Topics
7. Decimal operations
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AustraliaVIC
Level 6

7.11 Value of decimal expressions

Lesson
Practice
Lesson

Are you ready?

In earlier lessons we have learned the  order of operations.  Have a look at the following problem and be sure to use the correct order of operations.

Examples

Example 1

Find the value of 12 \times \left(5+6\right) -35

Worked Solution
Create a strategy

Follow the order of operations by doing the brackets first, then the multiplication, then the subtraction.

Apply the idea
\displaystyle 12 \times \left(5+6\right) -35\displaystyle =\displaystyle 12 \times \left(11\right) -35Add the numbers in the brackets
\displaystyle =\displaystyle 132-35Multiply 12 and 11
\displaystyle =\displaystyle 97Subtract 35
Idea summary

The order of operations:

  1. Solve anything inside grouping symbols (brackets).

  2. Solve any multiplication or division, including powers, working from left to right.

  3. Solve any addition or subtraction, working from left to right.

Order of operations with decimals

This video shows some examples of working out the value of an expression that involve decimals and 2 operations.

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Examples

Example 2

Find the value of 8.5+0.65\times10.

Worked Solution
Create a strategy

Follow the order of operations by doing the multiplication first and then the addition.

Apply the idea

To find 0.65\times 10 we can multiply the numbers without the decimal point.

\displaystyle 65\times 10\displaystyle =\displaystyle 650Add a 0

Since 0.65\times 10 has two decimal places we insert the decimal point into our answer so that there are two decimal places: \begin{aligned} 0.65\times 10&=6.50 \\ &=6.5 \end{aligned}

Now we can add our answer to 8.5 using a vertical algorithm: \begin{array}{c} &&&{}^18&.&5 \\ &+&&6&.&5 \\ \hline &&1&5&.&0 \end{array} So we have:

\displaystyle 8.5+0.65\times10\displaystyle =\displaystyle 15
Idea summary

The order of operations is the same for decimals as it is for whole numbers.

Order of operation with decimals: addition, subtraction and multiplication

This video shows some examples of working out the value of an expression that involve decimals and 3 operations.

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Examples

Example 3

Find the value of 0.6\times8-0.6\times4, giving your answer as a decimal.

Worked Solution
Create a strategy

Follow the order of operations and do the multiplications from left to right, and then the subtraction.

Apply the idea

0.6 \times 8 has one decimal place. So first we can multiply the whole numbers to get 6\times 8 = 48 and now we can add a decimal point so that there is one decimal place to get: 0.6 \times 8=4.8

0.6 \times 4 also has one decimal place. So we can multiply the whole numbers to get 6\times 4 = 24 and now we can add a decimal point so that there is one decimal place to get: 0.6 \times 4=2.4

Now we can subtract our results using a vertical algorithm: \begin{array}{c} &&4&.&8 \\ &-&2&.&4 \\ \hline &&2&.&4 \\ \hline \end{array} So we have:

\displaystyle 0.6\times8-0.6\times4\displaystyle =\displaystyle 2.4
Idea summary

When solving expressions with more than one operation, we have to follow the order of operations, this includes expressions with decimals.

Decimal calculations with a calculator

This video shows how to enter calculations with decimals on a calculator.

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Examples

Example 4

Consider 11.59\div4.44

a

Estimate the answer by rounding both values to the nearest whole number.

Worked Solution
Create a strategy

Round both numbers to the nearest whole number and then divide them.

Apply the idea

11.59 to the neares whole number is 12 and 4.44 to the neares whole number is 4.

\displaystyle \text{Estimate}\displaystyle =\displaystyle 12\div4Divide the rounded numbers
\displaystyle =\displaystyle 3
b

Use a calculator to evaluate 11.59\div4.44 and write the answer below. Give your answer to two decimal places.

Worked Solution
Create a strategy

Use a calculator and then round the answer.

Apply the idea

You should get the answer 2.61036036. You may get more or less decimal places depending on your calculator.

To round this to 2 decimal places, we need to look at the third decimal place which is 0. Since this is less than 5 we round down to get: 11.59\div4.44 = 2.61

c

What is the difference between the estimate and the actual answer?

Worked Solution
Create a strategy

Subtract the calculated value from part (b) from the estimated value from part (a).

Apply the idea
\displaystyle \text{Difference}\displaystyle =\displaystyle 3-2.61Subtract 2.61 from 3
\displaystyle =\displaystyle 0.39Use your calculator
Idea summary

We can use calculators to make calculations with decimals easier. The calculator also uses the order of operations.

Outcomes

VCMNA214

Add and subtract decimals, with and without digital technologies, and use estimation and rounding to check the reasonableness of answers

VCMNA215

Multiply decimals by whole numbers and perform divisions by non-zero whole numbers where the results are terminating decimals, with and without digital technologies

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