Percentage Change, Rates and Scale | How Multiplicative Reasoning Extends Beyond Ratio

Mathematics Learning Guide · Percentage change, rates, scale and proportional reasoning

Ratios, percentages, rates and scale factors are not four unrelated school topics. They are different ways of describing multiplicative relationships. Once students learn to identify the base quantity, the comparison quantity and the scale connecting them, a large family of problems becomes easier to understand, model and verify.

Contents

Part I — The multiplicative idea

1. One structure, many representations
2. The base quantity
3. Fractions and percentages
4. Ratios and percentages
5. Multiplicative factors
6. Benchmarking before calculation

Part II — Percentage increase and decrease

7. Percentage increase
8. Percentage decrease
9. Reverse percentages
10. Percentage points
11. Successive changes
12. Why equal percentage increases and decreases do not cancel

Part III — Rates and unit rates

13. What makes a rate different
14. Unit rates
15. Speed as a rate
16. Unit price
17. Density and other compound rates
18. Proportional and non-proportional rates

Part IV — Scale and similarity

19. Scale factor
20. Map scales
21. Similar figures
22. Area scale factors
23. Volume scale factors
24. Common scale errors

Part V — Diagnosis, transfer and examination control

25. The five-step multiplicative model
26. Common error families
27. Language that changes the relationship
28. Mixed practice
29. Verification
30. Worked casebook
31. What to remember

1. One structure, many representations

Suppose 18 students out of a group of 30 complete a task. The same relationship can be expressed as the fraction 18/30, the simplified fraction 3/5, the decimal 0.6, or the percentage 60%. Each representation highlights something different, but the underlying part-whole relationship is unchanged.

Now suppose another class has 24 completers out of 40. The raw counts have changed, but 24/40 also simplifies to 3/5 and therefore 60%. This is the power of multiplicative representation: it lets us compare structures rather than only absolute counts.

Ratios, percentages, rates and scale factors all depend on this kind of structural thinking. The student must decide which quantities are being compared, identify the reference or base quantity, and then preserve the relationship as the representation changes.

This is why a student who can calculate 15% of a number may still struggle with a reverse percentage problem. The arithmetic itself is not the whole skill. The learner must recognise which quantity represents 100%, which quantity represents the changed amount, and how the multiplier connects them.

A useful habit is to write a relationship sentence before calculation: “new = factor × original,” “cost = rate × quantity,” “drawing length = scale factor × real length,” or “part = percentage × whole.” These sentences expose the multiplicative skeleton of the problem.

The long-term goal is not to memorise one sentence for every chapter. It is to recognise a recurring pattern: one quantity is generated from another through multiplication by a factor.

Contents · Next: the base quantity

2. The base quantity

Percentage problems are easiest when the 100% quantity is identified first. This quantity is the base or reference. Every percentage is measured relative to something, and a large proportion of percentage errors are really base-selection errors.

If a shirt costs $80 and is discounted by 25%, the original $80 is the base. Twenty-five percent of $80 is $20, so the sale price is $60. If the sale price is $60 after a 25% discount and the original price is unknown, then $60 represents 75% of the base. The direction of reasoning has changed.

Students often apply a percentage to the number closest to it in the sentence. That can fail. “A is 20% more than B” uses B as the base, so A=1.2B. “A is 20% of B” gives A=0.2B. “B is 20% less than A” uses A as the base, so B=0.8A. Small language changes move the reference quantity.

A base can also be a subgroup. If 40% of the students who passed chose option A, the 100% reference set is the students who passed, not the entire class. This mirrors the denominator decision in fraction work.

Before any percentage calculation, ask: what quantity is being treated as the whole or original reference? Mark it as 100%. This simple move prevents many errors.

Previous · Contents · Next: fractions and percentages

3. Fractions and percentages

A percentage is a fraction with a standard denominator of 100. The fraction 3/5 can be converted to a percentage by writing an equivalent fraction out of 100: 3/5 = 60/100 = 60%. Alternatively, divide 3 by 5 to obtain 0.6 and multiply by 100%.

The representation “60%” does not mean sixty physical objects unless the whole actually contains 100 objects. It means sixty hundredths of the reference quantity. In a class of 30, 60% corresponds to 18 students; in a class of 40, it corresponds to 24.

Benchmark fractions support quick conversions. One half is 50%, one quarter is 25%, three quarters is 75%, one fifth is 20%, one tenth is 10%, and one eighth is 12.5%. These benchmarks help students estimate before calculating exactly.

Percentages greater than 100% are valid. If a quantity becomes 1.4 times its original size, it is 140% of the original. This is especially important in growth problems because students sometimes assume percentages must remain below 100.

Likewise, percentages can describe change rather than state. “The new value is 140% of the original” and “the value increased by 40%” are two ways of describing the same multiplier 1.4. The reference quantity is the original in both statements.

Previous · Contents · Next: ratios and percentages

4. Ratios and percentages

If red:blue = 3:2, the combined whole contains five ratio units. Red is therefore 3/5 of the whole, which is 60%. Blue is 2/5, which is 40%.

A common error is to divide 3 by 2 and say red is 150% of the whole. In fact, 3/2 = 150% expresses red relative to blue. That may be a meaningful comparison, but it is not red as a percentage of the combined group.

The phrase “percentage of what?” is therefore essential. Ratio-to-percentage conversion requires a deliberate choice of reference set. If the whole is all ratio parts, sum the parts first.

With a three-part ratio 2:3:5, the whole contains ten units. The three categories represent 20%, 30% and 50%. This conversion is useful in data interpretation because it connects relative composition to a standard scale.

Students should also recognise that the reverse process is possible. If two exhaustive categories are 60% and 40% of a whole, their ratio is 60:40 = 3:2. Percentage and ratio representations are alternative views of the same partition.

Previous · Contents · Next: multiplicative factors

5. Multiplicative factors

Percentage change becomes simpler when expressed as a multiplicative factor. Increasing by 20% means multiplying by 1.20. Decreasing by 20% means multiplying by 0.80. Increasing by 7% means multiplying by 1.07. Decreasing by 7% means multiplying by 0.93.

The factor contains the original 100% plus or minus the change. This prevents the student from calculating the percentage change separately when it is not necessary.

If a quantity of 250 increases by 12%, the new value is 250×1.12 = 280. If a quantity of 250 decreases by 12%, the new value is 250×0.88 = 220.

The factor form is especially powerful in reverse problems. If a value after a 12% increase is 280, then original×1.12=280, so original=250. The multiplier tells us exactly what percentage of the original the final amount represents.

This language also connects percentage work to scale factors. A 12% increase is a scale factor of 1.12. Percentage change and geometric scaling therefore share a common multiplicative structure.

Previous · Contents · Next: benchmark before calculation

6. Benchmarking before calculation

Before calculating a percentage exactly, estimate its size. Ten percent of 480 is 48, so 21% should be a little more than 96. An answer of 9.6 or 960 should immediately look suspicious.

Benchmarks reduce dependence on calculators. One percent is one hundredth. Five percent is half of ten percent. Twenty-five percent is one quarter. Fifty percent is one half. Seventy-five percent is three quarters.

For percentage increase, estimate direction as well as magnitude. A 30% increase must produce a value greater than the original, but less than one and a half times the original. A 70% decrease leaves only 30% of the original.

Benchmarking is valuable because it creates an independent check. Re-entering the same expression into a calculator may repeat a modelling error; comparing the result with an expected range tests plausibility differently.

This habit is transferable to rates and scale. A map distance of 3 cm representing 30 km implies 10 km per centimetre. If a later calculation produces 3000 km for 4 cm, the result conflicts with the scale estimate.

Previous · Contents · Next: percentage increase

7. Percentage increase

A percentage increase compares the amount of increase with the original quantity. If a value rises from 80 to 100, the increase is 20. Relative to the original 80, the increase is 20/80 = 25%. The new value is therefore 125% of the original.

This distinction between amount and percentage matters. An increase of 20 units does not imply a 20% increase unless the original quantity is 100.

Students should identify three quantities: original, change, and new. They are linked by new = original + change, while percentage increase = change/original × 100%.

If the percentage increase is known instead, the multiplier form is often shortest: new = original × (1 + rate). For a 25% increase, new = original×1.25.

Percentage increase can exceed 100%. If a population grows from 50 to 125, the increase is 75, which is 150% of the original. The new population is 250% of the original.

Previous · Contents · Next: percentage decrease

8. Percentage decrease

A percentage decrease compares the amount lost with the original quantity. If a price falls from $120 to $90, the decrease is $30. Relative to $120, that is 30/120 = 25%.

The new value is 75% of the original, so the multiplier is 0.75. This is the complement relationship: after a 25% decrease, 75% remains.

A common mistake is to divide the decrease by the new value. That would answer a different question: how large the decrease is relative to the final amount.

Again, the base determines the percentage. School problems usually define percentage change relative to the original unless stated otherwise.

When several decreases occur successively, do not simply add the percentages unless the question only asks for an approximation. Each change acts on a new base, so multipliers must be composed.

Previous · Contents · Next: reverse percentages

9. Reverse percentages

Reverse percentage problems ask for the original quantity when a changed quantity is known. These problems expose whether students understand the base rather than only the forward calculation.

If a price after a 20% discount is $72, then $72 represents 80% of the original. Let the original be x. Then 0.8x=72, so x=90.

A common wrong method is to add 20% of $72 to $72. That produces $86.40, but the 20% discount was calculated from the original price, not the discounted price. The bases differ.

The multiplier method makes the relationship explicit. Forward: original×0.8=final. Reverse: final÷0.8=original.

The same logic applies to increases. If a quantity after a 15% increase is 230, then original×1.15=230, so original=200.

A useful self-check is to run the original forward again. If 200 increases by 15%, the result should return to 230. Reverse solutions should reconstruct the given final state.

Previous · Contents · Next: percentage points

10. Percentage points

Percentage points describe the absolute difference between two percentages. If a success rate rises from 40% to 55%, the change is 15 percentage points.

Percentage increase asks a different question. Relative to the original 40%, the increase is 15/40 = 37.5%.

Both statements are correct but describe different comparisons. “Up 15 percentage points” means subtracting the two rates. “Up 37.5%” means measuring the increase relative to the original rate.

This distinction is common in statistics, finance and public reporting. Students should not assume that a percentage changing by ten percentage points has changed by ten percent.

A strong habit is to write the original rate, new rate and difference before deciding which comparison the question requests.

Previous · Contents · Next: successive changes

11. Successive changes

Successive percentage changes are multiplicative because each change acts on the current value. A 10% increase followed by another 10% increase uses the multiplier 1.1×1.1=1.21, producing an overall 21% increase.

Likewise, a 20% discount followed by a 10% discount gives 0.8×0.9=0.72, so the final value is 72% of the original, an overall 28% decrease.

Adding the discounts to get 30% would be wrong because the second 10% is applied to the already reduced value, not the original.

Multipliers make compound change transparent. In finance, repeated growth over time produces compounding. In depreciation, repeated percentage losses shrink the base each time.

The general principle is that multiplicative changes compose by multiplication, not by adding their percentage labels unless the bases remain identical.

Previous · Contents · Next: why changes do not cancel

12. Why equal percentage increases and decreases do not cancel

A 20% increase followed by a 20% decrease does not return a quantity to its starting value. Start with 100. Increase by 20% to 120. Then decrease 120 by 20%, which removes 24 and leaves 96.

The reason is that the two percentages use different bases. The increase uses 100; the decrease uses 120.

In multiplier form, 1.2×0.8=0.96. The combined effect is a 4% decrease.

To reverse a 20% increase exactly, divide by 1.2, which is equivalent to multiplying by 5/6, approximately 0.8333. That is a decrease of about 16.67% from the increased value, not 20%.

This is an important example of how additive intuition can fail in multiplicative systems. Equal-looking percentage labels do not guarantee symmetric effects.

Previous · Contents · Next: rates

13. What makes a rate different

A rate compares quantities measured in different units. Examples include kilometres per hour, dollars per kilogram, litres per minute and words per second.

Because the units differ, the quotient carries a compound unit. Sixty kilometres divided by two hours gives thirty kilometres per hour. The unit itself communicates the comparison.

Rates are closely related to ratios, but they are often treated separately because unit conversion matters and because the two quantities may vary continuously rather than counting categories.

A rate need not be constant. A car may have an average speed of 60 km/h over a journey while travelling faster and slower at different moments. The average rate summarises total distance divided by total time.

Understanding rates prepares students for slope, density, probability density, concentration and derivatives. The question “how much of one quantity per unit of another?” is mathematically powerful.

Previous · Contents · Next: unit rates

14. Unit rates

A unit rate expresses one quantity per one unit of another. If 12 oranges cost $9, the unit price is $9÷12=$0.75 per orange.

Unit rates make comparison easier. If another shop sells 8 oranges for $6.40, the unit price is $0.80 per orange, so the first offer is cheaper per orange.

The unit rate is a normalised comparison. It places different packages, distances or times onto a common denominator of one unit.

Students should retain units throughout the calculation. A result of 0.75 without “dollars per orange” loses the meaning that makes the comparison useful.

When quantities have inconvenient units, convert before comparing. A 750 g packet and a 1.2 kg packet should be compared using consistent mass units or directly via cost per kilogram.

Previous · Contents · Next: speed

15. Speed as a rate

Speed is distance travelled per unit time. Average speed = total distance ÷ total time. The phrase “per hour” indicates the unit rate.

If a cyclist travels 45 km in 3 hours, the average speed is 15 km/h. If the cyclist travels the same distance in 2.5 hours, the average speed becomes 18 km/h.

Students often average speeds incorrectly. If a journey includes one hour at 40 km/h and one hour at 60 km/h, the average is 50 km/h because the times are equal. But if equal distances are travelled at 40 and 60 km/h, the time spent at each speed differs, so the simple arithmetic mean is not generally correct.

The correct principle is always total distance divided by total time. This returns attention to the underlying quantities rather than a memorised averaging rule.

Speed problems therefore combine rate reasoning, state tracking and unit conversion. Minutes must be converted consistently with hours when necessary.

Previous · Contents · Next: unit price

16. Unit price

Unit price is a rate connecting cost to quantity. It is useful because total price alone can be misleading when package sizes differ.

If 1.5 kg of rice costs $6, then the unit price is $4 per kilogram. If 2 kg costs $7.60, the unit price is $3.80 per kilogram, making the larger package cheaper per kilogram.

Real pricing can include promotions, membership discounts or fixed charges, so unit price may not always scale proportionally. Mathematics problems often assume a simple proportional model unless stated otherwise.

Students should separate the mathematical model from real-world assumptions. A constant unit price means cost = rate × quantity. A fixed delivery fee changes the model to cost = fixed fee + rate × quantity.

This distinction anticipates linear functions. Pure proportional relationships pass through the origin; relationships with a fixed starting cost do not.

Previous · Contents · Next: density

17. Density and other compound rates

Density compares mass with volume: density = mass ÷ volume. If a material has mass 540 g and volume 200 cm³, its density is 2.7 g/cm³.

The units show that this is not a part-whole fraction. Grams and cubic centimetres are different quantities. The quotient describes how much mass exists per unit volume.

Other compound rates include population density, energy consumption per hour, rainfall per day and fuel economy. Each rate creates a new derived quantity.

Students should not cancel units casually unless they understand the operation. Unit algebra becomes increasingly useful in science because it can reveal impossible expressions and guide conversions.

The same multiplicative reasoning remains present: one quantity is linked to another through a rate factor.

Previous · Contents · Next: proportional rates

18. Proportional and non-proportional rates

A proportional relationship has a constant multiplicative factor and passes through the origin when graphed. If five notebooks cost $10 and every notebook has the same price, ten notebooks cost $20 and zero notebooks cost $0.

A non-proportional linear relationship may still have a constant rate of change but include a fixed starting amount. A taxi fare with a $4 flag-down charge plus $2 per kilometre has equation cost = 4 + 2d.

Doubling distance does not double total cost because the fixed $4 remains. The ratio cost:distance therefore changes with distance.

This distinction is important because students sometimes assume every table with steady increases is proportional. Check whether the ratio between corresponding quantities is constant and whether zero corresponds to zero.

Recognising proportionality prepares students for functions, graph interpretation and algebraic modelling.

Previous · Contents · Next: scale

19. Scale factor

A scale factor multiplies every corresponding linear dimension by the same amount. If a shape is enlarged by scale factor 3, every corresponding length becomes three times as large.

This is the same multiplicative structure seen in equivalent ratios. If an original length of 4 cm becomes 12 cm, the scale factor is 3.

Scale factors smaller than one produce reductions. A scale factor of 0.5 halves every linear dimension.

Students should distinguish scale factor from added length. Enlarging a 4 cm side by scale factor 3 gives 12 cm, not 7 cm. Multiplicative and additive changes are different.

The language “three times as long” is multiplicative; “three centimetres longer” is additive. This is the same distinction that appears in ratio and percentage work.

Previous · Contents · Next: map scales

20. Map scales

A map scale connects a distance on the map with the corresponding real-world distance. A scale of 1:50,000 means one unit on the map represents 50,000 of the same units in reality.

If the map uses centimetres, 1 cm represents 50,000 cm = 500 m = 0.5 km. Therefore 6 cm represents 3 km.

Unit conversion is part of the problem. The scale ratio itself is dimensionless only when both lengths use the same unit.

A common error is to multiply before converting and then forget what the result measures. Writing units through each step prevents this.

Map scale problems are ratio problems in disguise. The scale factor links representation to reality.

Previous · Contents · Next: similar figures

21. Similar figures

Similar figures have equal corresponding angles and proportional corresponding side lengths. A single linear scale factor links all corresponding lengths.

If triangle A has sides 3, 4 and 5 and triangle B is similar with scale factor 2, its corresponding sides are 6, 8 and 10.

Students should match corresponding sides before applying ratios. Using a correct scale factor on non-corresponding sides can still produce a wrong answer.

Similarity is therefore both a ratio problem and a correspondence problem. Labels, angle markings and side order matter.

Algebraic similarity questions often create equations from equal ratios. The multiplicative relationship is the same one learned earlier through ratio units.

Previous · Contents · Next: area scale factors

22. Area scale factors

If every linear dimension is multiplied by scale factor k, area is multiplied by k². This happens because area involves two dimensions.

A square with side 3 cm has area 9 cm². Enlarge the side by factor 2 to 6 cm. The area becomes 36 cm², which is four times the original. Four = 2².

Students sometimes use the linear scale factor directly on area. This fails because both length and width have been scaled.

The same principle applies to all similar plane figures. If the length scale factor is 5, the area scale factor is 25.

Reverse problems require square roots. If two similar figures have area ratio 49:81, the corresponding linear ratio is 7:9.

Previous · Contents · Next: volume scale factors

23. Volume scale factors

If every linear dimension is multiplied by k, volume is multiplied by k³ because three dimensions are scaled.

A cube with side 2 cm has volume 8 cm³. Enlarge it by scale factor 3 to side 6 cm. The volume becomes 216 cm³. The volume scale factor is 27 = 3³.

Students should avoid assuming that doubling lengths doubles volume. Doubling each dimension creates eight times the volume.

This principle matters in real modelling because large objects do not scale in all properties linearly. Surface area and volume respond differently to changes in size.

Reverse volume problems use cube roots. If similar solids have volume ratio 64:125, their linear ratio is 4:5.

Previous · Contents · Next: scale errors

24. Common scale errors

Error 1: additive scaling. A student adds the same amount to every side instead of multiplying by a common factor.

Error 2: wrong correspondence. The correct factor is applied between non-matching sides.

Error 3: unit mismatch. A map scale is used before centimetres, metres and kilometres are made consistent.

Error 4: linear factor used for area. Area should scale by k².

Error 5: linear factor used for volume. Volume should scale by k³.

Each error can be diagnosed by asking what quantity the scale factor applies to. Length, area and volume are related, but they do not scale identically.

Previous · Contents · Next: examination control

25. The five-step multiplicative model

1. Name the quantities. What is being compared?
2. Identify the base or reference. Which quantity is 100%, one unit, or the original scale?
3. Choose the factor. Percentage multiplier, unit rate, ratio unit or scale factor.
4. Apply the factor in the correct direction. Forward or reverse?
5. Verify using the original relationship. Does the answer reconstruct the stated percentage, rate or scale?

This model covers a wide range of questions. The surface language changes, but the same decisions recur.

For a percentage increase, the base is the original and the factor might be 1.2. For unit price, the base unit is one kilogram and the factor might be $4/kg. For a map, the factor connects drawing length to real length.

The most important distinction is between identifying the relationship and executing the arithmetic. Correct arithmetic cannot repair a wrong base or wrong direction.

Under examination pressure, use a short annotation rather than a long explanation: “100%=80,” “×1.15,” “$3.80/kg,” or “length SF=4, area SF=16.” These notes preserve the structure.

Previous · Contents · Next: error families

26. Common error families

Base errors: the percentage is applied to the wrong reference quantity.

Direction errors: a reverse problem is treated as a forward problem.

Additive errors: a multiplicative relationship is replaced by adding a fixed amount.

Unit errors: quantities with incompatible units are compared without conversion.

State errors: before-and-after values are mixed.

Dimension errors: a linear scale factor is incorrectly used for area or volume.

Computation errors: the model is sound but arithmetic fails.

Classifying errors by family helps students repair the actual weak link rather than repeating entire chapters.

Previous · Contents · Next: language

27. Language that changes the relationship

Small wording changes can create different mathematics. “20% of” means multiply by 0.2. “20% more than” means multiply by 1.2. “20% less than” means multiply by 0.8.

“Per” often signals a rate. “Three dollars per kilogram” means cost divided by mass. “One centimetre represents two kilometres” signals scale.

“Compared with the original” establishes a base. “Among those who passed” creates a subgroup reference set. “After the increase” describes a changed state.

Students should avoid relying on isolated keywords because words can appear in many contexts. The whole sentence should be translated into a relationship statement.

One useful question is: if I replaced the numbers with variables, what equation would this sentence create? That forces meaning to be preserved while surface numbers disappear.

Previous · Contents · Next: mixed practice

28. Mixed practice

Mixed practice should force students to choose among related multiplicative structures rather than repeat one method mechanically.

• A quantity of 240 increases by 15%. Find the new value.
• A value after a 15% increase is 276. Find the original.
• Six notebooks cost $14.40. Find the unit price.
• A map scale is 1:25,000. What real distance is represented by 8 cm?
• Two similar figures have linear scale factor 3. Find the area scale factor.
• A success rate rises from 48% to 60%. Find the change in percentage points and the relative percentage increase.

The value of this set lies in method selection. The numbers are not especially difficult. The learner must identify the base, units or scale before calculation.

After completion, classify any error. Was the wrong factor chosen? Was the base wrong? Was the direction reversed? Was a unit conversion missed?

This turns mixed practice into diagnosis rather than a score alone.

Previous · Contents · Next: verification

29. Verification

Multiplicative problems benefit from independent verification.

For percentage change, run the solved original forward and check that it recreates the final value. For rates, multiply the unit rate by the quantity and recover the total. For scale, apply the factor to a known dimension and compare with the corresponding dimension.

Check direction. An increase should move the value upward; a discount should move it downward. A scale factor greater than one should enlarge linear dimensions.

Check units. Dollars per kilogram should not end as kilograms per dollar unless the reciprocal was deliberately requested.

Check magnitude. A 5% change should normally be modest relative to the base. An area factor should be more dramatic than the corresponding linear factor when the scale factor exceeds one.

These checks are quick once practised and catch modelling errors that a calculator cannot detect.

Previous · Contents · Next: worked casebook

30. Worked casebook

Case A — percentage increase

A quantity of 360 increases by 12%. Multiplier = 1.12. New value = 360×1.12 = 403.2. Check: increase = 43.2; 43.2/360 = 0.12.

Case B — reverse discount

A sale price of $68 is after a 15% discount. The sale price represents 85% of the original. Original = 68÷0.85 = $80. Check: 15% of 80 is 12, and 80−12=68.

Case C — successive changes

A value increases by 10% and then decreases by 10%. Combined multiplier = 1.1×0.9=0.99. Final value is 99% of original, an overall 1% decrease.

Case D — unit rate

2.4 kg of fruit costs $9.60. Unit price = 9.60÷2.4 = $4/kg. At the same rate, 3.5 kg costs $14.

Case E — map scale

Scale 1:50,000. A 7 cm map distance represents 350,000 cm = 3.5 km. The important step is keeping units consistent before converting.

Case F — similar figures

Two similar figures have corresponding side lengths 6 cm and 15 cm. Linear scale factor = 15/6 = 2.5. Area scale factor = 2.5² = 6.25. Volume scale factor for corresponding similar solids would be 2.5³ = 15.625.

Case G — percentage points

A rate rises from 32% to 40%. Change = 8 percentage points. Relative increase = 8/32 = 25%. Both descriptions are correct but answer different questions.

Case H — non-proportional price

A delivery service charges $5 plus $2 per kilometre. Cost for 4 km = $13; cost for 8 km = $21. Doubling distance does not double cost because the fixed fee remains. The relationship is linear but not proportional.

Across all these cases, the same discipline applies: identify the quantities, locate the base, choose the correct multiplicative relationship, calculate, and verify.

Previous · Contents · Next: summary

31. What to remember

Percentages, rates and scale factors are connected by multiplicative reasoning.

For percentages, identify the 100% base. For percentage change, compare change with the original unless the problem states another reference. For reverse percentages, divide by the multiplier that created the final value.

For rates, keep units visible and reduce to a useful unit rate when comparison is needed. Do not assume every rate relationship is proportional if fixed charges or changing conditions exist.

For scale, distinguish additive change from multiplicative enlargement. Linear dimensions scale by k, areas by k² and volumes by k³.

The central question remains the same throughout: what quantity is being compared with what reference, and what multiplier connects them?

Once that structure is clear, calculation becomes shorter, checking becomes stronger, and ideas that once looked like separate chapters begin to form one coherent mathematical system.

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