnitudes).
Multiplying and dividing rational positive and negative numbers
We follow the exact same rules as before, we just need to take care when dealing with negative numbers.
We can divide fractions with keep, change, flip.
- Keep the first fraction the same
- Change division to multiplication
- Flip the second fraction to the reciprocal
Worked Example
Evaluate $-\frac{7}{10}\div\frac{4}{9}$−710÷49.
Think: Following the steps above, we want to keep the first fraction $-\frac{7}{10}$−710 as is, change the division operation to multiplication, and finally take the reciprocal of (or flip) $\frac{4}{9}$49. We also know that we are dividing a negative number by a positive number, so the result will be a negative number.
Do: The reciprocal of $\frac{4}{9}$49 is $\frac{9}{4}$94. We then want to multiply by this giving us $-\frac{7}{10}\div\frac{4}{9}=-\frac{7}{10}\times\frac{9}{4}$−710÷49=−710×94.
We can now multiply the numerators and denominators, $-\frac{7\times9}{10\times4}$−7×910×4 and evaluate the multiplications taking care to use the right signs, $-\frac{63}{40}$−6340. So $-\frac{7}{10}\div\frac{4}{9}=-\frac{63}{40}$−710÷49=−6340.
Reflect: By combining all we know about rational numbers and positive and negative numbers, we can follow a series of simple steps to arrive at the correct answer. We knew before calculating anything that our answer was going to be a negative number. It is useful to know whether to expect a positive or negative answer before evaluating, as this can help us know if we have made a mistake if we end up with an answer that is not the expected sign.
Adding fractions with the same denominator
Suppose we want to find $\frac{2}{7}+\frac{3}{7}$27+37. Here we have a circle with two sevenths shaded and a circle with three sevenths shaded.
Notice that the parts of each circle are the same size. We can place one circle on top of the other.
Now five sevenths of the circle are shaded in. So we can conclude that $\frac{2}{7}+\frac{3}{7}=\frac{5}{7}$27+37=57, or $2$2 sevenths plus $3$3 sevenths equals $5$5 sevenths.
When the denominators are the same, we are adding quantities of the same amount. So we can add the numerators and keep the same denominator.
Subtracting fractions with the same denominator
Suppose we want to find $\frac{3}{7}-\frac{2}{7}$37−27. Using the same circles as above, we can take two sevenths away from three sevenths.
The part that remains is one seventh of the circle. So we can conclude that $\frac{3}{7}-\frac{2}{7}=\frac{1}{7}$37−27=17, or $3$3 sevenths minus $2$2 sevenths equals $1$1 seventh.
When the denominators are the same, we are subtracting quantities of the same amount. So we can subtract the numerators and keep the same denominator.
This Geogebra applet can help you explore adding and subtracting fractions with the same denominator:
Adding fractions with different denominators
If the denominators are different then we are not adding quantities of the same amount. Consider $\frac{2}{9}+\frac{3}{4}$29+34. These two fractions look like this.
Before we can add these two fractions we rewrite them with the same denominator. Since we can change the denominator by multiplying the numerator and denominator by the same number, we want to first find a common multiple of the two denominators.
The denominators here are $9$9 and $4$4, so a common multiple of the denominators is $4\times9=36$4×9=36.
In this case, the denominators, $4$4 and $9$9 have no common factors. This means that $36$36 is the lowest common multiple of the denominators. This is sometimes called the lowest common denominator.
If the denominators were $4$4 and $6$6 instead, then we could find a common multiple the same way. That is, $4\times6=24$4×6=24. However, $24$24 is not the lowest common multiple of $4$4 and $6$6, because the lowest common multiple is $12$12.
In such a case, we could use either number, because they are both multiples.
This gives us two methods for adding fractions with different denominators. We can always find a common multiple by multiplying the two denominators. However, this will also mean that we will need to simplify the fraction resulting from the addition. Which method is better is a matter of preference.
Now we can rewrite the fractions. Multiplying the numerator and denominator of $\frac{2}{9}$29 gives $\frac{2\times4}{9\times4}=\frac{8}{36}$2×49×4=836. Multiplying the numerator and denominator of $\frac{3}{4}$34 gives $\frac{3\times9}{4\times9}=\frac{27}{36}$3×94×9=2736. Now these fractions look like this.
And now that the denominators are the same, we can add the fractions together.
We can see that $\frac{2}{9}+\frac{3}{4}=\frac{8}{36}+\frac{27}{36}=\frac{35}{36}$29+34=836+2736=3536.
So when the denominators are different, we rewrite the fractions with the same denominator, and then we can follow the procedure for fractions with the same denominator.
Subtracting fractions with different denominators
Suppose we want to find $\frac{3}{4}-\frac{2}{9}$34−29. Since the denominators are different, we rewrite the fractions with the same denominator before we subtract them. From the previous example we know that $\frac{3}{4}=\frac{27}{36}$34=2736 and $\frac{2}{9}=\frac{8}{36}$29=836.
When we take $\frac{8}{36}$836 away from $\frac{27}{36}$2736 we are left with $\frac{19}{26}$1926. So $\frac{3}{4}-\frac{2}{9}=\frac{27}{36}-\frac{8}{26}=\frac{19}{36}$34−29=2736−826=1936.
So when the denominators are different, we rewrite the fractions with the same denominator, and then we can follow the procedure for fractions with the same denominator.
This Geogebra applet can help you explore adding and subtracting fractions with the different denominators:
Adding and subtracting mixed numbers
Mixed numbers have a whole number part and a fraction part. The best way to add and subtract mixed numbers is to convert the mixed numbers into improper fractions. Then we can rewrite the improper fractions with the same denominator and add and subtract the fractions.
For example, to find $2\frac{3}{4}+1\frac{5}{6}$234+156 we can start by rewriting as improper fractions, $2\frac{3}{4}=\frac{8}{4}+\frac{3}{4}=\frac{11}{4}$234=84+34=114 and $1\frac{5}{6}=\frac{6}{6}+\frac{5}{6}=\frac{11}{6}$156=66+56=116.
Rewriting these improper fractions with the same denominator gives $\frac{11}{4}=\frac{11\times3}{4\times3}=\frac{33}{12}$114=11×34×3=3312 and $\frac{11}{6}=\frac{11\times2}{6\times2}=\frac{22}{12}$116=11×26×2=2212.
So then $2\frac{3}{4}-1\frac{5}{6}=\frac{33}{12}-\frac{22}{12}=\frac{11}{12}$234−156=3312−2212=1112.
When two fractions have the same denominator we can add or subtract them by adding or subtracting the numerators over the same denominator.
When two fractions have different denominators we first rewrite the fractions with the same denominator. Then we can add or subtract the numerators over the same denominator.
To add or subtract mixed numbers, we first write them as improper fractions and then we can use the same process.
Multiplying and dividing positive and negative decimals
As with fractions, we follow the same rules as before, taking into account if our numbers are positive and/or negative to decide whether our answer will be positive or negative.
Worked Example
Evaluate $4.83\times\left(-5.7\right)$4.83×(−5.7)
Think: We are multiplying a positive number by a negative number, so we know that the product will be negative. Ignoring the signs we can now calculate $4.83\times5.7$4.83×5.7 as we normally would, remembering to add the negative sign back to our answer.
Do: Before we even begin to calculate the answer it can be a good idea to have an estimation of the answer, especially when dealing with decimals. This will help us confirm our final answer is of the right magnitude.
Rounding both numbers to the nearest whole, gives us the much simpler calculation $5\times6$5×6, which we can evaluate to get $30$30, so we would expect our answer to be close to this value, taking into account the negative, we can expect our answer to be roughly equal to $-30$−30.
Now, to start the process, we ignore the decimal points. In this case we get $483$483 and $57$57. We can then multiply these together using the vertical algorithm:
| $4$4 | $8$8 | $3$3 | |||||
| $\times$× | $5$5 | $7$7 | |||||
| $3$3 | $3$3 | $8$8 | $1$1 |
Evaluating $483\times7$483×7 |
|||
| $+$+ | $2$2 | $4$4 | $1$1 | $5$5 | $0$0 |
Evaluating $483\times5\times10$483×5×10 |
|
| $2$2 | $7$7 | $5$5 | $3$3 | $1$1 |
Adding the two products together |
Now, to account for the decimal point, we add the total number of decimal places in the original numbers together.
In this case the original numbers are $4.83$4.83, which has two decimal places, and $5.7$5.7, which has one decimal place. So their product will have $2+1=3$2+1=3 decimal places.
Then to find the answer, we take the product that we calculated before and insert the decimal point such that there are $3$3 decimal places.
Finally, we need to account for the fact that we were actually multiplying by $-5.7$−5.7. So our final answer will be $-27.531$−27.531.
This is very close to our original estimate of $-30$−30.
The reciprocal of a number is $1$1 divided by that number.
- The reciprocal of a whole number is 1 over that number.
- The reciprocal of a fraction can be found by swapping the numerator and denominator.
The magnitude of a number is its distance from zero.
- For example, $6$6 and $-6$−6 have the same magnitude.
Practice questions
Question 1
Which decimal is greater?
$-0.74$−0.74
A$0.8$0.8
B
Question 2
Evaluate $-\frac{2}{5}\times\left(-\frac{9}{7}\right)$−25×(−97).
Question 3
Evaluate $\frac{2}{6}+\frac{2}{6}$26+26 and simplify your answer.
Question 4
Evaluate $\frac{3}{4}-\frac{1}{8}$34−18.
Question 5
Evaluate $2\frac{3}{11}+4\frac{7}{11}$2311+4711.
Question 6
Evaluate the quotient $7.36\div\left(-0.08\right)$7.36÷(−0.08)