Proportional Reasoning-1 | CBSE Class 8 Maths Notes

This note covers proportional changes in images, ratios and their simplest forms, equivalent ratios, missing terms, the rule of three, comparisons with consistent units, sharing in a given ratio, mixtures, and unit conversions.

How can changes in an image remain proportional?

Proportional changes multiply related measurements by the same factor. A factor of change is the number by which an original measurement is multiplied to obtain its new value. Enlarging or reducing an image involves comparing both its width and its height.

Consider five versions of a tiger photograph. Images A, C and D look similar despite their different sizes. Images B and E are slightly distorted: the tiger appears elongated in B, and compressed and fatter in E.

The measurements below use millimetres, written mm, for both width and height. Keeping the measurement units consistent allows the changes in corresponding dimensions to be compared.

Property: Equal factors preserve the width-to-height relationship

From A to C, both measurements become half their original values. The common factor is 1/2, meaning one half; the slash denotes division. From A to D, both measurements increase by the same factor. This matching multiplicative change explains why these images look similar.

Five differently sized versions of the tiger photograph are labelled A, B, C, D and E. A, C and D retain similar proportions; B looks elongated and E looks compressed and fatter.

Reference: NCERT Class 8, page 159, unnumbered

Why does subtracting the same amount fail here?

From A to B, both width and height decrease by 20 mm. However, the height becomes half its earlier value while the width does not. Equal subtraction therefore does not preserve the image's proportions in this example.

The relevant question is whether the two dimensions change by the same factor. Merely noticing that both become smaller, or that both decrease by the same amount, does not establish proportional change.

What is a ratio, and how do we simplify it?

Definition: A ratio compares two quantities in a specified order. In a : b, the letters a and b represent the first and second terms, and the colon means “to”. For every a units of the first quantity, there are b units of the second.

The terms of a ratio are its two numbers. Image A has width-to-height ratio 60 : 40. This means that 60 mm of width corresponds to 40 mm of height. The order “width to height” tells us which measurement comes first.

A ratio can be reduced using the highest common factor, abbreviated HCF. This is the greatest number that divides both whole-number terms exactly. Dividing both terms by their HCF gives the ratio in its simplest form.

Property: Ratios with the same simplest form are proportional

Two ratios are in proportion when their simplest forms are the same. The symbol :: means “is in proportion to”. In a : b :: c : d, c and d are the first and second terms of the second ratio.

Worked example 1. Are 3 : 4 and 72 : 96 proportional?

Answer: The ratio 3 : 4 is already in simplest form. The HCF of 72 and 96 is 24. Dividing both terms by 24 gives 3 : 4. Thus, 3 : 4 :: 72 : 96.

For image A, dividing 60 and 40 by their HCF, 20, gives 3 : 2. For D, dividing 90 and 60 by 30 also gives 3 : 2. Their ratios are therefore proportional.

Image B instead gives 40 : 20, or 2 : 1, while E gives 60 : 60, or 1 : 1. Neither has the same simplest form as 3 : 2. Simplification provides a systematic way of checking the visual comparison.

How can we generate equivalent ratios and find missing terms?

Equivalent ratios express the same proportional relationship. To obtain one from another, multiply or divide both terms by the same factor. The factor can be a whole number or a fraction, depending on the required term.

The symbols × and ÷ mean multiplication and division respectively. An equals sign, =, states that the expressions on its two sides have the same value. In a fraction, the numerator is the number above the fraction line and the denominator is the number below it.

Worked example 2. Complete the ratios with second term 42, first term 6, and first term 2, each proportional to 14 : 21.

Answer: Since 42 ÷ 21 = 2, multiply 14 by 2 to obtain 28 : 42. Since 6 ÷ 14 = 3/7, multiply 21 by 3/7 to obtain 6 : 9. Dividing both 14 and 21 by 7 gives 2 : 3.

How do we choose the factor?

1. Identify a term whose original and required values are both known.
2. Divide the required value by the original value to find the factor.
3. Apply that factor to the other original term.
4. Check that both terms have changed by the same factor.

In the middle calculation above, 3/7 means three sevenths. A fractional factor is necessary because changing 14 to 6 reduces the term without dividing it by a whole number to give 6 directly.

For further practice, ratios proportional to 4 : 9 include 8 : 18, 12 : 27 and 16 : 36. Both terms have been multiplied by 2, 3 and 4 respectively. Each ratio expresses the original relationship despite using larger numbers.

For 18 : 24, the completed ratios are 3 : 4, 12 : 16, 20 : 80/3 and 27 : 36. The third answer shows that a missing term need not be a whole number. Do not replace a fractional result merely to make the ratio look simpler.

How does proportional reasoning help with recipes and quantities?

When a recipe is increased while its taste is kept the same, the related ingredient quantities must change together. First identify the two quantities being compared. Then use their ratio to work out the change needed in each.

Worked example 3. Kesang uses 10 spoons of sugar for 6 glasses of lemonade. How much sugar should she use for 18 more glasses with the same sweetness?

Answer: The ratio of glasses to spoons is 6 : 10. The factor for the additional batch is 18 ÷ 6 = 3. The sugar must also be multiplied by 3, giving 10 × 3 = 30 spoons. Thus, 6 : 10 :: 18 : 30.

The answer refers to the 18 additional glasses. The ratio compares the original batch with the new batch being prepared. Keeping this distinction clear prevents the sugar required for one batch from being confused with a combined amount.

How does a reduction work?

A kilogram, written kg, is a unit of mass. The same proportional reasoning applies when a smaller amount of food is needed, provided the amount assigned to each person is maintained.

Worked example 4. A school cook usually makes 15 kg of rice for 120 students. Only 80 students attend on a rainy day. How much rice is required at the same amount per student?

Answer: The number of students changes by the factor 80/120 = 2/3. Multiply the rice by the same factor: 15 × 2/3 = 10. The cook should make 10 kg of rice, giving 120 : 15 :: 80 : 10.

Both examples begin with a known pair of quantities and a new value for one member of the pair. The task is to preserve their relationship. In the lemonade example the factor increases both quantities; in the rice example it decreases both.

Why do equal additions and different mixtures change ratios?

An additive change adds or subtracts an amount. A multiplicative change multiplies by a factor. These are different operations, even when both quantities are affected. Age comparisons show why the distinction matters.

Worked example 5. When Neelima was 3 years old, her mother was 10 times her age. Compare their age ratio then with the ratio when Neelima is 12.

Answer: Initially, the mother's age is 30 years, so Neelima's age to her mother's age is 3 : 30 = 1 : 10. Nine years later, their ages are 12 and 39. The ratio is 12 : 39 = 4 : 13, which differs from 1 : 10.

Note: Adding or subtracting the same number from both terms does not necessarily produce a ratio proportional to the original. Compare the resulting ratios rather than treating equal additions as equal factors.

How do coffee mixtures illustrate different ratios?

Coffee decoction is the coffee liquid mixed with milk to make filter coffee. A millilitre, written mL, is a unit of volume, the amount of space occupied by a substance. Manjunath usually mixes 15 mL of decoction with 35 mL of milk for regular coffee.

For stronger coffee, he mixes 20 mL of decoction with 30 mL of milk. For lighter coffee, he mixes 10 mL of coffee with 40 mL of milk. The decoction-to-milk ratios are therefore 15 : 35, 20 : 30 and 10 : 40 respectively.

Compare the amount of decoction relative to the milk, rather than deciding from the total drink alone. The stronger mixture has more decoction and less milk than the regular mixture; the lighter mixture has less decoction and more milk.

The quantities identify which ingredient is first in each ratio. Reversing the ingredient order while keeping the terms unchanged would describe a different mixture. This is why the labels matter just as much as the calculation.

What is the rule of three, and why does cross multiplication work?

Trairasika, the rule of three, finds a fourth quantity when three quantities are known in a proportional relationship. Write the relationship as a : b :: c : d, where a and b form the known first ratio, c is known, and d is the required fourth term.

Property: Proportional ratios have equal cross products

Let f represent the common factor of change. Multiplication written without a sign, as in fa or bc, means f × a or b × c respectively. The following derivation explains the rule for the positive quantities used here.

1. Applying the common factor to the first terms gives c = fa.
2. Applying it to the second terms gives d = fb.
3. Both c/a and d/b equal f, so c/a = d/b.
4. Multiplying both sides by ab gives bc = ad, or ad = bc.

The products ad and bc are called cross products. Equating them is cross multiplication. Thus, ratios are proportional when these cross products are equal. To find the fourth term, divide bc by a: d = bc/a.

What do the traditional terms mean?

The rule associated with Āryabhaṭa multiplies the phala by the ichchhā and divides by the pramāṇa. It gives the ichchhāphala, the required yield. These names describe the same positions represented by the letters in the formula.

Cross multiplication is a way of calculating within a proportional relationship. Before using it, establish what the two ratios represent and why they should be proportional. A correct calculation cannot repair a proportion that misrepresents the situation.

Why must corresponding quantities use consistent units?

A ratio problem can connect different kinds of quantities, such as time and distance. However, the time measurements must use a consistent unit across the two ratios, and so must the distance measurements. Converting corresponding quantities prevents unlike numerical scales from being compared.

Parentheses group an expression to be evaluated together. A kilometre, written km, measures distance. A car travelling at the same speed covers distances proportional to the time spent travelling. This condition is part of the problem, not something to omit from its solution.

Worked example 6. A car travels 90 km in 150 minutes. How far does it travel in 4 hours at the same speed?

Answer: Convert 4 hours to 240 minutes. Let x be the required distance in kilometres. Then 150 : 90 :: 240 : x. Cross multiplication gives 150x = 240 × 90. Therefore, x = (240 × 90)/150 = 144. The distance is 144 km.

Writing 150 : 90 :: 4 : x would mix minutes and hours in corresponding positions. Notice that the ratio order remains time to distance in both pairs after the conversion.

How do we compare prices for different packet sizes?

A gram, written g, is a unit of mass, and 1 kg equals 1,000 g. The symbol ₹ represents rupees. Prices for unequal packet sizes need to be compared at the same weight.

Worked example 7. Tea from a small farmer in Himachal Pradesh costs ₹200 for 200 g. Tea from a large estate in Meghalaya costs ₹800 for 1 kg. Are their weight-to-price ratios proportional, and which is more expensive?

Answer: In grams and rupees, the ratios are 200 : 200 = 1 : 1 and 1000 : 800 = 5 : 4. They are not proportional. At the Himachal rate, 1 kg costs ₹1,000, compared with ₹800 in Meghalaya. The Himachal tea is more expensive.

The cheaper packet price alone does not answer the question. Comparing equal weights separates the effect of packet size from the amount charged for the tea.

When should we reject a proposed proportion?

Some problems resemble proportional situations but do not preserve the relationship being proposed. Check the context before setting up the four terms. Ask what increases, what decreases, and whether corresponding quantities change by the same factor.

Why is speed to travel time an unsuitable ratio here?

Puneeth's father takes 2 hours to travel from Lucknow to Kanpur at 50 km/h. Here km/h means kilometres per hour. The question asks about the same journey at 75 km/h.

It cannot be modelled as 50 : 2 :: 75 : an unknown travel time using the rule of three in this form. For the same journey, increasing speed makes travel time decrease. That behaviour conflicts with increasing both terms by the same factor.

Are product prices necessarily proportional to their volumes?

The following shampoo example gives volume, the amount of space occupied by the shampoo, and price for different containers. Read each row as a matched pair; do not assume the price scales with container size.

Compare sachet volume to small-bottle volume, 6 : 180, with sachet price to small-bottle price, 2 : 154. These are not proportional. The example shows why a price relationship must be checked using the data.

Note: Three known numbers are not sufficient reason to apply the rule of three. The quantities must first fit a proportional relationship, with a consistent order and consistent units for corresponding measurements.

This check is especially useful when a question asks whether ratios are proportional. “Not proportional” is a meaningful result supported by the comparison, rather than an obstacle to be removed by changing the data.

How do we share a whole in a given ratio?

Sharing in a ratio divides a total into parts whose sizes have a specified relationship. The ratio describes the parts relative to each other. To find their actual sizes, the total quantity must also be known.

If 12 counters, meaning countable objects such as seeds or pebbles, are shared equally between two people, each person gets 6. The ratio is 6 : 6, or 1 : 1. If a partner gets 5, the other person gets 7, giving partner-to-other ratio 5 : 7.

Worked example 8. Share 12 counters in the ratio 3 : 1.

Answer: Give the first person 3 counters and the second person 1 counter in each round. After three rounds, the first person has 9 and the second has 3. All 12 counters have been shared, and 9 : 3 simplifies to 3 : 1.

How does the equal-group method work?

The whole is the total being shared. The symbol + means addition. For a ratio 4 : 3, divide that whole into 4 + 3 equal groups. One person receives four groups and the other receives three.

Worked example 9. Share 42 counters in the ratio 4 : 3.

Answer: There are 4 + 3 = 7 equal groups. Each group contains 42 ÷ 7 = 6 counters. The first share is 4 × 6 = 24 counters. The second is 3 × 6 = 18 counters.

An oval labelled 42 branches to boxes labelled 4 and 3. Arrows marked × 6 connect these boxes to 24 and 18. The annotations connect the whole to seven groups.

Reference: NCERT Class 8, page 174, unnumbered

The group size links the ratio terms to the actual quantities. This avoids repeatedly handing out counters while preserving the same sharing rule. Check both the total and the ratio, because satisfying just one does not fully answer the problem.

What formula gives the shares, and how does it apply to profit?

Let x now represent the whole quantity being divided, and let m and n represent the positive terms of the required ratio m : n. The first part receives m equal groups, and the second part receives n equal groups.

Property: The sum of the ratio terms determines the group size

1. Add the ratio terms to obtain m + n, the total number of equal groups.
2. Divide the whole by this number. Each group has size x/(m + n).
3. Multiply the group size by m to obtain the first part.
4. Multiply the same group size by n to obtain the second part.

The shares are m × x/(m + n) and n × x/(m + n). Parentheses indicate that m + n is added before the division. Both ratio terms use the same group size, so the resulting parts preserve the required ratio.

How is a profit shared according to investment?

An investment is money put into a business; profit is the gain to be shared in this example. The partners' agreement specifies that their profit shares follow the same ratio as their investments.

Worked example 10. Prashanti invests ₹75,000 and Bhuvan invests ₹25,000 in a food cart business. They agree to share a ₹4,000 profit in their investment ratio. Find the shares.

Answer: The investment ratio 75000 : 25000 simplifies to 3 : 1. Its terms add to 4, so each group of profit is ₹4,000 ÷ 4 = ₹1,000. Prashanti receives 3 × ₹1,000 = ₹3,000. Bhuvan receives ₹1,000.

The investment amounts establish the sharing ratio, while ₹4,000 establishes the whole to be divided. Keeping those roles separate prevents an investment amount from being mistaken for a share of the profit.

The same method gives ₹1,800 and ₹2,700 when ₹4,500 is divided in the ratio 2 : 3. For 6 cups of an idli mixture with rice to urad dal in the ratio 2 : 1, the quantities are 4 cups and 2 cups.

How do we solve a mixture problem when one ingredient is added?

A mixture ratio compares the amounts of its ingredients. The symbol − means subtraction. When one ingredient is added, distinguish the original amount, the required final amount and the amount to add. First divide the original whole to discover how much of each ingredient it contains.

Worked example 11. A 40 kg mixture contains sand and cement in the ratio 3 : 1. How much cement must be added to make the sand-to-cement ratio 5 : 2?

Answer: The original sand is (3/4) × 40 = 30 kg, and the original cement is (1/4) × 40 = 10 kg. The sand remains 30 kg. For the new ratio, 5 : 2 :: 30 : 12, so 12 kg of cement is required in total. Add 12 − 10 = 2 kg of cement.

Here, subtraction is used after finding the final cement amount. The unchanged 30 kg of sand connects the old mixture with the new one; the original total is not the final total after an addition.

How does the same reasoning apply to paint?

Blue and yellow paints are mixed in the ratio 3 : 5 to make 40 mL of green paint. The shares are 15 mL of blue and 25 mL of yellow. These quantities describe the original mixture before any extra paint is added.

Adding 20 mL of yellow changes the yellow amount to 45 mL while the blue remains 15 mL. The new ratio is 15 : 45, or 1 : 3. It is the yellow ingredient, rather than both ingredients, that has increased.

In both examples, the first calculation is a sharing problem and the next is a comparison of ingredient amounts. Writing the ingredient names beside the numbers helps identify which quantity remains unchanged and which quantity must be updated.

Finish by checking precisely what is requested. An answer for the final quantity of cement is different from an answer for the extra cement. Likewise, a new paint ratio must use the amounts after the addition.

Which unit conversions are useful in proportional reasoning?

Solving proportionality problems often requires conversion between units. A conversion gives the same measurement in another unit. Length, area and volume must each use the appropriate conversion, because they describe different types of measurement.

Area measures surface extent and uses square units. Volume measures occupied space and uses cubic units or units such as litres. A foot, plural feet, is a length unit; an acre and a hectare are area units.

Here cc abbreviates cubic centimetre, a unit of volume. Use the volume conversion when moving between litres and millilitres, and the area conversion when a land measurement changes between acres and square feet.

Worked example 12. A tap fills a 500 mL mug in 15 seconds. At the same filling rate, how long does it take to fill a 10-litre bucket?

Answer: Since 1 litre = 1,000 mL, the bucket holds 10,000 mL. This is 20 times the mug's volume. The required time is 15 × 20 = 300 seconds.

How are Celsius and Fahrenheit related?

Celsius and Fahrenheit are temperature scales. The symbols °C and °F mean degrees Celsius and degrees Fahrenheit. Their conversion includes an added or subtracted number, making it more complicated than multiplication alone.

Fahrenheit = (9/5) × Celsius + 32. In this formula, “Celsius” means the numerical Celsius reading and “Fahrenheit” means the numerical Fahrenheit reading. The reverse conversion is Celsius = (5/9) × (Fahrenheit − 32).

The corresponding readings include 0 °C = 32 °F and 25 °C = 77 °F. When using the reverse formula, subtract 32 before multiplying by 5/9. The parentheses specify that order.

Glossary

• Ratio — A comparison expressing how much of one quantity corresponds to another in a specified order.
• Terms of a ratio — The two numbers that represent the compared quantities in their stated order.
• Factor of change — The number multiplying an original term to produce its corresponding new term.
• Highest common factor — The greatest whole number that divides both whole-number terms exactly, used when simplifying a ratio.
• Simplest form — A ratio obtained by dividing both whole-number terms by their highest common factor.
• Proportion — A relationship between two ratios that have the same simplest form.
• Equivalent ratios — Ratios describing the same proportional relationship, with corresponding terms changed by the same factor.
• Cross multiplication — Equating the products of opposite terms in a proportion to check or calculate a term.
• Rule of three — Finding a fourth quantity from three known quantities linked by a proportional relationship.
• Whole — The complete quantity that is divided into parts according to a specified ratio.
• Group size — The whole divided by the sum of the ratio terms when sharing proportionally.
• Unit conversion — Expressing a measurement in another unit so corresponding quantities can be compared consistently.

Common errors and misconceptions

• Misconception: Subtracting the same amount from image width and height preserves its appearance. Correct: A to B loses 20 mm in each dimension, but the factors differ and the image is slightly distorted.
• Misconception: Equivalent ratios need larger whole-number terms. Correct: Scaling can reduce the terms and can use fractions, as 14 : 21 becomes 6 : 9 using 3/7.
• Misconception: Equal increases in ages preserve the age ratio. Correct: Neelima's age ratio changes from 1 : 10 to 4 : 13 after nine years.
• Misconception: The car problem can compare 150 minutes directly with 4 hours as numerical terms. Correct: Convert 4 hours to 240 minutes before constructing the proportion.
• Misconception: Any three given numbers justify the rule of three. Correct: Check the relationship first; the proposed speed-to-time proportion fails for the same journey.
• Misconception: Sharing 42 in 4 : 3 requires dividing by either 4 or 3. Correct: Divide by their sum, 7, to find each equal group's size.
• Misconception: The mixture needs 12 kg of extra cement. Correct: It needs 12 kg in total; subtract the existing 10 kg to find the 2 kg addition.

Exam-style questions with model answers

Q1. Are 3 : 4 and 72 : 96 proportional? Justify your answer using simplest forms. [2 marks]

1. The HCF of 72 and 96 is 24. Dividing both terms by 24 gives 72 : 96 = 3 : 4.
2. The other ratio is already 3 : 4. Their simplest forms match, so the two ratios are proportional.

Q2. Kesang uses 10 spoons of sugar for 6 glasses of lemonade. How many spoons are needed for 18 more glasses with the same sweetness? Show the factor and the resulting proportion. [3 marks]

1. The original glasses-to-sugar ratio is 6 : 10. To maintain the same sweetness, the two quantities for the additional batch must change by the same factor.
2. The number of glasses changes from 6 to 18, giving the factor 18 ÷ 6 = 3.
3. Multiply 10 spoons by 3 to obtain 30 spoons for the additional batch. The resulting proportion is 6 : 10 :: 18 : 30.

Q3. A car travels 90 km in 150 minutes. At the same speed, how far will it travel in 4 hours? Use 4 hours = 240 minutes, define your unknown and apply cross multiplication. [4 marks]

1. Let x be the required distance in kilometres. Use the given conversion to express the second travel time as 240 minutes.
2. Since speed remains the same, the time-to-distance ratios are proportional: 150 : 90 :: 240 : x.
3. Cross multiplication gives 150x = 240 × 90. Dividing both sides by 150 gives x = (240 × 90)/150.
4. Evaluating this expression gives x = 144. Therefore, the car covers 144 km in the stated four hours.

Q4. Neelima is 3 years old and her mother is 10 times her age. Find their age ratio now and when Neelima is 12. Explain whether equal additions preserve this ratio. [3 marks]

1. Her mother's current age is 3 × 10 = 30 years. Neelima's age to her mother's age is 3 : 30, simplifying to 1 : 10.
2. Neelima reaches 12 after nine years. Her mother is then 39, so the ratio becomes 12 : 39, simplifying to 4 : 13.
3. The simplest forms differ. Adding nine to both ages does not preserve this ratio; equal additions do not necessarily produce proportional ratios.

Q5. Prashanti invests ₹75,000 and Bhuvan invests ₹25,000. They agree to share a ₹4,000 profit in the ratio of their investments. Find each share, explaining the ratio, group total and group size. [5 marks]

1. Use Prashanti's investment as the first term and Bhuvan's as the second. Their investment ratio is 75000 : 25000.
2. Divide both terms by 25,000 to simplify the ratio to 3 : 1. This is the agreed ratio for sharing the profit.
3. Add the ratio terms: 3 + 1 = 4. The ₹4,000 profit must therefore be divided into four equal groups.
4. Each group is worth ₹4,000 ÷ 4 = ₹1,000. Prashanti receives three groups, so her profit share is ₹3,000.
5. Bhuvan receives one group, worth ₹1,000. The two shares together distribute the full ₹4,000 in the required ratio of 3 : 1.

Q6. A 40 kg mixture contains sand and cement in the ratio 3 : 1. Only cement is added until the sand-to-cement ratio becomes 5 : 2. Find the original quantities, the required final cement and the cement to add. [5 marks]

1. The original ratio has 3 + 1 = 4 equal groups. Each group represents 40 ÷ 4 = 10 kg of the mixture.
2. Sand occupies three groups, so its original mass is 3 × 10 = 30 kg. This amount remains unchanged because only cement is added.
3. Cement occupies one original group, so there are initially 10 kg of cement. Keep this amount separate from the final cement requirement.
4. For the new ratio, 5 : 2 :: 30 : 12. Since the sand stays at 30 kg, the final cement must be (2/5) × 30 = 12 kg.
5. Subtract the cement already present from the final requirement: 12 − 10 = 2 kg. Therefore, add 2 kg of cement to the original mixture.

Q7. A tap fills a 500 mL mug in 15 seconds. How long will it take to fill a 10-litre bucket at the same filling rate? Use 1 litre = 1,000 mL. [3 marks]

1. Convert the bucket's volume using the supplied conversion: 10 litres = 10 × 1,000 = 10,000 mL. Both volumes now use the same unit.
2. The bucket's volume is 10,000 ÷ 500 = 20 times the mug's volume. At the same filling rate, the time increases by this factor.
3. The filling time is therefore 15 × 20 = 300 seconds. This is the time required for the bucket.

Key takeaways

• Proportional changes multiply corresponding quantities by the same factor, preserving the relationship between them.
• A ratio has an order: identify which quantity is represented by each term before calculating.
• Dividing both whole-number terms by their highest common factor gives the ratio in simplest form.
• Ratios with the same simplest form are proportional, and their cross products are equal.
• Keep corresponding units consistent, as in converting four hours to 240 minutes before comparing travel times.
• When sharing a whole, add the ratio terms and divide the whole by this total.
• For mixture additions, distinguish the original ingredient amount, its required final amount and the amount to add.
• Check the context before applying the rule of three; quantities that change together are not necessarily proportional.

Test yourself

What does the ratio a : b mean, where a and b are its first and second terms?

For every a units of the first quantity, there are b units of the second quantity.

Why do image dimensions 60 mm by 40 mm and 30 mm by 20 mm remain proportional?

Both width and height are multiplied by the same factor, one half, so their relationship is preserved.

What is the simplest form of 60 : 40?

Divide both terms by their HCF, 20, to obtain the simplest form 3 : 2.

What first term completes a ratio with second term 42 that is proportional to 14 : 21?

The first term is 28 because both terms of 14 : 21 are multiplied by 2.

What cross-product relation follows from a : b :: c : d?

The relation is ad = bc, meaning a multiplied by d equals b multiplied by c.

How are 42 counters shared in the ratio 4 : 3?

Divide 42 into seven groups of six. The resulting shares are 24 counters and 18 counters.

A mixture already has 10 kg of cement and needs 12 kg in total. How much should be added?

Add 2 kg of cement, found by subtracting the existing 10 kg from the required 12 kg.

Why cannot a faster speed for the same journey be modelled by increasing travel time proportionally?

The time for the same journey decreases when speed increases, so the proposed proportional increase misrepresents the situation.