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Variation and financial models

Read the idea, work independently, then explain what changed.

2027 JM01 考試大綱 · 2, 3 and 8. Percentage, variations, exponential growth · PDF 2 / printed page 2

TOPIC 01

Variation and financial models

Identify direct, inverse, joint and partial variation and apply stated interest or depreciation models.

What you will be able to explain

Identify direct, inverse, joint and partial variation and apply stated interest or depreciation models.
Justify the method and check the conditions in a new situation.

Model or definition

Identify direct, inverse, joint and partial variation and apply stated interest or depreciation models.

y=kx;\ y=k/x;\ y=kxz;\ y=a+kx;\ A=P(1+r)^n

Conditions

Rates are decimals; all values are idealized models with the stated period and compounding rule.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Do a 10% increase and a 10% decrease restore the starting value?

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

Illustrative principal 100, annual rate 10%, 3 years: simple=130, annually compounded=133.1. A 10% rise followed by a 10% fall gives 99. This model excludes fees and taxes.

Explain: Compare two admissible cases and explain their different results using the stated model.

Transfer: Compare simple, discrete compound and continuous growth under stated hypothetical rates.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Worked example

y varies directly with x; y=t when x=2. Find y when x=6.

t=2

Rates are decimals; all values are idealized models with the stated period and compounding rule.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Express the cash flow or proportional relation before calculating.

Hint 2

Use this intermediate relation.

y=kx;\quad k=t/2

Worked solution

Express the cash flow or proportional relation before calculating.
Apply the stated relation and retain its conditions.
$y=(2/2)6=6$
The constant ratio y/x is preserved.

The requested value is 6.

Checks and common pitfalls: The constant ratio y/x is preserved.

Think first. Reveal a hint when the class is ready.

02 / Standard#Worked example

A hypothetical balance starts at 100t and compounds annually at 10% for 2 years. Find final balance.

t=3

Rates are decimals; all values are idealized models with the stated period and compounding rule.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Express the cash flow or proportional relation before calculating.

Hint 2

Use this intermediate relation.

A=P(1+r)^2

Worked solution

Express the cash flow or proportional relation before calculating.
Apply the stated relation and retain its conditions.
$A=100(3)(1.1)^2=363$
This is a stated classroom model, with no extra deposits, charges or rounding policy.

The requested value is 363.

Checks and common pitfalls: This is a stated classroom model, with no extra deposits, charges or rounding policy.

Think first. Reveal a hint when the class is ready.

03 / Transfer#Worked example

A hypothetical amount 100t grows continuously at rate ln2 per year. Find its value after one year.

t=4

Rates are decimals; all values are idealized models with the stated period and compounding rule.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Express the cash flow or proportional relation before calculating.

Hint 2

Use this intermediate relation.

A=Pe^{ru}

Worked solution

Express the cash flow or proportional relation before calculating.
Apply the stated relation and retain its conditions.
$A=100(4)e^{\ln2}=800$
A continuous rate is used in an exponential, not in the annual discrete-compound factor.

The requested value is 800.

Checks and common pitfalls: A continuous rate is used in an exponential, not in the annual discrete-compound factor.

Think first. Reveal a hint when the class is ready.

04 / Foundation#Your turn

y varies directly with x; y=t when x=2. Find y when x=6.

t=5

Rates are decimals; all values are idealized models with the stated period and compounding rule.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Express the cash flow or proportional relation before calculating.

Hint 2

Use this intermediate relation.

y=kx;\quad k=t/2

Worked solution

Express the cash flow or proportional relation before calculating.
Apply the stated relation and retain its conditions.
$y=(5/2)6=15$
The constant ratio y/x is preserved.

The requested value is 15.

Checks and common pitfalls: The constant ratio y/x is preserved.

Think first. Reveal a hint when the class is ready.

05 / Foundation#Your turn

y varies inversely with x; y=t at x=2. Find y at x=4.

t=6

Rates are decimals; all values are idealized models with the stated period and compounding rule.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Express the cash flow or proportional relation before calculating.

Hint 2

Use this intermediate relation.

xy=k=2t

Worked solution

Express the cash flow or proportional relation before calculating.
Apply the stated relation and retain its conditions.
$y=12/4=3$
Doubling x halves y in an inverse model.

The requested value is 3.

Checks and common pitfalls: Doubling x halves y in an inverse model.

Think first. Reveal a hint when the class is ready.

06 / Foundation#Your turn

y varies jointly with x,z and equals t at x=z=1. Find y at x=2,z=3.

t=7

Rates are decimals; all values are idealized models with the stated period and compounding rule.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Express the cash flow or proportional relation before calculating.

Hint 2

Use this intermediate relation.

y=kxz;\quad k=t

Worked solution

Express the cash flow or proportional relation before calculating.
Apply the stated relation and retain its conditions.
$y=7\cdot2\cdot3=42$
Both independent multiplicative factors enter the model.

The requested value is 42.

Checks and common pitfalls: Both independent multiplicative factors enter the model.

Think first. Reveal a hint when the class is ready.

07 / Foundation#Your turn

y=a+kx; y=t at x=0 and y=t+6 at x=2. Find y at x=3.

t=8

Rates are decimals; all values are idealized models with the stated period and compounding rule.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Express the cash flow or proportional relation before calculating.

Hint 2

Use this intermediate relation.

a=t;\quad 2k=6

Worked solution

Express the cash flow or proportional relation before calculating.
Apply the stated relation and retain its conditions.
$k=3,\quad y(3)=8+9=17$
Partial variation has a fixed part as well as a proportional part.

The requested value is 17.

Checks and common pitfalls: Partial variation has a fixed part as well as a proportional part.

Think first. Reveal a hint when the class is ready.

08 / Standard#Your turn

y=a+kx; y=t at x=0 and y=t+6 at x=2. Find y at x=3.

t=9

Rates are decimals; all values are idealized models with the stated period and compounding rule.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Express the cash flow or proportional relation before calculating.

Hint 2

Use this intermediate relation.

a=t;\quad 2k=6

Worked solution

Express the cash flow or proportional relation before calculating.
Apply the stated relation and retain its conditions.
$k=3,\quad y(3)=9+9=18$
Partial variation has a fixed part as well as a proportional part.

The requested value is 18.

Checks and common pitfalls: Partial variation has a fixed part as well as a proportional part.

Think first. Reveal a hint when the class is ready.

09 / Standard#Your turn

A hypothetical balance starts at 100t and compounds annually at 10% for 2 years. Find final balance.

t=10

Rates are decimals; all values are idealized models with the stated period and compounding rule.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Express the cash flow or proportional relation before calculating.

Hint 2

Use this intermediate relation.

A=P(1+r)^2

Worked solution

Express the cash flow or proportional relation before calculating.
Apply the stated relation and retain its conditions.
$A=100(10)(1.1)^2=1210$
This is a stated classroom model, with no extra deposits, charges or rounding policy.

The requested value is 1210.

Checks and common pitfalls: This is a stated classroom model, with no extra deposits, charges or rounding policy.

Think first. Reveal a hint when the class is ready.

10 / Standard#Your turn

An asset worth 100t loses 20% each year for 2 years. Find final value.

t=11

Rates are decimals; all values are idealized models with the stated period and compounding rule.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Express the cash flow or proportional relation before calculating.

Hint 2

Use this intermediate relation.

V=P(1-0.2)^2

Worked solution

Express the cash flow or proportional relation before calculating.
Apply the stated relation and retain its conditions.
$V=100(11)(0.8)^2=704$
Depreciation applies to the remaining value each year.

The requested value is 704.

Checks and common pitfalls: Depreciation applies to the remaining value each year.

Think first. Reveal a hint when the class is ready.

11 / Standard#Your turn

A hypothetical amount 100t grows continuously at rate ln2 per year. Find its value after one year.

t=12

Rates are decimals; all values are idealized models with the stated period and compounding rule.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Express the cash flow or proportional relation before calculating.

Hint 2

Use this intermediate relation.

A=Pe^{ru}

Worked solution

Express the cash flow or proportional relation before calculating.
Apply the stated relation and retain its conditions.
$A=100(12)e^{\ln2}=2400$
A continuous rate is used in an exponential, not in the annual discrete-compound factor.

The requested value is 2400.

Checks and common pitfalls: A continuous rate is used in an exponential, not in the annual discrete-compound factor.

Think first. Reveal a hint when the class is ready.

12 / Transfer#Your turn

An asset worth 100t loses 20% each year for 2 years. Find final value.

t=13

Rates are decimals; all values are idealized models with the stated period and compounding rule.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Express the cash flow or proportional relation before calculating.

Hint 2

Use this intermediate relation.

V=P(1-0.2)^2

Worked solution

Express the cash flow or proportional relation before calculating.
Apply the stated relation and retain its conditions.
$V=100(13)(0.8)^2=832$
Depreciation applies to the remaining value each year.

The requested value is 832.

Checks and common pitfalls: Depreciation applies to the remaining value each year.

Think first. Reveal a hint when the class is ready.

13 / Transfer#Your turn

A hypothetical amount 100t grows continuously at rate ln2 per year. Find its value after one year.

t=14

Rates are decimals; all values are idealized models with the stated period and compounding rule.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Express the cash flow or proportional relation before calculating.

Hint 2

Use this intermediate relation.

A=Pe^{ru}

Worked solution

Express the cash flow or proportional relation before calculating.
Apply the stated relation and retain its conditions.
$A=100(14)e^{\ln2}=2800$
A continuous rate is used in an exponential, not in the annual discrete-compound factor.

The requested value is 2800.

Checks and common pitfalls: A continuous rate is used in an exponential, not in the annual discrete-compound factor.

Think first. Reveal a hint when the class is ready.

End-of-lesson check and correction

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Teacher preparation and assessment

Question sequence

Identify direct, inverse, joint and partial variation and apply stated interest or depreciation models.
Which condition is essential in variation and financial models?
Do a 10% increase and a 10% decrease restore the starting value?

Board plan

Model or definition: Identify direct, inverse, joint and partial variation and apply stated interest or depreciation models.
$y=kx;\ y=k/x;\ y=kxz;\ y=a+kx;\ A=P(1+r)^n$
Conditions: Rates are decimals; all values are idealized models with the stated period and compounding rule.

Anticipated thinking

Successive percentage changes multiply their factors rather than add their rates.

Assessment checklist

1 mark: choose the correct representation and conditions.
1 mark: establish the intermediate relation.
1 mark: complete a connected calculation or proof.
1 mark: interpret and check the conclusion.