1.04 Operations with fractions
We've already learned how to add, subtract, multiply and divide fractions. Similarly, we've looked at each of these operations with negative numbers.
The process is just the same when we have questions with negative fractions - we'll just combine these two skills and their rules to complete the operation.
- Adding and subtracting: be sure there is a common denominator, and then add or subtract the numerators and keep the denominator
- Multiplying: Multiply the numerators, multiply the denominators, and simplify.
- Dividing: Take the reciprocal of the divisor and multiply. Then, follow the rules for multiplication.
- Mixed numbers: Change the mixed number to an improper fraction, and proceed as if this were a normal fraction operation. You may be expected to change the solution back to a mixed number.
- Subtracting a negative number is the same as adding a positive number, e.g. $5-\left(-3\right)=5+3$5−(−3)=5+3
- Adding a negative number is the same as subtracting a positive number, e.g. $2+\left(-7\right)=2-7$2+(−7)=2−7.
- The product or quotient of two negative terms is a positive answer, e.g. $-6\times\left(-9\right)=54$−6×(−9)=54 or $\left(-144\right)\div\left(-12\right)=12$(−144)÷(−12)=12
- The product or quotient of a positive and a negative term is a negative answer, e.g. $-5\times10=-50$−5×10=−50 or $20\div\left(-4\right)=-5$20÷(−4)=−5
Using the number line or zero pairs may be helpful with working positive and negative fractions.
Worked examples
Question 1
Calculate: $3+4-\left(-\frac{4}{5}\right)$3+4−(−45).
Think: We will evaluate the addition and subtraction, working from left to right.
Do:
| $3+4-\left(-\frac{4}{5}\right)$3+4−(−45) | $=$= | $7-\left(-\frac{4}{5}\right)$7−(−45) |
| $=$= | $7+\frac{4}{5}$7+45 | |
| $=$= | $7\frac{4}{5}$745 |
Reflect: What if this problem were written as a decimal? Is $3+4-\left(-0.8\right)$3+4−(−0.8) easier to compute? Do you still end up with the same solution when you convert the decimal back to a mixed number?
Practice questions
Question 2
Evaluate $-10\cdot\left(-2\frac{1}{4}\right)$−10·(−214), giving your answer as a mixed number in simplest form.
Question 3
Evaluate $-8\frac{4}{7}+3\frac{3}{7}$−847+337, writing your answer as a mixed number in its simplest form.
Question 4
Evaluate $4\frac{2}{3}\div\left(-1\frac{2}{5}\right)$423÷(−125), giving your answer as a mixed number in simplest form.