Topics
2. Rational Numbers
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United States of America
Grade 7

2.02 Operations with decimals

Lesson
Practice

Introduction

Previously, we have learned how to  add and subtract integers  as well as to  multiply and divide integers  . We will now extend previous understandings of operations on integers to operations in decimal numbers.

Decimal operations

There are some key differences we encounter when adding, subtracting, multiplying, and dividing decimals as opposed to other types of numbers. Here are some strategies that we can use with decimals:

  • When adding or subtracting decimals, we must line up the decimal points and place values of the numbers.
  • When multiplying decimals, multiply the numbers as if there were no decimal points at all. Then, count the numbers after the decimal points in the original numbers, and place the decimal in the total number of places to the left.
  • To divide decimals, we use the fact that is multiplying by 10 moves the decimal one point to the right. Take the divisor and multiply it by 10 until it is a whole number, then multiply the dividend by that same power of 10. We can then use long division rules for whole numbers.

When we have negative decimals, the same rules of negative and positive numbers apply. For example, when we multiply two negative decimal numbers together, we will have a positive answer, just as we would for any other type of number.

Examples

Example 1

Evaluate: 8.5+\left(-4.1\right)

Worked Solution
Create a strategy

Recall that when a plus and a minus sign are next to each other it becomes subtraction.

Apply the idea
\displaystyle 8.5+\left(-4.1\right)\displaystyle =\displaystyle 8.5-4.1Combine adjacent signs

\begin{array}{c} & 8 & . & 5 \\ - & 4 & . & 1 \\ \hline & 4 & . & 4 \\ \end{array}

Example 2

Evaluate: 7.4\times \left(-4.1\right)

Worked Solution
Create a strategy

Recall that positive times negative equals negative.

Apply the idea

Multiply the numbers without the decimal points and apply the correct sign: \begin {array}{c} & & & 7 & 4 \\ &\times & & -4 & 1 \\ \hline & & & 7 & 4 \\ &2 & 9 & 6 & \\ \hline & -3 & 0 & 3 & 4 \\ \end{array}

The final answer should have the same number of decimal places as the total number of decimal places of the factors, which is two:

\displaystyle 7.4\times(-4.1)\displaystyle =\displaystyle -30.34

Example 3

Evaluate: -9.3-\left(-2.2\right)

Worked Solution
Create a strategy

Subtracting a negative value is equivalent to adding the positive of that value. We can use the number line to illustrate the process.

Apply the idea
\displaystyle -9.3-\left(-2.2\right)\displaystyle =\displaystyle -9.3+2.2Combine adjacent signs

First, plot -9.3 on the number line.

-10-9-8-7-6-5

Then, count up 2.2 units, 2 wholes and 2 tenths, from -9.3 by moving to the right.

-10-9-8-7-6-5

So, -9.3 - (-2.2) = -7.1.

Reflect and check

To evaluate the sum, it may be easier to notice that the answer to -9.3+2.2 and the answer to 9.3-2.2 are opposite in sign.

So we can evaluate 9.3-2.2 and change the sign.

Example 4

Evaluate: -6.3 \div 0.15

Worked Solution
Create a strategy

We represent the division vertically so that it becomes a fraction. Eliminate the decimal point by multiplying both parts of fraction by powers of 10.

Apply the idea
\displaystyle \dfrac{-6.3}{0.15}\displaystyle =\displaystyle \dfrac{-6.3\times100}{0.15\times100}Multiply both parts by 100
\displaystyle =\displaystyle \dfrac{-630}{15}Evaluate the multiplication
\displaystyle =\displaystyle -42Evaluate
Idea summary

  • If you have adjacent positive (plus) and negative (minus) signs, this will become a minus sign.

  • If you have two adjacent negative (minus) signs, this will become an addition sign.

  • When adding two numbers with different signs, we can use the number line to illustrate the process.

  • When multiplying and dividing, if one of your numbers is negative and the other is positive, your answer will be negative.

  • When multiplying and dividing, if both numbers are negative, your answer will be positive.

Outcomes

7.NS.A.1

Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

7.NS.A.1.D

Apply properties of operations as strategies to add and subtract rational numbers.

7.NS.A.2

Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

7.NS.A.2.C

Apply properties of operations as strategies to multiply and divide rational numbers.

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