Ratio and Proportion | ICSE Class 10 Maths Notes

This note covers ratios, proportion, continued proportion, mean proportion, invertendo, alternendo, componendo, dividendo, combinations of these properties, and simple applications involving cost, distance, time, mass and shadows.

What does a ratio tell us about two quantities?

A ratio compares two quantities by division. Let a and b stand for positive numerical quantities. Their ratio, written a : b, means a/b. The colon means “is to”, while the slash indicates division. In a/b, a is the numerator, the term being divided; b is the denominator, the term that divides it. Order identifies which quantity is compared with which.

The first term, a, is the antecedent; the second term, b, is the consequent. A term is an individual quantity in the ratio. The consequent cannot be zero because division by zero is undefined. We use positive original quantities throughout the explanations below.

How do order and units affect a ratio?

For two quantities of the same kind, express both in the same unit before reducing their ratio. A comparison of masses, for instance, needs both masses in grams or both in kilograms. The resulting numerical ratio has no unit because the common units cancel.

A ratio linking different kinds of quantities, such as distance and time, represents a rate. Its numerical value depends on the units used. Preserve the same unit choices when comparing such ratios. Reversing the order of just one ratio changes what is being compared.

What makes two ratios equivalent?

Equivalent ratios have the same value. Multiplying or dividing both terms by the same non-zero number preserves that value. Reducing a ratio means dividing both terms by a common factor, a number that divides each term exactly.

The symbols =, × and ≠ mean “equals”, “multiplied by” and “is not equal to”, respectively. Let k be a positive common multiplier. Then ka : kb = a : b, where adjacent letters indicate multiplication.

This does not mean that the individual terms remain unchanged. Both quantities change by the same factor, while their comparison stays fixed. Adding the same quantity to both terms is a different operation and does not generally preserve a ratio.

How do we recognise and verify a proportion?

Definition: A proportion is an equality of two ratios. Let c and d be two further positive quantities. Then a : b = c : d means a/b = c/d.

Read the four terms in order: a, b, c, d. The outside terms, a and d, are the extremes. The inside terms, b and c, are the means. These names identify positions in the written proportion; “means” here does not mean arithmetic averages.

Property: The product of the extremes equals the product of the means

From a/b = c/d, multiply both sides by bd. The denominators cancel, giving ad = bc. This procedure is called cross multiplication. It supplies a direct test for proportionality and a way to find an unknown term.

1. Write the two ratios with the quantities in a consistent order.
2. Check that the denominators are non-zero and that the units match appropriately.
3. Multiply the first and fourth terms to obtain the product of the extremes.
4. Compare this with the product of the second and third terms.

The converse, the statement in the reverse direction, also holds: if ad = bc and b and d are non-zero, divide both sides by bd to recover a/b = c/d. Thus the product test and the ratio statement describe the same relationship under these conditions.

How is an unknown fourth term found?

If a, b and c are known positive quantities and d is the unknown fourth term, cross multiplication gives ad = bc. Dividing by a gives d = bc/a. State what the unknown represents before using this expression.

After calculating, substitute the result into the original proportion. This checks both arithmetic and placement. A calculation may be arithmetically correct yet answer a different question if the unknown was put opposite the wrong quantity. The relationship must be correct before cross multiplication begins.

What are continued proportion and mean proportion?

Three positive quantities a, b and c are in continued proportion when a : b = b : c. The middle quantity b appears twice: as the consequent of the first ratio and the antecedent of the second. This repeated position is the defining feature.

Do not confuse this statement with simply writing three quantities beside one another. A three-term ratio gives their relative sizes, but continued proportion adds the requirement that the first-to-second ratio equals the second-to-third ratio.

Property: The square of the middle term equals the product of the outer terms

Cross multiplication in a/b = b/c gives b² = ac. Here b², read “b squared”, means b multiplied by itself. The term b is called the mean proportional between a and c.

For positive a and c, the positive mean proportional is b = √(ac). The symbol √ means the non-negative square root: the number whose square is the expression beneath it. Since the original quantities here are positive, the required mean is positive.

How does the third proportional differ from the mean proportional?

The third proportional to a and b is c in the relationship a : b = b : c. Rearranging ac = b² gives c = b²/a. Here the first two terms are known and the last term is required.

The arithmetic mean of a and c is their sum divided by two, written (a + c)/2. The plus sign means addition. The mean proportional is instead obtained by multiplying the outer terms and then taking the positive square root.

To check a proposed mean proportional, square it and compare the result with ac. Alternatively, compare a/b with b/c. Both checks follow from the definition, so neither requires a separate rule. For a third proportional, check the same relationship after placing the calculated term last.

How does invertendo transform a proportion?

Property: Invertendo

Invertendo means taking the reciprocal of each of two equal ratios. A reciprocal is the result of dividing one by a non-zero number. In fraction form, taking a reciprocal interchanges the numerator and denominator.

For positive a, b, c and d, a/b = c/d gives b/a = d/c. Both ratios have been inverted. The original numerators a and c now become denominators, which is why they must be non-zero.

Why does invertendo work?

Start with the product relation ad = bc. Dividing both sides by ac gives d/c = b/a. Writing the two sides in the opposite order gives b/a = d/c, precisely the invertendo result. The same cross-product equality supports both forms.

Invertendo is useful when the required answer places each original consequent before its antecedent. It changes a comparison of a to b into a comparison of b to a, while making the corresponding reversal on the other side.

Reversing one ratio alone is not invertendo. Before using the property, identify both complete ratios. After using it, read each denominator aloud or mark it in your working. This helps prevent a correct operation on one side being paired with an unchanged expression on the other.

If an algebraic problem allows zero numerators, the original equality might still be meaningful, but its reciprocals would not be. Check the new denominators when extending these rules beyond the positive quantities considered here.

How does alternendo rearrange corresponding terms?

Property: Alternendo

Alternendo changes a comparison within each pair into a comparison between corresponding terms. For positive a, b, c and d, a/b = c/d gives a/c = b/d. The middle terms in the four-term proportion exchange positions.

The first antecedent is now compared with the second antecedent; the first consequent is compared with the second consequent. This is particularly useful when a question gives one arrangement of a proportion but asks for another.

How can the rearrangement be proved?

Cross multiply the original equality to obtain ad = bc. Divide this equality by cd, which is non-zero. The left side becomes a/c and the right side becomes b/d. Thus the rearrangement follows directly from the original proportionality.

Alternendo and invertendo perform different tasks. Alternendo regroups corresponding quantities. Invertendo reverses each existing ratio. Write the transformed equality before inserting numbers so that the intended arrangement remains visible.

In an application involving cloth, one valid arrangement compares length with cost in each purchase. Alternendo gives another arrangement that compares the two lengths with each other and the two costs with each other. Consistent units are needed in either arrangement.

This explains why more than one proportion can solve the same problem. The expressions may look different while sharing the same cross-product equation. To decide whether a rearrangement is valid, compare its cross products with those of the original relationship rather than judging by appearance.

How does componendo introduce a sum?

Property: Componendo

Componendo adds the consequent to the antecedent in each of two equal ratios, keeping each denominator unchanged. For positive a, b, c and d, a/b = c/d gives (a + b)/b = (c + d)/d.

The brackets show that the entire sum is divided by the denominator. Without them, a + b/b would describe a different calculation: only b would be divided by b. Clear grouping is part of the mathematical statement.

Why is adding one the key step?

Add one to each side of a/b = c/d. On the left, write one as b/b; on the right, write it as d/d. Fractions sharing a denominator can be added by adding their numerators. This gives the componendo formula.

The equality survives because the same number has been added to both sides. We have not claimed that the new ratio equals the old ratio. Each original ratio has increased by one, while the two new ratios remain equal to each other.

Componendo is helpful when a sum appears in the required expression. First identify the original antecedent and consequent. Then add those two terms in the numerator and retain the original consequent in the denominator.

When applying this algebra to measured quantities, add terms only when their units and meanings allow addition. A sum of two lengths is meaningful; adding a length to a monetary cost is not. Numerical ratio manipulations and physical quantities must be interpreted consistently.

How does dividendo introduce a difference?

Property: Dividendo

Dividendo subtracts the consequent from the antecedent in each equal ratio while keeping the denominator unchanged. For positive a, b, c and d, a/b = c/d gives (a − b)/b = (c − d)/d. The symbol − means subtraction.

Why does subtracting one preserve equality?

Subtract one from both sides of a/b = c/d. Express one as b/b on the left and d/d on the right. Subtracting the numerators over their existing common denominators produces the dividendo result.

The order of subtraction must match. The property above uses antecedent minus consequent on both sides. Reversing both differences also preserves equality, but reversing just one difference generally changes the sign on only one side.

A difference can be zero without making this dividendo formula undefined. The denominators are still b and d, which are non-zero. However, a later step that divides by either difference needs a fresh non-zero check.

Although the original terms here are positive, their differences need not be positive. If a is smaller than b, a − b is negative. Because the original ratios are equal, the corresponding comparison of c with d follows the same direction.

Do not discard a minus sign because the original quantities were positive. Keep the subtraction order visible in brackets and carry its sign through the calculation. This becomes especially important when dividendo is combined with another property.

How are the proportion properties combined?

Property: Componendo and dividendo together

Suppose a/b = c/d for positive a, b, c and d, with a ≠ b and c ≠ d. Componendo gives equal ratios involving sums; dividendo gives equal ratios involving differences. Dividing the first equality by the second gives (a + b)/(a − b) = (c + d)/(c − d).

1. Use componendo to write (a + b)/b = (c + d)/d.
2. Use dividendo to write (a − b)/b = (c − d)/d.
3. Check that a − b and c − d are non-zero before dividing.
4. Divide the corresponding expressions and cancel the common denominators b and d.

This combined property is useful when a sum-to-difference ratio is requested. It is not formed by adding an arbitrary quantity to one term. Each sum and difference must come from the same antecedent and consequent in the original ratio.

How can the common ratio explain the same result?

Let k denote the common value a/b = c/d. Then a = kb and c = kd. Substituting into (a + b)/(a − b) gives (kb + b)/(kb − b), which simplifies to (k + 1)/(k − 1).

The same substitution on the other side gives (kd + d)/(kd − d), again simplifying to (k + 1)/(k − 1). The denominator requires k ≠ 1. This is the common-ratio form of the condition that the original terms in each ratio must be unequal.

Check every new denominator. A valid starting proportion does not automatically make every transformed expression defined. After choosing a property, retain its conditions alongside the calculation. These conditions are part of the reasoning, rather than an optional comment after the answer.

How do proportions solve cost and quantity problems?

Two quantities are in direct proportion when their ratio remains constant as their values change. A constant is a value that stays unchanged within the relationship. For quantities represented by x and y, with y non-zero, this is written x/y = k, where k is their constant ratio.

Here x and y denote varying quantities. The fact that both increase is not sufficient: their ratio must remain fixed. In a cost problem, the same price per unit supplies this condition. The unitary method means finding the value for one unit before scaling to the required quantity.

How is the cost of cloth calculated?

Consider cloth of the same type costing ₹210 for 5 metres, with the price per metre unchanged. The symbol ₹ denotes rupees, and m denotes metres. Define C as the required cost in rupees and L as the required length in metres.

The proportion is C/210 = L/5, giving C = 210 × L/5. It compares cost with cost and length with length. Alternatively, 5/210 = L/C compares length with cost on both sides. Alternendo connects these arrangements.

Worked example 1. If 5 metres of cloth cost ₹210, find the cost of 2 metres at the same price per metre.

Answer: C/210 = 2/5, so C = 210 × 2/5 = 84. The cost is ₹84. The required length is smaller, and its cost falls in the same ratio.

Worked example 2. If 5 metres of cloth cost ₹210, find the cost of 4 metres at the same price per metre.

Answer: C/210 = 4/5. Therefore C = 210 × 4/5 = 168, giving ₹168. Substituting into the original comparison gives 168/210 = 4/5.

Worked example 3. If 5 metres of cloth cost ₹210, find the cost of 10 metres at the same price per metre.

Answer: C/210 = 10/5, so C = 420. The cost is ₹420. Doubling the length doubles the cost because the price per metre stays fixed.

Worked example 4. If 5 metres of cloth cost ₹210, find the cost of 13 metres at the same price per metre.

Answer: C/210 = 13/5, giving C = 210 × 13/5 = 546. Thus the cost is ₹546. Retain the exact fraction until multiplication and division are complete.

These calculations use the same relationship with different required lengths. The known purchase remains a valid reference throughout because the cloth type and price per metre have not changed. If the pricing rule changed, that condition would need checking before reusing the proportion.

How do we apply proportions to distance, mass and shadows?

Begin a direct application by identifying the fixed relationship, choosing consistent units and naming the unknown. A table of paired quantities can help keep corresponding values together. Then write the proportion, solve it and check the answer in the original context.

How does a fixed travel relationship work?

The abbreviation km means kilometres. A litre measures volume. In the following example, assume that the distance travelled per litre of petrol remains unchanged. Let D denote the required distance in kilometres.

Worked example 5. A car travels 60 km using 4 litres of petrol. How far will it travel using 12 litres at the same distance per litre?

Answer: D/60 = 12/4, so D = 60 × 12/4 = 180. The distance is 180 km. Petrol use triples, so the distance also triples under the stated condition.

How must masses be converted?

The unit symbols g and kg mean grams and kilograms. One kilogram equals 1000 grams. The notation 2½ means two and a half. Compare masses using one common unit before setting up the proportion. Let n be the unknown number of sheets in the following calculation.

Worked example 6. Twelve sheets of the same thick paper weigh 40 g. How many sheets of that paper weigh 2½ kg?

Answer: Convert 2½ kg to 2500 g. Then n/12 = 2500/40, so n = 12 × 2500/40 = 750. The required number is 750 sheets, using the same mass per sheet.

What condition is needed for the shadow comparison?

Compare object heights with their shadow lengths under similar conditions. That qualification must accompany the relationship: shadow length is not determined by object height alone. Let h denote the unknown tree height in metres.

Worked example 7. An electric pole 14 m high casts a shadow 10 m long. Under similar conditions, a tree casts a shadow 15 m long. Find the tree's height.

Answer: h/15 = 14/10. Therefore h = 14 × 15/10 = 21. The tree is 21 m high. Checking gives 21/15 = 14/10, so the height-to-shadow ratios agree.

How does uniform speed connect distance and time?

Uniform speed means that equal distances are covered in equal time intervals. The notation km/hour means kilometres per hour. One hour equals 60 minutes. Let t denote time in minutes; continue to use D for distance in kilometres.

Worked example 8. A train travels at a uniform speed of 75 km/hour. Find its distance in 20 minutes and the time required to travel 250 km.

Answer: Use 75 km in 60 minutes. Then D/20 = 75/60, giving D = 25 km. For the second part, 250/t = 75/60, giving t = 200 minutes, or 3 hours 20 minutes.

The train calculation depends on uniform speed; it is not justified merely because a longer journey takes more time. The paper calculation depends on the same paper, and the cloth calculation depends on unchanged pricing. Stating the relevant condition makes each proportion meaningful.

Keep units in the final answer and use the question's requested quantity. A numerical result alone does not distinguish a distance from a time, a mass or a count. A final substitution checks the equality; interpreting the answer checks that the correct unknown was found.

Glossary

• Ratio — A comparison of two quantities by division, written in a stated order.
• Antecedent — The first term of a ratio, forming the numerator in fraction notation.
• Consequent — The second term of a ratio, which must be non-zero when used as a denominator.
• Proportion — An equality of two ratios, expressing the same numerical comparison in both.
• Extremes — The first and fourth terms of a proportion written with four ordered terms.
• Means — The second and third terms of a proportion written with four ordered terms.
• Continued proportion — A relationship among three quantities in which the first-to-second and second-to-third ratios are equal.
• Mean proportional — The positive middle quantity in continued proportion, whose square equals the product of the outer quantities.
• Invertendo — The property that permits taking reciprocals of both equal ratios when those reciprocals are defined.
• Alternendo — The property that compares corresponding antecedents and corresponding consequents in a proportion.
• Componendo — The property that adds each consequent to its antecedent while retaining the original denominator.
• Dividendo — The property that subtracts each consequent from its antecedent while retaining the original denominator.
• Direct proportion — A relationship in which the ratio of two varying quantities remains constant.

Common errors and misconceptions

• Misconception: Reversing one ratio preserves a proportion. Correct: Invertendo reverses both ratios. Reversing only one generally changes the equality, so write both reciprocals explicitly.
• Misconception: The mean proportional is the arithmetic mean. Correct: For positive outer terms a and c, it is √(ac), found from b² = ac, rather than (a + c)/2.
• Misconception: Any two quantities increasing together are directly proportional. Correct: Their ratio must remain constant. A shared direction of change does not establish this relationship.
• Misconception: Componendo keeps the original ratio unchanged. Correct: It adds one to each ratio. The two resulting ratios remain equal to each other, but differ from the originals.
• Misconception: The combined property can be used when the difference is zero. Correct: A difference in a denominator must be non-zero; otherwise the resulting expression is undefined.
• Misconception: Masses in grams and kilograms can be compared using their written numbers immediately. Correct: Convert them to the same unit before forming a numerical mass ratio.
• Misconception: Any tree and pole have proportional heights and shadows. Correct: The comparison requires similar conditions. Keep that qualification when forming and explaining the proportion.

Exam-style questions with model answers

Q1. For positive quantities a, b, c and d, a : b = c : d. Identify the means and extremes, and state their product relation. [2 marks]

1. The means are b and c, the two inside terms. The extremes are a and d, the two outside terms.
2. The product of the extremes equals the product of the means, so ad = bc.

Q2. Positive quantities a, b and c are in continued proportion. Derive an expression for the mean proportional b in terms of a and c, and distinguish it from their arithmetic mean. [3 marks]

1. Continued proportion means a : b = b : c. The middle term b is repeated as a consequent and then as an antecedent.
2. Cross multiplication gives b² = ac. Since b is positive, taking the positive square root gives b = √(ac).
3. The arithmetic mean is (a + c)/2. It uses addition and division, whereas the mean proportional uses the product and its positive square root.

Q3. Given positive quantities a, b, c and d with a/b = c/d, derive invertendo and alternendo using cross multiplication. [3 marks]

1. Multiplying the given equality by bd gives ad = bc. This is the product relation from which both rearrangements follow.
2. Divide by ac to obtain d/c = b/a. Reordering the equality gives b/a = d/c, the invertendo property.
3. Divide ad = bc by cd to obtain a/c = b/d, the alternendo property. All these divisions are allowed because the given quantities are positive.

Q4. Let a, b, c and d be positive quantities satisfying a/b = c/d, with a ≠ b and c ≠ d. Prove (a + b)/(a − b) = (c + d)/(c − d), explaining the denominator conditions. [5 marks]

1. Start from a/b = c/d. Since b and d are positive, the two original ratios are defined and may be transformed by adding or subtracting the same number.
2. Add one to each side. Writing one as b/b and d/d gives (a + b)/b = (c + d)/d by componendo.
3. Subtract one from the original equality. This gives (a − b)/b = (c − d)/d by dividendo.
4. The given conditions a ≠ b and c ≠ d ensure that both differences, and therefore both dividendo ratios, are non-zero.
5. Divide the componendo equality by the dividendo equality. Cancelling b and d within the respective quotients gives (a + b)/(a − b) = (c + d)/(c − d).

Q5. Five metres of cloth cost ₹210. At the same price per metre, find the cost of 13 metres and verify the proportion. [3 marks]

1. Let C be the cost of 13 metres in rupees. Since the price per metre is unchanged, cost is directly proportional to length.
2. Write C/210 = 13/5. Cross multiplication gives 5C = 210 × 13, so C = 546. The cost is ₹546.
3. Check by substitution: 546/210 = 13/5. The cost ratio equals the length ratio, confirming the required direct proportion.

Q6. Twelve sheets of the same thick paper weigh 40 g. Find how many sheets weigh 2½ kg, using 1 kg = 1000 g. State why proportion applies. [4 marks]

1. Convert the required mass to the same unit as the known mass: 2½ kg = 2500 g.
2. Let n be the required number of sheets. Using the same paper means that mass per sheet remains unchanged, so number and mass are directly proportional.
3. Write n/12 = 2500/40. Cross multiplication gives 40n = 12 × 2500.
4. Dividing by 40 gives n = 750. Therefore 750 sheets of this paper weigh 2½ kg under the stated condition.

Q7. A pole 14 m high casts a shadow 10 m long. Under similar conditions, a tree casts a shadow 15 m long. Find the height of the tree and check your result. [3 marks]

1. Let h be the tree's height in metres. Under similar conditions, use equal height-to-shadow ratios: h/15 = 14/10.
2. Cross multiplication gives 10h = 14 × 15. Therefore h = 21, so the tree is 21 m high.
3. Substituting the result gives 21/15 = 14/10. The ratios agree, confirming the calculated height while retaining the similar-conditions requirement.

Q8. A train travels at a uniform speed of 75 km/hour. Find its distance in 20 minutes and the time needed to cover 250 km. Use 1 hour = 60 minutes, and express the second answer in hours and minutes. [5 marks]

1. At the stated uniform speed, the train covers 75 km in 60 minutes. Distance and time are directly proportional because the speed remains unchanged.
2. Let D be the distance in kilometres covered in 20 minutes. Write D/20 = 75/60, comparing distance per minute on both sides.
3. Multiply by 20 to obtain D = 75 × 20/60 = 25. The train therefore travels 25 km in 20 minutes.
4. Let t be the time in minutes for 250 km. Then 250/t = 75/60, so t = 250 × 60/75 = 200 minutes.
5. Convert 200 minutes into hours and remaining minutes: 200 minutes = 3 hours 20 minutes. This is the required time for the second journey.

Key takeaways

• A ratio compares quantities in a stated order; use consistent units and keep its denominator non-zero.
• In a proportion, the product of the extremes equals the product of the means.
• Continued proportion repeats the middle quantity, whose square equals the product of the outer quantities.
• For positive quantities, the mean proportional is the positive square root of their product.
• Invertendo reverses both ratios, whereas alternendo compares corresponding antecedents with one another and corresponding consequents with one another.
• Componendo adds one to both equal ratios; dividendo subtracts one, producing corresponding sums and differences.
• Combining componendo and dividendo requires non-zero differences wherever those differences become denominators in the resulting ratios.
• Direct applications require a constant ratio, appropriate units and the relevant conditions, such as uniform speed or similar shadow conditions.

Test yourself

What does a : b mean, and which term cannot be zero?

It means a/b. The consequent b cannot be zero because it is the denominator of the division.

What product relation follows from a : b = c : d?

Cross multiplication gives ad = bc: the product of the extremes equals the product of the means.

If positive a, b and c are in continued proportion, how is b found from a and c?

Use b² = ac and take the positive square root, giving b = √(ac).

How does alternendo transform a/b = c/d for positive terms?

It gives a/c = b/d, comparing the two antecedents and the two consequents with one another.

Why does componendo give (a + b)/b = (c + d)/d?

Adding one to each original ratio adds its denominator to its numerator while preserving equality between the two sides.

Why must a − b and c − d be non-zero in the combined sum-to-difference property?

They become denominators in the transformed ratios. Division by zero would make those expressions undefined.

Does increasing together prove that two quantities are directly proportional?

No. Their ratio must remain constant; simply increasing together does not establish a direct proportion.

What conditions support the cloth, train and shadow examples?

The cloth uses an unchanged price per metre, the train has uniform speed, and the shadows are compared under similar conditions.