5. Applications of Decimals

Why Are Decimals Important?

Decimals allow us to describe quantities that fall between whole numbers.

A distance might be:

4.7 km

A mass might be:

2.35 kg

An item might cost:

$8.99

A temperature might be:

21.6°C

Decimals are especially useful when quantities need to be measured or recorded with greater precision than whole numbers provide.

They appear throughout everyday life, science, business, engineering, technology, and statistics.

---

Decimals and Place Value

Understanding decimal applications begins with place value.

Consider:

24.583

This means:

• 2 tens
• 4 ones
• 5 tenths
• 8 hundredths
• 3 thousandths

The decimal point separates the whole-number part from the fractional part.

The value of each digit depends on its position.

---

Decimals and Money

Money is one of the most familiar applications of decimals.

For example:

$12.75

means:

12 dollars and 75 cents

because:

$1 = 100 cents

Therefore:

$0.75 = 75 cents

The hundredths place is particularly important when working with most currencies divided into 100 smaller units.

---

Reading Money Correctly

Consider:

$8.05

This means:

8 dollars and 5 cents

It does not mean:

8 dollars and 50 cents

The zero is an important placeholder.

Similarly:

$8.50

means:

8 dollars and 50 cents

---

Adding Money

Suppose you buy:

• a sandwich for $6.75
• a drink for $2.40
• fruit for $1.85

Find the total cost.

Align the decimal points:

$6.75 + $2.40 + $1.85 = $11.00

Therefore:

Total cost = $11.00

---

Calculating Change

Suppose your purchase costs:

$17.65

and you pay:

$20.00

Calculate:

$20.00 − $17.65 = $2.35

Therefore:

Change = $2.35

A quick estimate also confirms the answer:

$20 − $18 ≈ $2

So $2.35 is reasonable.

---

Comparing Prices

Suppose two stores sell the same item.

Store A:

$14.95

Store B:

$13.89

Compare the prices:

13.89 < 14.95

So Store B has the lower listed price.

The difference is:

$14.95 − $13.89 = $1.06

---

Buying Several of the Same Item

Suppose one notebook costs:

$3.75

You buy:

6 notebooks

Calculate:

6 × $3.75 = $22.50

Therefore:

Total cost = $22.50

This combines decimal multiplication with a practical money calculation.

---

Finding Unit Price

A package containing 8 drinks costs:

$14.40

Cost per drink:

$14.40 ÷ 8 = $1.80

Therefore:

Unit price = $1.80 per drink

Unit prices can help compare products sold in different package sizes.

---

Decimals and Discounts

Suppose an item normally costs:

$60.00

It is discounted by:

$12.50

Sale price:

$60.00 − $12.50 = $47.50

Therefore:

Sale price = $47.50

Decimals are commonly used when calculating discounts, taxes, tips, and final prices.

---

Budgeting with Decimals

Suppose a weekly budget includes:

• Food: $62.75
• Transport: $24.50
• Entertainment: $18.25
• Other expenses: $12.80

Total spending:

$62.75 + $24.50 + $18.25 + $12.80

= $118.30

If the budget is:

$150.00

money remaining:

$150.00 − $118.30 = $31.70

---

Estimating Money Calculations

Estimation is useful when shopping.

Suppose your basket contains items costing:

$8.95

$12.20

$5.75

$16.10

Round:

$9 + $12 + $6 + $16

Estimated total:

$43

Exact total:

$43.00

Estimation helps determine whether you have enough money and provides a check for exact calculations.

---

Decimals and Measurement

Measurements often fall between whole-number values.

Examples include:

1.72 m

4.35 kg

2.6 L

12.4 cm

8.75 km

Decimals allow measurements to be recorded more precisely.

---

Measuring Length

Suppose a table is:

1.85 m

long.

This means:

1 metre + 0.85 metre

Since:

0.85 m = 85 cm

the length could also be described as:

1 m 85 cm

---

Adding Measurements

Suppose two boards have lengths:

2.45 m

and:

1.78 m

Total length:

2.45 + 1.78 = 4.23

Therefore:

Total length = 4.23 m

Always include the unit in the final answer.

---

Finding a Difference in Measurements

One plant is:

1.42 m

tall.

Another is:

0.87 m

tall.

Difference:

1.42 − 0.87 = 0.55

Therefore:

The taller plant is 0.55 m taller.

---

Mass and Decimals

A package might have a mass of:

2.75 kg

If 4 identical packages are used:

2.75 × 4 = 11.00

Therefore:

Total mass = 11 kg

Decimals allow masses between whole kilograms to be represented accurately.

---

Volume and Capacity

Suppose a container holds:

1.5 L

of water.

Four containers hold:

1.5 × 4 = 6.0 L

If:

2.35 L

is used:

6.00 − 2.35 = 3.65 L

remain.

---

Measuring Area

Decimals are also common in area calculations.

A rectangular room measures:

5.4 m × 3.8 m

Area:

5.4 × 3.8 = 20.52

Therefore:

Area = 20.52 m²

Notice that the unit for area is:

square metres (m²)

---

Measuring Perimeter

For the same room:

Length:

5.4 m

Width:

3.8 m

Perimeter:

2(5.4 + 3.8)

First:

5.4 + 3.8 = 9.2

Then:

2 × 9.2 = 18.4

Therefore:

Perimeter = 18.4 m

---

Decimals and Temperature

Temperature measurements often contain decimals.

For example:

21.5°C

37.2°C

−4.8°C

Suppose the temperature rises from:

18.6°C

to:

23.4°C

Increase:

23.4 − 18.6 = 4.8°C

---

Decimals and Distance

Suppose a cyclist travels:

12.75 km

in the morning and:

8.6 km

in the afternoon.

Total distance:

12.75 + 8.60 = 21.35 km

If the goal was:

25 km

distance remaining:

25.00 − 21.35 = 3.65 km

---

Decimals and Time

Time is sometimes expressed using decimals.

For example:

1.5 hours

means:

1 hour + 0.5 hour

Since:

0.5 × 60 = 30 minutes

we have:

1.5 hours = 1 hour 30 minutes

However, decimal time must be interpreted carefully because:

1 hour = 60 minutes

not 100 minutes.

---

Another Time Example

Convert:

2.25 hours

to hours and minutes.

Whole-number part:

2 hours

Decimal part:

0.25 hour

Calculate:

0.25 × 60 = 15 minutes

Therefore:

2.25 hours = 2 hours 15 minutes

---

Decimals in Data

Decimals are frequently used when collecting and reporting data.

For example, a science experiment might record:

Decimal notation allows small differences between measurements to be recorded.

---

Reading Decimal Data

When interpreting data, ask questions such as:

• What does each decimal represent?
• What units are being used?
• Which value is greatest?
• Which value is least?
• What is the difference between values?
• Is there a trend?
• How precise are the measurements?
• Are any values unusual?

---

Finding the Range

Consider the data:

4.2, 4.7, 4.4, 4.9, 4.3

Greatest value:

4.9

Least value:

4.2

Range:

4.9 − 4.2 = 0.7

Therefore:

Range = 0.7

---

Finding a Mean from Decimal Data

Suppose four measurements are:

2.4, 2.8, 2.6, 3.0

Add:

2.4 + 2.8 + 2.6 + 3.0 = 10.8

Divide by 4:

10.8 ÷ 4 = 2.7

Therefore:

Mean = 2.7

---

Interpreting Decimal Graphs

Graphs often use decimal scales.

For example, an axis might be labelled:

0, 0.5, 1.0, 1.5, 2.0

To interpret the graph correctly, you must understand the interval between values.

Always check:

• axis labels
• units
• scale intervals
• decimal place values

before interpreting data.

---

Decimals in Sports

Decimals are used to record:

• race times
• distances
• averages
• percentages
• speeds
• scores in some sports

Suppose three runners record:

12.84 s

12.71 s

12.93 s

The fastest runner has the smallest time.

Therefore:

12.71 s

is the fastest time.

---

Decimals in Science

Scientific measurements often require decimals.

A laboratory might record:

• mass = 24.68 g
• volume = 12.5 mL
• temperature = 22.4°C
• distance = 1.275 m
• time = 4.82 s

Decimals allow scientists to communicate measurements with appropriate precision.

---

Decimals in Engineering and Construction

A design might specify:

2.75 m

14.6 mm

8.25 kg

3.125 cm

Small differences can matter when parts need to fit together accurately.

For this reason, correct decimal notation and measurement are important in:

• construction
• manufacturing
• engineering
• architecture

---

Decimals in Maps and Navigation

Distances on maps and navigation systems frequently use decimals.

For example:

3.8 km

12.45 km

0.75 km

A journey consisting of:

3.8 km + 12.45 km + 0.75 km

has a total distance of:

17.00 km

---

Multi-Step Problem 1: Shopping

A customer buys:

• 3 books at $7.85 each
• 2 pens at $1.65 each

Books:

3 × $7.85 = $23.55

Pens:

2 × $1.65 = $3.30

Total:

$23.55 + $3.30 = $26.85

If the customer pays:

$30.00

Change:

$30.00 − $26.85 = $3.15

Therefore:

The customer receives $3.15 change.

---

Multi-Step Problem 2: Travel

A vehicle travels:

125.6 km

on Monday and:

98.75 km

on Tuesday.

Total:

125.60 + 98.75 = 224.35 km

If the vehicle used:

17.5 L

of fuel:

224.35 ÷ 17.5 ≈ 12.82

Therefore, the vehicle travelled approximately:

12.82 km per litre

---

Multi-Step Problem 3: Flooring

A rectangular room measures:

6.5 m × 4.2 m

Area:

6.5 × 4.2 = 27.3 m²

If each box of flooring covers:

2.1 m²

number of boxes:

27.3 ÷ 2.1 = 13

Therefore:

13 boxes

are required.

---

Multi-Step Problem 4: Recipe

A recipe requires:

0.75 kg

of flour for one batch.

For 4 batches:

0.75 × 4 = 3.00 kg

If you have:

3.5 kg

of flour:

3.50 − 3.00 = 0.50 kg

Therefore:

0.5 kg of flour remains.

---

Multi-Step Problem 5: Data Analysis

A student records these plant heights:

14.2 cm

15.1 cm

14.8 cm

15.5 cm

Total:

14.2 + 15.1 + 14.8 + 15.5 = 59.6 cm

Mean:

59.6 ÷ 4 = 14.9 cm

Therefore:

Mean height = 14.9 cm

---

Multi-Step Problem 6: Water Use

A tank contains:

48.5 L

of water.

During the day:

12.75 L

is used.

Then:

8.6 L

is added.

Calculate:

48.50 − 12.75 = 35.75

Then:

35.75 + 8.60 = 44.35

Therefore:

44.35 L of water remains in the tank.

---

Multi-Step Problem 7: Comparing Unit Prices

Package A contains:

6 items for $10.50

Unit price:

$10.50 ÷ 6 = $1.75

Package B contains:

8 items for $13.20

Unit price:

$13.20 ÷ 8 = $1.65

Package B has the lower unit price:

$1.65 per item

This demonstrates why unit price can be more informative than simply comparing total package prices.

---

Multi-Step Problem 8: Fuel Cost

A vehicle needs:

32.5 L

of fuel.

Fuel costs:

$2.14 per litre

Total cost:

32.5 × $2.14 = $69.55

Estimate:

33 × $2 ≈ $66

So:

$69.55

is reasonable.

---

Choosing the Correct Operation

Real-world problems do not always tell you directly which operation to use.

Look at what is happening in the situation.

Use addition when quantities are being combined.

Use subtraction when finding:

• a difference
• what remains
• change

Use multiplication for:

• equal groups
• repeated quantities
• area
• cost per item × number of items

Use division for:

• equal sharing
• unit rates
• number of groups
• amount per group

---

Multi-Step Problems Require Planning

Consider:

"A store has 18.5 kg of rice. It packs the rice equally into 5 bags. Two bags are sold. How much rice remains?"

First find the amount per bag:

18.5 ÷ 5 = 3.7 kg

Two bags contain:

2 × 3.7 = 7.4 kg

Remaining:

18.5 − 7.4 = 11.1 kg

Therefore:

11.1 kg of rice remains.

The challenge is not only performing decimal operations but deciding which operations are needed and in what order.

---

Estimating Real-World Answers

Before calculating exactly, estimate.

Suppose a meal costs:

$18.75

and a drink costs:

$4.85

Estimate:

$19 + $5 = $24

Exact:

$18.75 + $4.85 = $23.60

The exact answer is close to the estimate.

Therefore, it is reasonable.

---

Checking Reasonableness

Suppose a student calculates:

4.8 kg × 6 = 288 kg

Estimate:

5 × 6 = 30

The answer should be close to:

30 kg

not:

288 kg

The correct calculation is:

4.8 × 6 = 28.8 kg

Estimation helps identify the misplaced decimal point.

---

Communicating Decimal Solutions

A correct calculation is only part of a strong mathematical solution.

A clear solution should include:

• the calculation
• correct decimal notation
• appropriate units
• suitable precision
• a sentence answering the question when needed

For example:

Weak answer:

14.75

Better answer:

The total distance travelled was 14.75 km.

---

Always Include Units

Consider:

8.5 × 3.2 = 27.2

Without context, this is simply a number.

But if the values are the dimensions of a rectangle:

8.5 m × 3.2 m = 27.2 m²

The unit tells us what the answer represents.

---

Choosing Appropriate Decimal Notation

Different situations use different conventions.

Money is normally written to two decimal places:

$7.50

rather than:

$7.5

For a measurement, the number of decimal places may indicate precision:

2.4 cm

and:

2.40 cm

have the same numerical value, but they may communicate different measurement precision.

Context matters.

---

Rounding Practical Answers

Sometimes a calculation produces more decimal places than are useful.

Suppose:

10 ÷ 3 = 3.333333...

If the answer represents a measurement, we might report:

3.33 m

to the nearest hundredth.

If it represents people or containers, however, we cannot simply report:

3.33 people

The context determines how the result should be interpreted.

---

Whole Items and Decimal Results

Suppose:

47 students

must travel in vans holding:

8 students each

Calculate:

47 ÷ 8 = 5.875

But:

5.875 vans

does not make practical sense.

Five vans are not enough.

Therefore:

6 vans

are required.

A calculator result must always be interpreted in context.

---

Decimals and Calculators

Calculators are useful for complex decimal calculations, but mathematical understanding is still necessary.

Before using a calculator:

• identify the correct operation
• estimate the answer

After using it:

• check the decimal placement
• compare with the estimate
• interpret the result
• round appropriately
• include units

---

Worked Example 1: Money

A shirt costs:

$24.95

and trousers cost:

$38.75

Total:

$24.95 + $38.75 = $63.70

If you pay:

$70.00

change:

$70.00 − $63.70 = $6.30

---

Worked Example 2: Measurement

A piece of rope is:

12.8 m

long.

Three pieces measuring:

2.35 m

each are cut from it.

Length removed:

3 × 2.35 = 7.05 m

Remaining:

12.80 − 7.05 = 5.75 m

---

Worked Example 3: Data

Temperatures recorded during an experiment are:

22.4°C, 23.1°C, 22.8°C, 23.3°C

Mean:

(22.4 + 23.1 + 22.8 + 23.3) ÷ 4

= 91.6 ÷ 4

= 22.9°C

Therefore:

Mean temperature = 22.9°C

---

Worked Example 4: Unit Rate

A vehicle travels:

168.75 km

using:

12.5 L

of fuel.

Calculate:

168.75 ÷ 12.5 = 13.5

Therefore:

Fuel efficiency = 13.5 km/L

---

Worked Example 5: Area and Cost

A rectangular garden measures:

7.5 m × 4.8 m

Area:

7.5 × 4.8 = 36 m²

Grass seed costs:

$2.25 per m²

Cost:

36 × $2.25 = $81.00

Therefore:

The grass seed will cost $81.00.

---

Common Mistakes

Mistake 1: Ignoring units

Writing:

12.5

instead of:

12.5 kg

can make an answer unclear.

---

Mistake 2: Misaligning decimal points

When adding and subtracting, decimal points must be aligned.

---

Mistake 3: Misplacing the decimal point

Use estimation to check whether the answer has a sensible magnitude.

---

Mistake 4: Treating decimal hours like minutes

1.5 hours

does not mean:

1 hour 5 minutes

It means:

1 hour 30 minutes

---

Mistake 5: Reporting impossible decimal quantities

A calculation might produce:

4.6 buses

but the practical answer may need to be:

5 buses

---

Mistake 6: Using unnecessary precision

A shopping estimate usually does not need an answer such as:

$42.783649

Choose precision appropriate to the situation.

---

Mistake 7: Giving only a number

Communicate what the number means and include units where appropriate.

---

Error Analysis

A student calculates the total of:

$8.50 + $3.75

and writes:

$11.125

Estimate:

$9 + $4 ≈ $13

So $11.125 should immediately seem suspicious.

Correct calculation:

$8.50 + $3.75 = $12.25

The estimate helps reveal the error.

---

Another Error Analysis

A runner travels:

4.75 km

each day for:

5 days

A student calculates:

4.75 + 5 = 9.75 km

But the distance is repeated five times.

The correct operation is multiplication:

4.75 × 5 = 23.75 km

Therefore:

Total distance = 23.75 km

Choosing the correct operation is essential.

---

A Reliable Real-World Problem-Solving Strategy

Step 1: Understand the problem

Identify what is known and what must be found.

Step 2: Identify the units

Look for dollars, metres, litres, kilograms, seconds, and other units.

Step 3: Choose the operation

Decide whether addition, subtraction, multiplication, division, or several operations are required.

Step 4: Estimate

Predict approximately what the answer should be.

Step 5: Calculate

Perform the decimal operations accurately.

Step 6: Check

Compare the answer with the estimate.

Step 7: Interpret

Decide what the numerical result means in the situation.

Step 8: Communicate

State the final answer clearly using appropriate decimal notation and units.

---

Did You Know?

Decimals help us describe the real world because many quantities do not occur in exact whole-number amounts.

A runner does not always finish a race in exactly 12 seconds.

A bottle does not always contain exactly 2 litres.

A package does not always weigh exactly 5 kilograms.

Decimals allow us to represent these quantities with much greater precision.

This is why decimal notation is fundamental in measurement, science, finance, engineering, statistics, and technology.

---

Key Terms

• Decimal: Number that uses place value to represent whole quantities and parts of a whole.
• Decimal notation: Writing numbers using a decimal point.
• Place value: Value of a digit according to its position.
• Measurement: Process of assigning a numerical value and unit to a quantity.
• Unit: Standard quantity used for measurement.
• Data: Collected information or measurements.
• Unit price: Cost of one unit of an item.
• Unit rate: Comparison expressed for one unit of another quantity.
• Estimate: Approximate value used for prediction or checking.
• Mean: Sum of values divided by the number of values.
• Range: Difference between the greatest and least values.
• Precision: Level of detail used to express a value.
• Reasonableness: Whether a result makes sense in context.
• Multi-step problem: Problem requiring more than one mathematical operation.
• Interpretation: Explaining what a numerical result means in its context.

---

Key Relationships

Money:

$1 = 100 cents

Measurement:

1.5 m = 1 m + 0.5 m

Decimal time:

0.5 hour = 30 minutes

0.25 hour = 15 minutes

Area of a rectangle:

Area = length × width

Unit rate:

Unit rate = total quantity ÷ number of units

Mean:

Mean = sum of values ÷ number of values

Range:

Range = greatest value − least value

---

Key Takeaways

• Decimals are widely used to represent quantities between whole numbers.
• Money calculations frequently involve decimal addition, subtraction, multiplication, and division.
• Decimal notation distinguishes dollars from cents and other major and minor currency units.
• Unit prices can be calculated using decimal division.
• Decimals allow measurements to be recorded with greater precision.
• Length, mass, volume, temperature, area, distance, and time can all involve decimals.
• Decimal hours must be converted carefully because one hour contains 60 minutes.
• Scientific measurements commonly use decimal values.
• Decimal data can be compared, ordered, averaged, and analyzed.
• Graphs may contain decimal scales that must be interpreted carefully.
• Real-world decimal problems often require more than one operation.
• Choosing the correct operation is as important as performing the calculation accurately.
• Estimation helps predict and check decimal calculations.
• Calculator answers should be checked for reasonableness.
• Numerical answers must be interpreted according to their context.
• Some practical situations require rounding to whole quantities.
• Units should be included when communicating measurements and practical answers.
• Appropriate precision depends on the situation.
• Money is usually communicated using two decimal places.
• Clear mathematical communication includes calculations, decimal notation, units, and a final statement explaining the result.
• Decimals are essential tools for working with money, measurement, data, science, business, engineering, and everyday problem-solving.