Mathematics in the Real World

Fractions, decimals, ratios, and percentages are closely connected.

In everyday situations, we often need to move between these different representations.

For example:

3/4 = 0.75 = 75%

and the ratio:

3 : 4

can be represented by the fraction:

3/4

when comparing the first quantity directly with the second.

However, the meaning of a number depends on its context.

A fraction might describe part of a pizza.

A decimal might describe a price.

A ratio might describe ingredients in a recipe.

A percentage might describe a discount.

The mathematics may be similar, but the interpretation changes.

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Connecting Fractions, Decimals, and Percentages

Fractions, decimals, and percentages can often represent the same quantity.

Examples:

1/2 = 0.5 = 50%

1/4 = 0.25 = 25%

3/4 = 0.75 = 75%

1/5 = 0.2 = 20%

3/5 = 0.6 = 60%

Choosing the most useful form can make a problem much easier.

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Choosing the Best Representation

Suppose a survey finds that:

18 out of 30 people

prefer option A.

We could represent this as:

Fraction:

18/30 = 3/5

Decimal:

0.6

Percentage:

60%

Ratio of A to total:

18 : 30 = 3 : 5

For communicating survey results, 60% may be especially easy to understand.

For calculations, 0.6 may be convenient.

For exact comparison, 3/5 may be useful.

There is no single representation that is always best.

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Ratios and Fractions Are Related but Different

Suppose a class contains:

12 students wearing glasses

and:

18 students not wearing glasses

Ratio:

glasses : no glasses = 12 : 18

Simplify:

2 : 3

But the fraction of the whole class wearing glasses is:

12/(12 + 18)

= 12/30

= 2/5

So:

2 : 3

does not mean:

2/3 of the class

The total number of ratio parts is:

2 + 3 = 5

Therefore, the first group represents:

2/5 of the total

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Mathematics as a Problem-Solving Toolkit

Real-world problems rarely tell us which operation to use.

A problem may require:

• adding fractions
• multiplying decimals
• finding a percentage
• simplifying a ratio
• calculating a unit rate
• setting up a proportion
• estimating
• reading a graph
• comparing several options
• performing several calculations in sequence

The challenge is often not performing the calculation.

The challenge is deciding:

What mathematics should I use?

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A General Problem-Solving Process

A reliable strategy is:

1. Understand the problem.

What information is given?

What are you trying to find?

2. Identify the mathematical relationships.

Is this about fractions, percentages, ratios, rates, proportions, or several of these?

3. Choose a useful representation.

Would a table, diagram, equation, ratio, decimal, or percentage help?

4. Calculate carefully.

5. Check units.

6. Estimate your expected answer.

7. Check whether the exact answer is reasonable.

8. Interpret the result.

9. Explain your reasoning.

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Project Situation 1: Planning a Meal

Suppose you are organizing a meal for:

12 people

A recipe serves:

4 people

and requires:

• 300 g pasta
• 200 g vegetables
• 150 g sauce

First determine the scale factor:

12 ÷ 4 = 3

Multiply every ingredient by 3.

Pasta:

300 × 3 = 900 g

Vegetables:

200 × 3 = 600 g

Sauce:

150 × 3 = 450 g

The recipe has been scaled proportionally.

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Adding a Budget

Suppose the ingredients cost:

Pasta = $6.50

Vegetables = $8.20

Sauce = $7.80

Additional ingredients = $5.50

Total:

$6.50 + $8.20 + $7.80 + $5.50

= $28.00

Cost per person:

$28 ÷ 12 ≈ $2.33

Now the problem combines:

• ratios
• proportional reasoning
• decimals
• unit rates
• money

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Project Situation 2: Shopping and Discounts

Suppose you need to buy shoes and have a budget of:

$100

Store A:

Original price = $120

Discount = 25%

Store B:

Original price = $105

Discount = 15%

Store A:

$120 × 0.75 = $90

Store B:

$105 × 0.85 = $89.25

The final prices are:

$90.00

and:

$89.25

Difference:

$90 − $89.25 = $0.75

Both fit within the $100 budget.

The numerical comparison shows that the prices are very close, so other relevant factors such as quality, fit, durability, or return conditions could matter more than the small price difference.

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Project Situation 3: Planning a Trip

Suppose a map uses the scale:

1 cm : 15 km

A route measures:

8.4 cm

on the map.

Actual distance:

8.4 × 15 = 126 km

If average speed is:

70 km/h

estimated travel time:

time = distance / speed

126 ÷ 70 = 1.8 hours

Convert:

0.8 hour × 60 = 48 minutes

Estimated travel time:

1 hour 48 minutes

The problem combines:

• ratios
• scale
• decimals
• rates
• unit conversion

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Adding a Fuel Calculation

Suppose a vehicle travels:

14 km/L

Fuel required:

126 ÷ 14 = 9 L

If fuel costs:

$2.10/L

estimated fuel cost:

9 × $2.10 = $18.90

One journey now involves several mathematical ideas.

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Project Situation 4: Comparing Products

Suppose a supermarket offers:

Package A:

750 g for $5.40

Package B:

1.2 kg for $7.80

Package C:

500 g for $3.60

Convert all quantities to kilograms.

A:

750 g = 0.75 kg

Unit price:

$5.40 ÷ 0.75 = $7.20/kg

B:

$7.80 ÷ 1.2 = $6.50/kg

C:

500 g = 0.5 kg

$3.60 ÷ 0.5 = $7.20/kg

The unit prices are:

A = $7.20/kg

B = $6.50/kg

C = $7.20/kg

Unit price provides one useful measure for comparing the products, while total spending and the amount actually needed may also matter.

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Project Situation 5: Mixing a Drink

A sports drink uses:

concentrate : water = 1 : 4

You need:

15 L

altogether.

Total ratio parts:

1 + 4 = 5

One part:

15 ÷ 5 = 3 L

Therefore:

Concentrate:

1 × 3 = 3 L

Water:

4 × 3 = 12 L

Fraction concentrate:

3/15 = 1/5

Percentage concentrate:

1/5 × 100% = 20%

The same mixture can therefore be described as:

1 : 4 concentrate to water

or:

20% concentrate

when describing concentrate as a fraction of the entire mixture.

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Project Situation 6: Analyzing Sports Data

Suppose two players have the following results:

Player A:

42 successful shots from 60 attempts

Player B:

52 successful shots from 80 attempts

Player A:

42/60 = 0.70 = 70%

Player B:

52/80 = 0.65 = 65%

Looking only at successful shots gives:

52 > 42

But comparing success rates gives:

65% versus 70%

Different statistics answer different questions.

This is an important part of data interpretation.

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Project Situation 7: Probability and Prediction

Suppose historical data show that a particular event occurred:

18 times in 60 trials

Experimental probability:

18/60

Simplify:

3/10

Decimal:

0.3

Percentage:

30%

If similar conditions continue for another:

200 trials

expected frequency:

0.30 × 200 = 60

Prediction:

about 60 occurrences

Actual results may differ because probability-based predictions are not guarantees.

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Project Situation 8: Savings

Suppose:

Principal = $2,000

Simple interest rate = 4%

Time = 3 years

Use:

I = Prt

I = 2000 × 0.04 × 3

I = $240

Final amount:

$2,000 + $240 = $2,240

This problem combines:

• decimals
• percentages
• multiplication
• financial mathematics

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Project Situation 9: Planning an Event

Suppose you have a budget of:

$600

for an event.

Expected expenses:

Venue = $180

Food = $240

Decorations = $60

Equipment = $75

Total:

$180 + $240 + $60 + $75

= $555

Money remaining:

$600 − $555 = $45

Percentage of the budget spent:

555/600 × 100%

= 92.5%

Percentage remaining:

7.5%

This gives us more information than simply knowing that $45 remains.

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Interpreting Tables

Real-world information is frequently presented in tables.

Consider:

A strong analysis might involve:

• calculating each sale price
• calculating each unit price
• comparing quantities
• considering the amount needed
• checking a budget constraint

A table gives us information.

Mathematical reasoning turns the information into evidence.

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Interpreting Graphs

Graphs can show:

• prices
• spending
• percentages
• survey results
• probability
• changes over time
• comparisons between groups

When reading a graph, ask:

What does the graph measure?

What units are being used?

What does each axis represent?

What is the scale?

What patterns are visible?

Are we looking at totals, rates, or percentages?

Does the graph support the conclusion being made?

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Watch for Misleading Graphs

Graphs can sometimes create misleading impressions.

For example, suppose a price changes from:

$100 to $105

That is an increase of:

$5

or:

5%

If a graph's vertical axis starts at $99 rather than $0, the visual difference between the bars could appear extremely large.

Always examine:

• axis starting points
• intervals
• labels
• units
• missing data
• time periods

Do not rely only on how dramatic a graph looks.

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Absolute vs Relative Change

Suppose:

Product A increases from:

$20 to $30

Increase:

$10

Percentage increase:

10/20 × 100% = 50%

Product B increases from:

$100 to $120

Increase:

$20

Percentage increase:

20/100 × 100% = 20%

Product B increased by more dollars.

Product A increased by a larger percentage.

Both statements are correct.

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Estimation Before Calculation

Strong problem solvers often estimate before calculating.

Suppose a product costs:

$198

and has:

21% off

We can estimate:

20% of $200:

≈ $40

So we expect the sale price to be roughly:

$160

Exact calculation:

$198 × 0.79 = $156.42

The result is reasonably close to our estimate.

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Reasonableness

A mathematically correct-looking calculation can still produce an unreasonable answer if the wrong operation was chosen.

Suppose:

A $100 product has 20% off.

If your answer says:

Final price = $120

something is wrong.

A discount should make the price decrease.

Ask:

Does my answer make sense in the situation?

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Units Matter

Suppose:

A car travels:

180 km in 3 hours

Rate:

180 ÷ 3 = 60

But:

60 what?

The correct answer is:

60 km/h

Units give numbers meaning.

Real-world calculations may involve:

• dollars
• kilograms
• grams
• litres
• kilometres
• metres
• hours
• minutes
• dollars per kilogram
• kilometres per hour
• percentages

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Convert Units Before Comparing

Suppose:

Package A:

500 g for $4

Package B:

1.5 kg for $10.50

We should compare using the same quantity.

Package A:

500 g = 0.5 kg

$4 ÷ 0.5 = $8/kg

Package B:

$10.50 ÷ 1.5 = $7/kg

Unit conversion makes the comparison meaningful.

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Multi-Step Problems

Real-world problems frequently require several steps.

For example:

A jacket costs:

$160

It is:

25% off

and you have a:

$130 budget

Step 1: Find the discount.

$160 × 0.25 = $40

Step 2: Find the sale price.

$160 − $40 = $120

Step 3: Compare with the budget.

$130 − $120 = $10

The jacket fits within the stated budget with:

$10 remaining

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Communicating Mathematical Thinking

A strong mathematical solution should show more than an answer.

It should communicate:

What information was used?

What strategy was chosen?

What calculations were performed?

What does the result mean?

For example:

Instead of writing only:

$72

write:

20% of $90 = $18

$90 − $18 = $72

Therefore, the final price after the 20% discount is $72.

The second response communicates the reasoning clearly.

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Explain Why You Chose a Strategy

Suppose you compare two packages.

A strong explanation might be:

"I calculated the unit price because the packages contain different quantities. Converting both prices to cost per kilogram allows them to be compared using the same unit."

This demonstrates mathematical reasoning rather than simply calculator use.

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Use Evidence in Conclusions

Avoid conclusions such as:

"A is better."

Instead, explain the numerical evidence:

"Option A costs $1.20 less per kilogram and would cost $6 less for the quantity required."

The numbers support the conclusion.

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Real-World Project 1: Smart Shopper Challenge

Scenario

You have:

$150

to purchase several items.

Your task is to compare different stores, package sizes, and discounts.

Your Project

Choose at least 5 products.

For each product, record:

• original price
• quantity
• discount, if any
• sale price
• unit price

Then compare at least two options for each product.

Your final budget must remain within:

$150

Mathematics to Include

Your project should demonstrate:

• decimals
• percentages
• discounts
• unit rates
• addition
• budgeting
• comparisons

Final Analysis

Explain:

• which options you selected
• how much you spent
• how much remained
• which calculations influenced your decisions
• whether the largest discount always produced the lowest price

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Real-World Project 2: Plan a Meal

Scenario

Plan a meal for:

8 people

with a maximum budget of:

$80

Your Project

Choose a recipe originally designed for a different number of servings.

Calculate:

• scale factor
• required amount of each ingredient
• total quantity
• cost of each ingredient
• total cost
• cost per person
• percentage of the budget used

Challenge

Find one ingredient sold in two different package sizes.

Calculate the unit price and determine how the options compare for the amount required.

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Real-World Project 3: Design a Bedroom

Scenario

Create a scale drawing of a bedroom.

Suppose the actual room is:

4.8 m × 3.6 m

Choose an appropriate scale, such as:

1 cm : 0.4 m

Your Project

Include:

• room dimensions
• bed
• desk
• storage
• doorway
• other furniture

Every object must use the same scale.

Calculate:

• actual dimensions
• drawing dimensions
• scale factor
• floor area
• fraction or percentage of floor area occupied by selected furniture

Explain how ratios and scale factors allowed you to represent the room accurately.

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Real-World Project 4: Sports Analyst

Scenario

Compare the performance of several athletes or teams using a provided or teacher-approved dataset.

Calculate

For each competitor, calculate appropriate measures such as:

• successes per attempt
• points per game
• goals per game
• win percentage
• relevant ratios
• percentages

Present Your Data

Create:

• a data table
• at least one appropriate graph
• calculations
• a written interpretation

Your conclusion should explain what each statistic shows and any limitations of comparing performance using only those numbers.

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Real-World Project 5: Probability Investigation

Question

Does experimental probability become closer to theoretical probability as the number of trials increases?

Choose an experiment such as:

• tossing a coin
• rolling a die
• spinning a spinner
• selecting colored counters

Procedure

Perform:

10 trials

then:

50 trials

then:

100 trials

Record your cumulative results.

Calculate experimental probabilities as:

• fractions
• decimals
• percentages

Compare these with the theoretical probability.

Analysis

Explain:

• how the experimental probability changed
• whether it approached the theoretical probability
• why individual results were not exactly predictable
• why larger samples may provide more stable estimates

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Real-World Project 6: Personal Budget Challenge

Scenario

Imagine a hypothetical monthly income of:

$2,500

Create a realistic monthly budget.

Possible categories include:

• housing
• food
• transportation
• communication
• entertainment
• savings
• clothing
• miscellaneous expenses

Requirements

Calculate:

• total expenses
• money remaining
• percentage spent in each major category
• percentage saved
• one possible unexpected expense
• how the budget would change if income decreased by 10%

Explain the mathematical reasoning behind your adjustments.

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Real-World Project 7: Compare Savings Options

Suppose you have:

$5,000

to place in a hypothetical savings product for:

3 years

Option A:

3% simple interest

Option B:

4% simple interest with a $100 total fee

Option C:

2.5% simple interest plus a $150 end-of-term bonus

Calculate the final amount for each under the stated assumptions.

Your analysis should include:

• interest earned
• fees or bonuses
• final amounts
• differences between options
• explanation of which numerical factors most strongly affect the comparison

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Real-World Project 8: Create Your Own Business

Imagine you are starting a small business.

Examples could include:

• selling baked goods
• designing T-shirts
• tutoring
• making crafts
• selling plants
• running a small school event

Calculate

Your project should include:

Startup costs

Cost per item

Selling price

Expected number sold

Revenue

Profit or loss

Profit percentage

Discount promotion

For example:

Cost to produce one item:

$6

Selling price:

$10

Profit per item:

$10 − $6 = $4

If 50 are sold:

Revenue:

50 × $10 = $500

Cost:

50 × $6 = $300

Profit:

$500 − $300 = $200

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Real-World Project 9: Analyze an Advertisement

Find or use a teacher-provided advertisement containing numerical claims.

Examples:

30% OFF

SAVE $50

25% EXTRA

2 FOR $10

BUY 2, GET 1 FREE

Analyze:

• original price
• advertised saving
• final price
• percentage saving
• unit price
• conditions of the offer

Then explain what the advertisement communicates clearly and what additional information would be useful for evaluating the offer.

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Real-World Project 10: Mathematics in My Life

Choose one situation where you regularly encounter mathematics.

Examples include:

• cooking
• shopping
• gaming
• sports
• travel
• fitness
• saving money
• photography
• building
• art
• music

Identify at least three mathematical ideas involved.

For example, cooking might involve:

• fractions
• ratios
• proportions
• percentages
• unit conversions

Create a short report or presentation demonstrating how the mathematics is used.

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A Strong Project Should Include

A high-quality mathematical project should contain:

A clear problem or question

Explain what you are investigating.

Relevant data

Provide the numerical information needed.

Correct mathematics

Show calculations.

Appropriate representations

Use tables, graphs, diagrams, ratios, fractions, decimals, or percentages where useful.

Interpretation

Explain what the calculations mean.

Evaluation

Discuss what the numbers reveal and any limitations.

Communication

Present your reasoning clearly enough that another person could follow it.

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Checking Data Quality

Real-world mathematics depends on the quality of the information being used.

Before using data, ask:

• Where did the numbers come from?
• Are the units clear?
• Is the information current enough for the task?
• Are the quantities being compared fairly?
• Is the sample large enough to support the conclusion?
• Are important costs or conditions missing?
• Does the graph accurately represent the data?

Correct calculations cannot fix poor or inappropriate data.

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Using Technology

Technology can make real-world mathematical projects easier.

Useful tools can include:

• calculators
• spreadsheets
• graphing software
• digital maps
• online price lists
• data collection tools

A spreadsheet can automatically calculate:

• totals
• averages
• percentages
• differences
• unit prices
• budget balances

It can also turn data into graphs.

However, technology should support mathematical reasoning rather than replace it.

You should still understand what the calculation represents.

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Worked Challenge 1: Shopping, Discounts, and Budgeting

You have:

$250

You want to buy:

Shoes: $120 with 25% off

Jacket: $90 with 20% off

Bag: $65 with 15% off

Shoes:

120 × 0.75 = $90

Jacket:

90 × 0.80 = $72

Bag:

65 × 0.85 = $55.25

Total:

$90 + $72 + $55.25 = $217.25

Money remaining:

$250 − $217.25 = $32.75

Percentage of budget spent:

217.25 / 250 × 100%

= 86.9%

This single problem combines decimals, percentages, addition, budgeting, and interpretation.

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Worked Challenge 2: Recipe, Ratio, and Cost

A recipe for 6 people requires:

Flour = 450 g

Milk = 600 mL

Fruit = 300 g

You need to serve:

15 people

Scale factor:

15/6 = 2.5

Flour:

450 × 2.5 = 1125 g

Milk:

600 × 2.5 = 1500 mL

Fruit:

300 × 2.5 = 750 g

If the total cost is:

$37.50

cost per person:

$37.50 ÷ 15 = $2.50

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Worked Challenge 3: Data and Prediction

A player makes:

54 successful attempts from 75

Success rate:

54/75 = 0.72 = 72%

If the player makes another:

50 attempts

a simple prediction based on the historical rate is:

0.72 × 50 = 36

Therefore:

about 36 successes

might be expected under similar conditions.

This is a probability-based prediction rather than a guarantee.

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Worked Challenge 4: Combining Several Representations

A school club has:

48 members

Of these:

18 are new members

Fraction:

18/48 = 3/8

Decimal:

3 ÷ 8 = 0.375

Percentage:

37.5%

Ratio:

New : Returning

Returning members:

48 − 18 = 30

Therefore:

18 : 30 = 3 : 5

This demonstrates an important distinction:

Fraction of all members who are new:

3/8

Ratio of new to returning:

3 : 5

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Common Mistakes

Mistake 1: Choosing an operation because of a keyword

Real problems require understanding the relationship, not simply spotting words such as "of" or "more."

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Mistake 2: Comparing quantities with different units

Convert them to common units first.

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Mistake 3: Confusing part-to-part and part-to-whole relationships

For:

2 : 3

the first quantity represents:

2/5 of the total

not 2/3.

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Mistake 4: Forgetting what 100% represents

A percentage must always refer to some whole or reference quantity.

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Mistake 5: Assuming the largest discount means the lowest price

Original prices may differ.

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Mistake 6: Assuming the largest package is automatically best value

Calculate the unit price and consider how much is actually needed.

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Mistake 7: Reporting calculator results without interpretation

Always explain what the result means.

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Mistake 8: Giving excessive decimal places

Money usually requires appropriate rounding to the nearest cent.

Other measurements should be rounded according to the context.

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Mistake 9: Treating predictions as guarantees

Probability provides expected outcomes, not certainty.

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Mistake 10: Trusting a graph without examining its scale

Graphs should be interpreted critically.

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Communicating a Final Conclusion

A strong conclusion should answer the original question and support the answer with mathematical evidence.

For example:

Weak conclusion:

"Option B is better."

Stronger mathematical conclusion:

"Option B has a unit price of $6.50/kg compared with $7.20/kg for Option A. For the 3 kg required, this corresponds to a difference of $2.10, so Option B has the lower cost for the required quantity."

The second conclusion explains the reasoning and provides evidence.

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Final Review: Choosing the Mathematics

When you see...

Part of a whole

Consider a fraction, decimal, or percentage.

Two quantities being compared

Consider a ratio.

Different quantities such as kilometres and hours

Consider a rate.

Cost for one item or unit

Calculate a unit rate.

Equivalent ratios

Use a proportion.

A price reduction

Calculate a percentage discount.

Growth over time

Consider percentage growth or interest.

Uncertain outcomes

Use probability.

Several options

Convert them to comparable forms and analyze the numerical evidence.

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Did You Know?

Much of the mathematics used in everyday decision-making is built from relatively simple ideas used together.

A single shopping trip might involve:

• decimals for prices
• percentages for discounts
• ratios for comparing quantities
• unit rates for comparing packages
• fractions for portions
• estimation for checking totals
• budgeting for controlling spending

The power of mathematics comes from being able to combine these ideas appropriately.

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Key Terms

• Fraction: Number representing part of a whole or a division.
• Decimal: Number represented using place value based on powers of ten.
• Percentage: Quantity expressed per 100.
• Ratio: Comparison between quantities.
• Rate: Ratio comparing quantities, often with different units.
• Unit rate: Rate expressed per one unit.
• Proportion: Statement that two ratios or rates are equivalent.
• Scale factor: Multiplier used to enlarge or reduce proportional quantities.
• Discount: Reduction in an original price.
• Budget: Plan for income, spending, and saving.
• Interest: Money earned or charged based on an amount and rate.
• Probability: Measure of how likely an event is.
• Expected frequency: Predicted number of occurrences based on probability.
• Estimate: Approximate value used to judge or predict a result.
• Data: Information collected for analysis.
• Constraint: Limit or condition affecting a problem.
• Unit conversion: Changing a measurement into an equivalent measurement using another unit.
• Mathematical model: Mathematical representation of a real situation.
• Reasonableness: Whether an answer makes sense in context.

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Key Mathematical Relationships

Fraction to decimal:

Numerator ÷ Denominator

Decimal to percentage:

Decimal × 100%

Percentage to decimal:

Percentage ÷ 100

Percentage of an amount:

Amount × Percentage as a Decimal

Ratio scale factor:

New Quantity = Original Quantity × Scale Factor

Unit rate:

Quantity ÷ Number of Units

Discount:

Discount = Original Price × Discount Rate

Sale price:

Sale Price = Original Price − Discount

Simple interest:

I = Prt

Simple probability:

P(event) = Favorable Outcomes / Total Possible Outcomes

Expected frequency:

Expected Frequency = Probability × Number of Trials

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Key Takeaways

• Fractions, decimals, ratios, and percentages are different ways of representing and comparing quantities.
• The most useful representation depends on the situation.
• Real-world problems often require several mathematical skills rather than one formula.
• Ratios and proportions are useful for recipes, mixtures, maps, scale drawings, and comparisons.
• Unit rates allow quantities of different sizes to be compared fairly.
• Percentages are useful for discounts, budgets, growth, statistics, and probability.
• Fractions and decimals frequently provide alternative ways to express the same numerical information.
• Financial decisions may involve budgets, unit prices, discounts, percentages, and interest.
• Probability helps describe uncertainty and make predictions.
• Tables and graphs organize numerical information but must be interpreted carefully.
• Units must be checked before quantities are compared.
• Estimation is an important tool for checking whether answers are reasonable.
• Real-world data should be evaluated for relevance, quality, and context.
• A calculator or spreadsheet can perform calculations, but the user must still decide which calculation is appropriate.
• Strong mathematical solutions show calculations and explain what the results mean.
• Conclusions should be supported by numerical evidence.
• A mathematically useful project connects calculations to a genuine question or practical problem.
• Mathematics is most powerful when we can choose the right strategy, combine different ideas, interpret the results, and communicate the reasoning clearly.