Rates and variations

Overview

Rates compare two different quantities, such as kilometres per hour, shillings per dollar, or metres per second. Variations describe how one quantity changes when another quantity changes.

This topic matters because many Form II and CSEE-style problems are written as real situations: speed and time, buying and selling prices, exchange rates, mass supported by a beam, or dimensions of a solid. The main skill is to translate words into a mathematical relationship, find the constant, and then use the relationship carefully.

Prerequisites

• Ratios and proportions - Variation is built from proportional reasoning.
• Repeating decimals and fractions - Rates often require converting between fractions, decimals, and percentages.
• Algebraic expressions and equations - Variation statements are written as equations such as $y = kx$ or $y = \frac{k}{x}$.
• Linear simultaneous equations - Some rate problems can lead to more than one unknown.
• Approximations, rounding, significant figures, and decimal places - Real-context answers often need sensible units and rounding.

Learning Scope

This chapter covers unit rates, direct variation, inverse variation, joint variation, combined variation, constants of proportionality, and common applications involving speed, exchange rates, percentages, and interest-rate language.

This page does not fully teach accounting, profit and loss, compound interest, functions, or geometry. Those contexts may appear as applications, but the focus here is the rate or variation relationship.

Subtopics

Rates And Unit Rates

A rate compares two quantities measured in different units. For example, speed compares distance with time:

$$ \text{speed} = \frac{\text{distance}}{\text{time}} $$

If a car travels $180 \text{ km}$ in $3 \text{ hours}$, its speed is:

$$ \frac{180 \text{ km}}{3 \text{ h}} = 60 \text{ km/h} $$

A unit rate tells the amount for one unit. The speed $60 \text{ km/h}$ means $60 \text{ km}$ for each $1$ hour.

Key insight: Always read the unit. A rate of $60 \text{ km/h}$ and a rate of $60 \text{ m/s}$ are not the same rate.

Exchange Rates

An exchange rate is a unit rate between currencies. If $1$ US dollar is worth $2,300$ Tanzanian shillings, then:

$$ 1 \text{ USD} = 2,300 \text{ Tsh} $$

To convert Tanzanian shillings into US dollars, divide by $2,300$:

$$ \text{USD} = \frac{\text{Tsh}}{2,300} $$

To convert US dollars into Tanzanian shillings, multiply by $2,300$:

$$ \text{Tsh} = 2,300 \times \text{USD} $$

Key insight: Decide whether the answer should be a larger or smaller number before calculating. This often reveals whether to multiply or divide.

Direct Variation

Two quantities vary directly if one quantity is a constant multiple of the other. If $y$ varies directly as $x$, then:

$$ y \propto x $$

and:

$$ y = kx $$

where $k$ is the constant of proportionality.

For example, if buying price $B$ is directly proportional to selling price $S$, then:

$$ B = kS $$

Key insight: In direct variation, when $x$ is multiplied by a number, $y$ is multiplied by the same number.

Finding The Constant Of Proportionality

The constant of proportionality connects the variables. For direct variation:

$$ y = kx $$

So:

$$ k = \frac{y}{x} $$

If $y = 18,000$ when $x = 20,000$, then:

$$ k = \frac{18,000}{20,000} = 0.9 $$

The equation is:

$$ y = 0.9x $$

Key insight: Use the given pair of values to find $k$ before finding any new value.

Inverse Variation

Two quantities vary inversely if one increases while the other decreases in such a way that their product remains constant. If $y$ varies inversely as $x$, then:

$$ y \propto \frac{1}{x} $$

and:

$$ y = \frac{k}{x} $$

Equivalently:

$$ xy = k $$

For example, if time $t$ varies inversely with speed $v$, then:

$$ t = \frac{k}{v} $$

Key insight: In inverse variation, doubling one quantity halves the other, as long as the same task or distance is being described.

Joint Variation

A quantity varies jointly with two or more quantities if it is directly proportional to their product. If $y$ varies jointly as $a$ and $b$, then:

$$ y = kab $$

For example, if mass $M$ supported by a beam varies directly with breadth $b$ and thickness $d$, a possible model is:

$$ M = kbd $$

Key insight: Joint variation is still direct variation, but more than one factor is involved.

Combined Variation

Combined variation mixes direct and inverse variation. If $y$ varies directly as $a$ and inversely as $b$, then:

$$ y = \frac{ka}{b} $$

If $h$ varies directly as $V$ and inversely as the square of $r$, then:

$$ h = \frac{kV}{r^2} $$

Key insight: Words such as "directly", "inversely", and "square" must all be represented in the formula.

Rates In Percentage And Interest Contexts

An interest rate is a rate per time period. For simple interest:

$$ I = PRT $$

where $P$ is principal, $R$ is the rate per time period written as a decimal, and $T$ is time.

For compound interest, the amount after $n$ periods is commonly written as:

$$ A = P(1+r)^n $$

where $r$ is the rate per period as a decimal.

This page uses interest only as a rates context. Detailed financial mathematics should be reviewed separately before final use.

Key insight: A percentage rate such as $2\%$ must be written as $0.02$ when used in a formula.

Choosing A Model From Words

Many mistakes happen before calculation begins. Use the wording to choose the model:

• "$y$ is directly proportional to $x$" means $y = kx$.
• "$y$ varies inversely with $x$" means $y = \frac{k}{x}$.
• "$y$ varies directly with $a$ and inversely with $b$" means $y = \frac{ka}{b}$.
• "$y$ varies inversely with the square of $x$" means $y = \frac{k}{x^2}$.

Key insight: Translate the relationship first, then substitute numbers.

Key Terms

• Rate: A comparison of two quantities with different units, such as $\text{km/h}$ or $\text{Tsh/USD}$.
• Unit rate: A rate written for one unit of the second quantity.
• Variation: A relationship showing how one quantity changes when another changes.
• Direct variation: A relationship of the form $y = kx$.
• Inverse variation: A relationship of the form $y = \frac{k}{x}$.
• Joint variation: A direct variation involving the product of two or more quantities, such as $y = kab$.
• Combined variation: A relationship containing both direct and inverse variation, such as $y = \frac{ka}{b}$.
• Constant of proportionality: The fixed number $k$ in a variation equation.
• Exchange rate: A rate used to convert one currency into another.

Worked Examples

Example 1: Find A Unit Rate

A motorcyclist travels $150 \text{ km}$ in $2.5 \text{ h}$. Find the average speed.

Use:

$$ \text{speed} = \frac{\text{distance}}{\text{time}} $$

Then:

$$ \begin{aligned} \text{speed} &= \frac{150 \text{ km}}{2.5 \text{ h}} \\ &= 60 \text{ km/h} \end{aligned} $$

Final answer:

$$ 60 \text{ km/h} $$

Example 2: Direct Variation

The cost price $C$ of an item is directly proportional to its selling price $S$. If $C = 18,000$ when $S = 20,000$, write the equation connecting $C$ and $S$.

Since $C$ varies directly as $S$:

$$ C = kS $$

Use the given values:

$$ \begin{aligned} 18,000 &= k(20,000) \\ k &= \frac{18,000}{20,000} \\ k &= 0.9 \end{aligned} $$

Therefore:

$$ C = 0.9S $$

Final answer:

$$ C = 0.9S $$

Example 3: Inverse Variation

The time $t$ seconds needed to travel a fixed distance varies inversely with speed $v$ metres per second. If $t = 1,800$ when $v = 10$, find the equation for $t$ in terms of $v$.

Since $t$ varies inversely with $v$:

$$ t = \frac{k}{v} $$

Use the given values:

$$ \begin{aligned} 1,800 &= \frac{k}{10} \\ k &= 18,000 \end{aligned} $$

So:

$$ t = \frac{18,000}{v} $$

Final answer:

$$ t = \frac{18,000}{v} $$

Example 4: Use Inverse Variation To Find A New Value

Using the equation $t = \frac{18,000}{v}$, find the speed needed if the time is $900$ seconds.

Substitute $t = 900$:

$$ \begin{aligned} 900 &= \frac{18,000}{v} \\ 900v &= 18,000 \\ v &= \frac{18,000}{900} \\ v &= 20 \end{aligned} $$

Final answer:

$$ 20 \text{ m/s} $$

The speed doubles because the time is halved.

Example 5: Combined Variation

The height $h$ of a cylinder varies directly as its volume $V$ and inversely as the square of its radius $r$. Given that $h = 3$ when $V = 900$ and $r = 10$, find $h$ when $V = 1,200$ and $r = 5$.

Translate the statement:

$$ h = \frac{kV}{r^2} $$

Find $k$:

$$ \begin{aligned} 3 &= \frac{k(900)}{10^2} \\ 3 &= \frac{900k}{100} \\ 3 &= 9k \\ k &= \frac{1}{3} \end{aligned} $$

Now use $V = 1,200$ and $r = 5$:

$$ \begin{aligned} h &= \frac{\frac{1}{3}(1,200)}{5^2} \\ &= \frac{400}{25} \\ &= 16 \end{aligned} $$

Final answer:

$$ h = 16 $$

If the units are centimetres, this is $16 \text{ cm}$.

Example 6: Exchange Rate Conversion

If $1$ US dollar is worth $2,300$ Tanzanian shillings, convert $27,600,000$ Tanzanian shillings into US dollars.

Use:

$$ \text{USD} = \frac{\text{Tsh}}{2,300} $$

Then:

$$ \begin{aligned} \text{USD} &= \frac{27,600,000}{2,300} \\ &= 12,000 \end{aligned} $$

Final answer:

$$ 12,000 \text{ USD} $$

Common Mistakes

• Mistake: Writing a direct variation as $y = \frac{k}{x}$.

Correction: Direct variation has the form $y = kx$.

• Mistake: Writing an inverse variation as $y = kx$.

Correction: Inverse variation has the form $y = \frac{k}{x}$.

• Mistake: Forgetting to find $k$ before finding the required value.

Correction: Use the first set of values to find $k$, then substitute the new values.

• Mistake: Ignoring squared wording.

Correction: "Inversely as the square of $r$" means divide by $r^2$, not by $r$.

• Mistake: Dropping units from rate answers.

Correction: Write units such as $\text{km/h}$, $\text{m/s}$, or $\text{Tsh/USD}$.

• Mistake: Multiplying instead of dividing in currency conversion.

Correction: Use the direction of the exchange rate to decide the operation.

• Mistake: Using $2$ instead of $0.02$ for a $2\%$ rate.

Correction: Convert percentages to decimals before using them in formulas.

Practice Tasks

Direct Understanding

1. Explain what a rate compares.
2. State the general equation for direct variation.
3. State the general equation for inverse variation.
4. In $y = 5x$, identify the constant of proportionality.
5. Explain the difference between $y = kx$ and $y = \frac{k}{x}$.

Skill Practice

1. Find the speed of a cyclist who travels $72 \text{ km}$ in $4 \text{ h}$.
2. Convert $540 \text{ km/h}$ into kilometres per minute.
3. If $y$ varies directly as $x$ and $y = 24$ when $x = 6$, find $k$.
4. If $p$ varies inversely as $q$ and $p = 8$ when $q = 5$, write the equation connecting $p$ and $q$.
5. If $A$ varies jointly as $b$ and $h$, write the general variation equation.
6. If $z$ varies directly as $x$ and inversely as $y^2$, write the general variation equation.

Application Problems

1. A bus travels $210 \text{ km}$ in $3.5 \text{ h}$. Find its average speed.
2. If $1 \text{ USD} = 2,300 \text{ Tsh}$, convert $115,000 \text{ Tsh}$ into US dollars.
3. If the cost price $C$ varies directly as selling price $S$ and $C = 45,000$ when $S = 50,000$, find $C$ when $S = 80,000$.
4. Time $t$ varies inversely with speed $v$. If $t = 40$ minutes when $v = 12 \text{ m/s}$, find $t$ when $v = 16 \text{ m/s}$.
5. The mass $M$ supported by a beam varies directly as breadth $b$ and inversely as length $l$. If $M = 200$ when $b = 2$ and $l = 15$, find $M$ when $b = 3$ and $l = 20$.

Multi-Step Reasoning

1. A quantity $h$ varies directly as $V$ and inversely as $r^2$. Given $h = 4$ when $V = 500$ and $r = 5$, find $h$ when $V = 800$ and $r = 10$.
2. A school invests money at $2\%$ compound interest per year. Write the expression for the amount after $2$ years if the principal is $P$.
3. If $y$ varies directly as $x^2$ and inversely as $z$, and $y = 18$ when $x = 3$ and $z = 2$, find $y$ when $x = 4$ and $z = 8$.

Edge Cases

1. A learner says that inverse variation means "subtract one quantity from another." Explain why this is wrong.
2. Decide whether $y = 3x + 2$ is a direct variation. Give a reason.
3. Explain why the equation $t = \frac{k}{v}$ only models speed and time when the distance is fixed.
4. In a combined variation problem, explain why units can help detect an unreasonable answer.

Generated Question Layer

• Conceptual questions: Ask learners to classify a relationship as rate, direct variation, inverse variation, joint variation, or combined variation.
• Skill questions: Generate tasks requiring learners to find $k$, write a variation equation, and substitute a new set of values.
• Application problems: Use local contexts such as speed, transport, currency exchange, classroom quantities, prices, construction mixtures, and measurement.
• Progressive sets: Start with one-step unit rates, then direct variation, then inverse variation, then combined variation with squared terms.
• Edge cases: Include non-variation equations such as $y = mx + c$, percentage-rate traps, missing units, and wording involving "square" or "cube".

Exam-Derived Signals

The first automatic 2021-2025 Paper 1 mapping counted 10 primary Rates and variations records: 1 in 2021, 3 in 2022, 2 in 2023, 2 in 2024, and 2 in 2025. These records are unreviewed extraction signals, not verified official past-question links.

The 2022 examination format crosswalk maps the format group Units/Rates and variation to this topic with 1 item and a 7.14 percent weight, but the record is marked needs_manual_review because the term "Units" does not cleanly match the current 2023 topic registry.

Recent unreviewed extracted signals include:

| Year | Question ID | Signal | | ---: | --- | --- | | 2021 | csee_041_2021_p1_q06_b | Combined variation involving mass supported by a beam, breadth, and length. | | 2022 | csee_041_2022_p1_q06_b_i | Direct proportionality between cost price and selling price, including finding $k$. | | 2022 | csee_041_2022_p1_q08_b_i | Interest-rate context involving an accumulated amount after two years. | | 2022 | csee_041_2022_p1_q08_b_ii | Interest-rate context involving interest after two years. | | 2023 | csee_041_2023_p1_q06_b_i | Writing an equation for buying price directly proportional to selling price. | | 2023 | csee_041_2023_p1_q06_b_ii | Using a direct-proportion equation after a percentage increase. | | 2024 | csee_041_2024_p1_q06_b_i | Writing an inverse-variation equation connecting time and speed. | | 2024 | csee_041_2024_p1_q06_b_ii | Using inverse variation to find a new speed for a shorter time. | | 2025 | csee_041_2025_p1_q05_a | Exchange-rate conversion between Tanzanian shillings and US dollars. | | 2025 | csee_041_2025_p1_q05_b | Combined variation involving cylinder height, volume, and radius squared. |

Many of these records are marked needs_manual_review because they overlap with ratios, inverse relations, algebra, geometry, or missing-mark flags. They should be checked against the original papers before being used as final learner-facing past-question references.

Source And Review Notes

• Official syllabus status: The topic identity, form placement, competence, source topic ID, and hub come from the 2023 CSEE Mathematics syllabus through data/curriculum_map.json.
• Official source reference: The cited syllabus file is raw/syllabuses/csee/2023/csee_mathematics_syllabus_2023.pdf.
• Exam signal status: The 2021-2025 signals come from data/topic_frequency_2021_2025.json and data/question_map_2021_2025.jsonl; they are unreviewed and should not be treated as audited past-paper references.
• Exam format status: The 2022 format crosswalk maps Units/Rates and variation to this page, but the mapping itself is marked needs_manual_review.
• Content authorship status: Explanations, worked examples, and practice tasks are original learner-facing prose written from the syllabus topic and assessment signals, not copied from textbooks, Wikipedia, or extracted solutions.
• Renderer QA: This page uses $...$ and $$...$$ math notation for compatibility with Obsidian, KaTeX, and MathJax. Some plain Markdown viewers may show the raw delimiters.