2.01 Operations with decimals
You already know how to work with positive decimals using various combinations of operations. Now we will use and extend this knowledge by looking at questions that involve negative decimals.
Last year you learned the rules for adding, subtracting, multiplying, and dividing decimals. Review those rules here:
- The only thing to remember when adding or subtracting decimals is to line up the decimal points and place values in the numbers. This way, the decimal point in the answer will go directly below the two decimal points in the sum or difference.
- When multiplying decimals, multiply the numbers as if there were no decimal point at all. Then, count the numbers after the decimal points in the original numbers, and place the decimal the total number of places to the left.
- To divide decimals we use the fact that multiplying by ten moves the decimal one point to the right. Take the divisor and multiply it by ten until it is a whole number, and then multiply the dividend by that same power of ten. Then you can use your long division rules for whole numbers.
Next, remember the rules for when you add, subtract, multiply or divide with integers:
- Adding and subtracting - always consider the number line to help you
- If you have adjacent positive (plus) and negative (minus) signs (such as $1+\left(-6\right)$1+(−6)), this will become a minus sign.
- If you have two adjacent negative (minus) signs (e.g. $-5-\left(-6\right)$−5−(−6), this will become an addition sign.
- Multiplying and dividing
- If one of your number is negative and the other is positive (such as $3\times\left(-4\right)$3×(−4)), your answer will be negative.
- If both numbers are negative (such as $\left(-12\right)\div\left(-4\right)$(−12)÷(−4)), your answer will be positive.
Now we will put all of these rules together to complete operations on positive and negative decimals.
Worked examples
Question 1
Evaluate $3.4+\left(-5.2\right)$3.4+(−5.2)
Think: To evaluate a decimal sum, line up the decimals and perform the operation. Adding a negative number moves left on the number line, so can be rewritten as subtraction. $3.4+\left(-5.2\right)=3.4-5.2$3.4+(−5.2)=3.4−5.2. More importantly, we should notice that the answer must be negative as we are subtracting a larger number from a smaller one.
We will actually find $-\left(5.2-3.4\right)$−(5.2−3.4).
Do:
| $5$5 | . | $2$2 |
|
|
| $-$− | $3$3 | . | $4$4 |
(Rewrite in vertical form with the decimal points lined up) |
| $1$1 | . | $8$8 |
(Perform the subtraction) |
|
So this means that $3.4+\left(-5.2\right)=-1.8$3.4+(−5.2)=−1.8
Reflect: Round these decimals to the nearest whole number and estimate the solution. How does your estimate compare with the actual solution?
Question 2
Evaluate: $-5.4\times\left(-3.6\right)$−5.4×(−3.6)
Think: There are two numbers after the decimals in this problem, so our product will have two values to the right of its decimal point. The product of two negative numbers will be positive.
Do: $-5.4\times\left(-3.6\right)=19.44$−5.4×(−3.6)=19.44
Reflect: Suppose the second number were $-3.60$−3.60. Would this change your final answer?
Question 3
Evaluate: $\left(-6.3\right)\div0.15$(−6.3)÷0.15
Think: The quotient of a negative number and a positive number will be negative. We will need to multiply by ten times ten, moving the decimal point twice, because there are two values after the decimal in our divisor.
Do:
| $\left(-6.3\right)\div0.15$(−6.3)÷0.15 |
|
|
| $=-6.3\div0.15$=−6.3÷0.15 | $\left[\times10\text{ }\times10\right]$[×10 ×10] |
(Multiply the divisor by two powers of ten) |
| $=-630\div15$=−630÷15 |
(Do to the dividend exactly what we did to the divisor) |
|
| $=-42$=−42 |
(Perform the division) |
Reflect: Is this answer the same as the solution to $\left(-63\right)\div1.5$(−63)÷1.5?
Practice questions
QUESTION 4
Evaluate $2.5+\left(-8.1\right)$2.5+(−8.1).
QUESTION 5
Evaluate $7.4\cdot\left(-4.1\right)$7.4·(−4.1).
QUESTION 6
Evaluate $-9.3-\left(-2.2\right)$−9.3−(−2.2)
QUESTION 7
Jenny takes out a loan of $\$2200$$2200. She pays back $\$42.60$$42.60 each month and doesn't have to pay interest.
If she has made $5$5 repayments so far, how much does Jenny still owe?