Checking whether a model's answer makes practical sense | Year 9

01 / LEARN

Understand the idea.

Substitute and estimate

Checking a model requires more than repeating a calculator entry. Substitute a proposed answer into the original equation, then make an estimate from the size of the inputs. A few percent of a small principal should be a modest amount.

Check units and labels too: an annual rate needs time in years, a percentage rate needs conversion to a decimal, and interest is different from the final balance. A correct calculation of the wrong quantity remains an incorrect answer.

A practical answer needs a practical input

A practical solution must satisfy both the equation and the domain. Item counts are usually non-negative whole numbers; measurements and prices have their own restrictions. Check each algebraic root separately instead of accepting every calculator result.

Keep precision during calculations and round currency at the end unless instructed otherwise. Identify specific limits, such as capacity or the range where a price rule applies; extending a formula outside that range requires further information.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example

A calculation says 300 dollars at 4% simple interest earns 2,400 dollars in 2 years. Find and correct the likely rate-entry error.

Worked solution

1
Identify the likely entry error.
$4 % = \frac{4}{100} = 0.04$
Entering 4 instead of 0.04 makes the rate one hundred times too large.
2
Calculate with the decimal rate.
$I = 300 (0.04) (2) = 24 dollars$
Simple interest uses the original principal, annual rate and time in years.
3
Check the yearly scale.
$300 (0.04) = 12, 2 (12) = 24$
Twelve dollars each year is consistent with four percent of a 300-dollar principal.
4
Compare with the erroneous result.
$300 (4) (2) = 2400$
This reproduces the claimed error and confirms why the correct interest is AUD 24.00.

Answer

Correct interest AUD 24.00

EXAMPLE 02

Worked example

An item-count equation gives $n = - 8$ or $n = 5$ . The rule is $C = n^{2} + 3 n + 10$ and C=50 dollars. Interpret the result.

Worked solution

1
State the restriction on the input.
$n \in {0, 1, 2, \dots}$
The variable counts items, so negative or fractional counts are not allowed.
2
Reject the infeasible root.
$- 8 < 0$
The algebraic candidate -8 cannot represent an item count.
3
Check the remaining root in the original rule.
$C (5) = 5^{2} + 3 (5) + 10 = 25 + 15 + 10 = 50$
Five items satisfy the stated cost equation.
4
Report the practical answer.
$n = 5 items$
This root satisfies both the equation and the non-negative whole-number domain.

Answer

$n = 5$ items

03 / YOUR TURN

Try it yourself.

Check your understanding

A model permits 0 to 12 sessions. Does its formula justify a quote for 20 sessions?

Show answer & reasoning

No.

No: $20 > 12$ , so twenty sessions lies outside the allowed domain. Evaluating the formula there would produce a number, but the stated model does not justify that quote.

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

A principal of 400 dollars at 5% simple interest for 2 years is said to earn 4,000 dollars. Correct it.
2 marks
For an annual interest formula, convert 9 months to years.
2 marks
Interest is 28 dollars on a 350-dollar principal. Find the final amount.
2 marks
A ticket equation gives counts 4 and 11, but only 8 places are available. Which count is feasible?
2 marks
A charge is 7.50 dollars per item. Find the total for 3 items and report it to cents.
2 marks
A model assumes no booking fee, but the venue charges one. Explain how to improve the cost model.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

A principal of 400 dollars at 5% simple interest for 2 years is said to earn 4,000 dollars. Correct it.
2 marks
$I = 400 (0.05) (2) = 40$ .
AUD 40.00 interest
For an annual interest formula, convert 9 months to years.
2 marks
$\frac{9}{12} = 0.75$ .
$0.75$ years
Interest is 28 dollars on a 350-dollar principal. Find the final amount.
2 marks
$A = P + I = 350 + 28 = 378$ .
AUD 378.00
A ticket equation gives counts 4 and 11, but only 8 places are available. Which count is feasible?
2 marks
Eleven exceeds capacity; four does not.
$4$
A charge is 7.50 dollars per item. Find the total for 3 items and report it to cents.
2 marks
$7.50 \times 3 = 22.50$ .
AUD 22.50
A model assumes no booking fee, but the venue charges one. Explain how to improve the cost model.
3 marks
The fee is paid independently of the number of people/items.
A revised form is $C = F + r n$ when fee $F$ and per-person rate $r$ are supplied.
Add the supplied booking fee as a fixed cost.

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum Mathematics

Year 9 financial modelling within AC9M9A05 and AC9M9M05; supplied hypothetical rates and costs, not financial advice.

Official curriculum source ↗

In this lesson

Understand the idea.

Substitute and estimate

A practical answer needs a practical input

See the method in action.

Worked example

Identify the likely entry error.

Calculate with the decimal rate.

Check the yearly scale.

Compare with the erroneous result.

Worked example

State the restriction on the input.

Reject the infeasible root.

Check the remaining root in the original rule.

Report the practical answer.

Try it yourself.

Check your understanding

Keep practising