A chart makes it easier to see which categories account for the largest portions of the budget.

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Reading Budget Tables

Consider this budget:

Tables allow us to compare what was planned with what actually happened.

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Calculating Budget Differences

For food:

$280 − $250 = $30

Actual food spending was:

$30 above budget

Transportation:

$120 − $105 = $15

Actual transportation spending was:

$15 below budget

Entertainment:

$135 − $100 = $35

Actual entertainment spending was:

$35 above budget

These differences are called variances.

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Interpreting Budget Data

A budget should not only be calculated. It should also be interpreted.

For example, the previous table suggests:

• food spending exceeded the planned amount by $30
• transportation spending was $15 below the planned amount
• entertainment exceeded the plan by $35
• clothing was $20 below the plan
• savings were $30 below the planned target

This information can help someone decide what adjustments to make in the next budget.

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Planned vs Actual Spending

Charts can make differences easier to identify.

Graphs can reveal patterns that may be less obvious when looking only at individual numbers.

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Comparing Financial Options

Mathematics can help compare different financial choices.

Suppose two phone plans are available.

Plan A:

$30 per month

Plan B:

$25 per month + $50 setup fee

For one month:

Plan A:

$30

Plan B:

$25 + $50 = $75

For 12 months:

Plan A:

12 × $30 = $360

Plan B:

12 × $25 + $50 = $350

The comparison changes depending on how long the service is used.

This shows why comparing only one advertised number can be misleading.

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Comparing Unit Prices

Suppose:

Package A:

500 g for $4.50

Package B:

800 g for $6.40

Calculate cost per 100 g.

Package A:

500 g contains five groups of 100 g.

$4.50 ÷ 5 = $0.90 per 100 g

Package B:

800 g contains eight groups.

$6.40 ÷ 8 = $0.80 per 100 g

Unit rates make differently sized packages easier to compare.

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Discounts and Budgets

Suppose an item normally costs:

$80

and is discounted by:

25%

Discount:

0.25 × $80 = $20

Sale price:

$80 − $20 = $60

If the shopping budget is $70:

$70 − $60 = $10

would remain.

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Taxes and Final Prices

The displayed price is not always the final amount paid.

Suppose an illustrative purchase costs:

$120

and a tax of:

8%

is applied.

Tax:

0.08 × $120 = $9.60

Final cost:

$120 + $9.60 = $129.60

When budgeting for purchases, it is important to know whether additional charges are included in the listed price.

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Tips and Service Charges

Suppose a restaurant bill is:

$60

and someone calculates a 15% tip.

Tip:

0.15 × $60 = $9

Total:

$60 + $9 = $69

In real situations, whether a tip or service charge applies depends on local practices and the bill itself.

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Recurring Expenses

Small recurring expenses can become significant over time.

Suppose a subscription costs:

$12/month

Annual cost:

$12 × 12 = $144

Three similar subscriptions would cost:

3 × $144 = $432 per year

Converting recurring costs to annual amounts can make their financial impact easier to understand.

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Daily Spending Over Time

Suppose someone spends:

$5 each school day

on an optional purchase.

Over 5 days:

$5 × 5 = $25

Over 20 school days:

$5 × 20 = $100

Over 180 school days:

$5 × 180 = $900

A relatively small repeated expense can become a much larger total over time.

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Saving Toward a Goal

Suppose someone wants to save:

$1,200

and can save:

$150 per month

Calculate:

$1,200 ÷ $150 = 8

It would take:

8 months

assuming the same amount is saved each month.

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Working Backwards from a Savings Goal

Suppose someone wants to save:

$2,400 in 12 months

Required monthly savings:

$2,400 ÷ 12 = $200

Therefore:

$200 per month

must be set aside on average.

If monthly income is $1,600:

$200 / $1,600 × 100% = 12.5%

The savings goal represents:

12.5% of monthly income

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Emergency and Unexpected Expenses

Real budgets do not always follow the plan exactly.

Unexpected expenses can include:

• repairs
• replacement items
• medical or dental costs
• transportation problems
• urgent travel
• sudden changes in regular expenses

One way budgets can account for uncertainty is by including money for unexpected expenses or maintaining savings that are not already committed elsewhere.

The appropriate amount depends on the person's circumstances.

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Comparing Short-Term and Long-Term Costs

Suppose:

Option A costs:

$20 per month

Option B costs:

$180 per year

Compare annual costs.

Option A:

$20 × 12 = $240/year

Option B:

$180/year

Comparing both options over the same time period provides a meaningful mathematical comparison.

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The Cheapest Option Is Not Always the Best Decision

Cost is important, but financial decisions can involve more than price.

Other factors might include:

• quality
• reliability
• how long the product will last
• maintenance costs
• cancellation fees
• warranties
• convenience
• amount actually needed

For example, buying a larger package because it has a lower unit price may not save money if much of the product is wasted.

Mathematics provides evidence for a decision, but the context determines which factors matter.

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Advertising and Financial Decisions

Advertisements often emphasize numbers such as:

50% OFF

Only $10/month

Buy 2, Get 1 Free

Save $100

These numbers can be useful, but they should be interpreted carefully.

Useful questions include:

• What was the original price?
• What is the final price?
• Are there additional fees?
• How long does the offer last?
• Is a subscription required?
• How much will it cost over a full year?
• How much of the product is actually needed?

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Comparing Two Discounts

Store A offers:

20% off a $100 item

Discount:

$20

Final price:

$80

Store B sells the same item for:

$85 with a $10 coupon

Final price:

$75

The percentage advertised does not by itself determine the final cost.

Calculate the actual amount paid.

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Using Percentages to Analyze Spending

Suppose monthly income is:

$2,400

and food spending is:

$480

Percentage of income spent on food:

480 / 2400 × 100%

= 20%

If transportation costs:

$240

then:

240 / 2400 × 100% = 10%

Percentages make categories easier to compare.

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Comparing Spending Categories

Suppose someone spends:

Food = $500

Housing = $900

Transportation = $250

Entertainment = $150

Savings = $200

Total:

$2,000

Housing percentage:

900 / 2000 × 100% = 45%

Food percentage:

500 / 2000 × 100% = 25%

Transportation percentage:

12.5%

Entertainment percentage:

7.5%

Savings percentage:

10%

These percentages show how the total is distributed.

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Financial Information in Tables

Suppose three savings plans are presented:

A table helps us identify patterns and compare values quickly.

For example:

Plan C saves:

$1,200 − $600 = $600

more per year than Plan A.

But it also requires:

$50 more each month

The numbers describe the trade-off.

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Financial Information in Graphs

Suppose a person's spending changes over several months.

A graph can help identify:

• increases
• decreases
• unusually high months
• unusually low months
• overall patterns

However, the graph should be interpreted together with the actual values and context.

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Checking Graph Scales

Graphs can sometimes make differences appear larger or smaller depending on the scale used.

Suppose two expenses are:

$100 and $105

The difference is only:

$5

If a graph's vertical axis begins at $99 instead of $0, the visual difference may appear very large.

When interpreting financial graphs, always check:

• axis labels
• units
• scale
• time period
• categories
• whether the graph begins at zero when relevant
• whether important information has been omitted

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Percent Change in Expenses

Suppose an electricity bill increases from:

$80 to $100

Change:

$100 − $80 = $20

Percentage increase:

20 / 80 × 100% = 25%

Therefore:

the expense increased by 25%

Percentage changes can help compare increases involving different starting amounts.

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Comparing Changes

Expense A increases:

$20 → $30

Increase:

$10

Percentage increase:

10 / 20 × 100% = 50%

Expense B increases:

$100 → $120

Increase:

$20

Percentage increase:

20 / 100 × 100% = 20%

Expense B increased by more dollars, but Expense A had the larger percentage increase.

This demonstrates why both absolute and percentage changes can be useful.

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Making Decisions with Numerical Data

Suppose you have:

$500

available for a purchase.

Option A:

Purchase price = $420

Expected additional cost = $30

Total:

$450

Money remaining:

$50

Option B:

Purchase price = $390

Expected additional cost = $90

Total:

$480

Money remaining:

$20

Looking only at the purchase price would make Option B appear less expensive.

Calculating the total cost changes the comparison.

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Opportunity Cost

Financial choices often involve opportunity cost.

Opportunity cost is the value of what is given up when one option is chosen instead of another.

Suppose someone has:

$100

They could spend $100 on entertainment or save the $100 toward a future goal.

If they spend it, the opportunity cost includes the savings they could have kept.

Opportunity cost is not always a simple dollar calculation, but numerical information can help make the trade-off clearer.

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Budget Constraints

A constraint is a limit or condition.

Suppose you have:

$300

to organize an event.

Possible costs:

Food = $140

Decorations = $60

Equipment = $75

Total:

$140 + $60 + $75 = $275

Money remaining:

$300 − $275 = $25

The $300 budget acts as a constraint.

Any plan costing more than $300 would exceed the available budget.

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Worked Example 1: Monthly Budget

Income:

$1,800

Expenses:

Housing = $650

Food = $300

Transportation = $180

Phone = $50

Entertainment = $120

Other = $100

Total expenses:

650 + 300 + 180 + 50 + 120 + 100

= $1,400

Remaining:

$1,800 − $1,400 = $400

Answer:

$400 remains

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Worked Example 2: Savings

Income:

$2,500

Savings goal:

18% of income

Calculate:

0.18 × $2,500 = $450

Answer:

$450

If other expenses total $1,850:

$2,500 − $450 − $1,850 = $200

remains after savings and expenses.

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Worked Example 3: Comparing Offers

Option A:

$15/month for 12 months

Total:

15 × 12 = $180

Option B:

$120 for the first 6 months + $12/month for the next 6 months

Total:

$120 + (6 × $12)

= $192

Comparing total costs over the same 12-month period provides a consistent basis for comparison.

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Worked Example 4: Discount

A backpack costs:

$75

Discount:

20%

Discount amount:

0.20 × $75 = $15

Sale price:

$75 − $15 = $60

If the budget is $70:

$70 − $60 = $10

remains.

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Worked Example 5: Spending Percentage

Monthly income:

$3,000

Housing:

$1,050

Percentage:

1050 / 3000 × 100%

= 35%

Therefore:

35% of income is allocated to housing

in this example.

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Worked Example 6: Savings Goal

Goal:

$900

Current savings:

$300

Still needed:

$900 − $300 = $600

If $75 is saved each month:

600 ÷ 75 = 8 months

Answer:

8 additional months

assuming the saving rate remains constant.

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Worked Example 7: Budget Adjustment

Income:

$1,500

Expenses:

$1,620

Deficit:

$1,620 − $1,500 = $120

To balance the budget, the person would need some combination of:

• reducing planned expenses by $120
• increasing income by $120
• or making changes totaling $120

The mathematics identifies the size of the gap.

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Worked Example 8: Unit Price

Package A:

12 items for $18

Unit price:

18 ÷ 12 = $1.50/item

Package B:

20 items for $28

Unit price:

28 ÷ 20 = $1.40/item

The unit prices are:

$1.50/item versus $1.40/item

Other factors such as how many items are actually needed may also matter.

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Worked Example 9: Annual Cost

A service costs:

$8.99/month

Approximate annual cost:

$8.99 × 12 = $107.88

A monthly price that appears small can therefore represent a much larger annual expense.

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Worked Example 10: Multi-Step Purchase

A device costs:

$500

Discount:

15%

Discount amount:

0.15 × $500 = $75

Discounted price:

$500 − $75 = $425

Suppose an illustrative 8% tax is then applied:

0.08 × $425 = $34

Final price:

$425 + $34 = $459

Notice that the tax was calculated on the discounted price in this example.

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Estimation and Financial Decisions

Estimation is useful when checking financial calculations.

Suppose:

18% of $498

is needed.

Estimate:

20% of $500 ≈ $100

Exact calculation:

0.18 × 498 = $89.64

The exact value is reasonably close to the estimate.

If a calculator produced:

$896.40

we would know something was probably wrong.

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A Budget Is a Model

A budget is not a prediction that every expense will occur exactly as planned.

It is a mathematical model of expected income and spending.

Actual values can change.

For this reason, useful budgeting involves comparing:

planned values

with:

actual values

and then updating future plans.

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A Reliable Financial Problem-Solving Strategy

When solving a budgeting or money problem:

Step 1: Identify the financial goal or question.

Step 2: Identify the relevant income, costs, percentages, and time periods.

Step 3: Make sure values are being compared over the same period.

Step 4: Calculate total income.

Step 5: Calculate relevant expenses.

Step 6: Find unit rates, percentages, or total costs when needed.

Step 7: Compare the numerical results.

Step 8: Check whether the answer fits the budget constraint.

Step 9: Estimate to check reasonableness.

Step 10: Explain what the numbers show before making a decision.

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Common Mistakes

Mistake 1: Comparing monthly and annual costs directly

Convert them to the same time period first.

---

Mistake 2: Looking only at the advertised price

Additional fees, taxes, recurring charges, or required purchases may affect total cost.

---

Mistake 3: Confusing a discount with the final price

A $20 discount does not mean the item costs $20.

---

Mistake 4: Ignoring recurring costs

A small monthly expense can become significant over a year.

---

Mistake 5: Assuming the lowest unit price always means the lowest total spending

A larger package may have a lower unit price but require spending more money overall.

---

Mistake 6: Forgetting the original value when calculating percentage change

Use:

percentage change = change / original value × 100%

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Mistake 7: Treating planned spending as actual spending

Budgets are plans. Actual spending should be recorded and compared with the plan.

---

Mistake 8: Ignoring graph scales

Always examine labels, units, intervals, and the range shown.

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Did You Know?

Budgeting combines many mathematical skills rather than being a completely separate type of mathematics.

Budgeting can involve:

• addition and subtraction
• multiplication and division
• decimals
• fractions
• percentages
• ratios
• unit rates
• proportions
• tables
• graphs
• estimation
• data analysis

This makes budgeting an important real-world application of many mathematical ideas.

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Key Terms

• Budget: Plan for income, spending, and saving.
• Income: Money received.
• Expense: Money spent.
• Fixed expense: Expense that generally remains constant over a particular period.
• Variable expense: Expense that can change.
• Savings: Money kept for future use rather than currently spent.
• Savings rate: Savings expressed as a percentage of income.
• Surplus: Amount by which income exceeds expenses.
• Deficit: Amount by which expenses exceed income.
• Balance: Amount remaining after expenses are subtracted from income.
• Budget constraint: Limit on how much money is available.
• Unit price: Cost for one item or one standard unit.
• Discount: Reduction from an original price.
• Tax: Amount added or collected according to an applicable tax system.
• Recurring expense: Expense that occurs repeatedly.
• Variance: Difference between a planned amount and an actual amount.
• Opportunity cost: Value of an alternative given up when a choice is made.
• Percentage change: Change expressed relative to the original amount.

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Key Equations

Budget balance:

Balance = Income − Expenses

Savings rate:

Savings rate = Savings / Income × 100%

Percentage of income spent:

Spending percentage = Expense / Income × 100%

Unit price:

Unit price = Total cost / Quantity

Discount amount:

Discount = Original price × Discount rate

Sale price:

Sale price = Original price − Discount

Percentage change:

Percentage change = Change / Original value × 100%

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Key Takeaways

• A budget is a mathematical plan for managing income, expenses, and savings.
• Income is money received, while expenses are money spent.
• When income exceeds expenses, there is a surplus.
• When expenses exceed income, there is a deficit.
• Savings can be expressed as an amount or as a percentage of income.
• Fixed and variable expenses behave differently and can affect how easily a budget can be adjusted.
• Tables help organize financial information.
• Charts help reveal patterns and differences in spending.
• Planned and actual spending should be compared.
• Unit prices allow products of different sizes to be compared.
• Monthly and annual costs should be converted to the same time period before comparison.
• Discounts, taxes, fees, and recurring costs can affect the final amount paid.
• Percentage calculations can show how much of income is spent or saved.
• Percentage change helps compare increases and decreases.
• Advertised prices should be interpreted in the context of total cost.
• The least expensive option is not automatically the most appropriate option because quality, quantity, reliability, and other constraints may matter.
• Estimation helps identify unreasonable calculations.
• A budget is a model and may need to change when actual income or expenses change.
• Numerical data can clarify trade-offs and consequences.
• Financial mathematics does not make a decision for us; it provides evidence that helps us understand the available options.