Ratio and Proportion | ICSE Class 6 Maths Notes

This note covers comparison by division, the meaning and order of ratios, common units, equivalent ratios, fractions and ratios, proportion, missing values, division in a given ratio, the unitary method, basic percentage, and simple speed, distance and time problems.

What is a ratio, and how does it compare quantities?

Definition: A ratio compares two numbers or quantities by division. The symbol : means “to” when writing a ratio. The numbers on its two sides are its terms.

For example, the ratio of 8 books to 20 books is written 8 : 20. Read this as “8 to 20”. The first term refers to the first collection of books, and the second term refers to the second collection.

Division asks how one quantity compares with another in size. Subtraction, by contrast, finds the difference between them. The difference tells us how much more or less there is; a ratio expresses their relative sizes.

How are difference and ratio different?

For the collections of 8 and 20 books, subtraction gives a difference of 12 books. A ratio gives 8 : 20, which reduces to 2 : 5. These statements describe the same collections, but answer different questions about them.

Here, the symbol − means subtraction, ÷ means division, and = means “is equal to”. A calculation such as 20 − 8 = 12 gives an amount. A comparison such as 8 ÷ 20 gives a ratio's value.

A factor divides a number exactly. A common factor divides both ratio terms exactly. In lowest form, whole-number terms share no common factor other than 1. Whole numbers are zero and the counting numbers.

Worked example 1. Find the ratio of 8 books to 20 books in its lowest form.

Answer: Write 8 : 20. Divide both terms by 4: 8 ÷ 4 = 2 and 20 ÷ 4 = 5. The required ratio is 2 : 5.

Why does the order of a ratio matter?

A ratio follows the order in which the quantities are named. “First collection to second collection” puts the first collection before the colon. Reversing the wording reverses the terms. Read the words before simplifying the numbers.

Property: The order of the terms matters

A fraction represents a number as one number divided by another non-zero number. Non-zero means different from zero.

The ratio 5 : 4 differs from 4 : 5. In fraction form they are 5/4 and 4/5. The slash / denotes division in a fraction. In a fraction, the numerator is the number above the dividing line and the denominator is the number below it.

Thus, 5 : 4 compares a larger first quantity with a smaller second quantity. The reverse comparison, 4 : 5, compares the smaller quantity with the larger one. Reversing the terms changes the comparison being made.

A ratio is not necessarily greater than 1. Its fraction value can be smaller than, equal to or greater than 1. Do not change the requested order simply to put the larger number first.

How can you preserve the order?

1. Read the complete phrase naming the two quantities.
2. Write the first named quantity on the left of the colon.
3. Write the second named quantity on the right of the colon.
4. Simplify both terms together, keeping them in their original positions.

For the book collections, “8 books to 20 books” gives 2 : 5. Reversing the comparison gives 5 : 2. The collections have not changed; the question has changed. A clear written label prevents this common error.

Why must quantities be expressed in the same unit?

A unit is a standard used to measure a quantity. Metres and centimetres measure length; rupees and paise measure money. Before taking the ratio of measurements of the same kind, express both in the same unit.

The abbreviations m and cm stand for metre and centimetre. One metre equals 100 centimetres. The symbol ₹ stands for rupees, and one rupee equals 100 paise. The symbol × means multiplication.

Worked example 2. Find the ratio of ₹8 to 80 paise. Use 1 rupee = 100 paise.

Answer: ₹8 = 8 × 100 paise = 800 paise. The ratio is 800 : 80 = 10 : 1. Both quantities are now expressed in paise.

Writing 8 : 80 immediately would compare the printed numbers without accounting for their units. A rupee is worth more than a paisa, so the unit conversion is part of the comparison, not an optional final step.

How do you compare lengths written differently?

Worked example 3. A rectangular sheet is 1.2 m long and 21 cm wide. Find the ratio of its width to its length. Use 1 m = 100 cm.

Answer: Length = 1.2 × 100 cm = 120 cm. Width to length = 21 : 120 = 7 : 40, after dividing both terms by 3.

The word width identifies the measurement placed first in this example. Converting correctly is not enough if the order is then reversed. The answer 7 : 40 gives the width relative to the length.

After conversion, the same unit occurs on both sides. The simplified ratio is therefore written as a comparison of numbers. Keep units visible during the conversion so that the reasoning can be checked.

How do equivalent ratios and simplification work?

Equivalent ratios express the same comparison. Two ratios are equivalent when their corresponding fractions are equivalent, meaning equal in value. Their written terms can differ without changing the relationship between the quantities.

Property: Equal scaling preserves a ratio

Multiplying both terms by the same positive number, meaning a number greater than zero, gives an equivalent ratio. Dividing both terms by the same positive common factor also gives an equivalent ratio. Changing just one term changes the relationship.

The ratios 3 : 8 and 12 : 32 are equivalent. Multiplying 3 and 8 by 4 gives 12 and 32. Dividing 12 and 32 by 4 returns 3 and 8. Both operations preserve the comparison.

Worked example 4. Reduce 12 : 32 to its lowest form and explain its connection with 3 : 8.

Answer: Divide both terms by 4. Since 12 ÷ 4 = 3 and 32 ÷ 4 = 8, the ratio becomes 3 : 8. The terms 3 and 8 have no common factor other than 1.

What does lowest form tell you?

Simplification changes the way a ratio is written. It does not change the original amounts being compared. A ratio of 8 books to 20 books can be expressed as 2 : 5 without saying that the collections actually contain only 2 and 5 books.

To simplify, first check units. Then look for a common factor, divide both terms by it, and check whether a further common factor remains. Stop when the whole-number terms share no factor other than 1.

The same reasoning helps compare ratios: reduce each to lowest form and see whether the results match. Keep the order of the terms fixed throughout. Equal-looking terms in different positions do not establish equivalence.

How is a ratio different from a fraction of a whole?

A ratio may be written as a fraction, but the quantities being compared still need to be identified. A part-to-part ratio compares two parts of a collection. A part-to-whole ratio compares a part with the complete collection.

In a fraction describing part of a whole, the denominator refers to the whole. In a part-to-part ratio written as a fraction, the denominator refers to the second part. These are different comparisons, even when the same collection is involved.

A rectangular grid has six columns and eight rows of equal squares. Blue hatching marks 15 squares; 33 squares are unshaded. The two kinds of squares together make the complete grid.

See Fig. 8.1 in your NCERT textbook

Which quantity belongs in the denominator?

The shaded-to-unshaded ratio is 15 : 33. The fraction of the whole that is shaded uses all the squares as its denominator. The total is 15 + 33 = 48, where + means addition.

Worked example 5. A grid contains 15 shaded squares and 33 unshaded squares. Find the shaded-to-unshaded ratio and the fraction of all squares that are shaded.

Answer: Shaded to unshaded = 15 : 33 = 5 : 11. The total is 48 squares, so the shaded fraction is 15/48 = 5/16. The second comparison uses the whole grid.

The table shows calculations from the same counts. It does not describe different grids. Before writing a fraction or ratio, ask whether the second quantity is the other part or the whole. That decision determines the denominator.

What is proportion, and how can you test it?

Definition: A proportion is an equality of two ratios. Four quantities are in proportion when the first-to-second ratio equals the third-to-fourth ratio.

The notation 3 : 8 = 24 : 64 states a proportion. It can also be written 3 : 8 :: 24 : 64. The symbol :: means that the ratio on its left equals the ratio on its right.

Property: Equivalent ratios form a proportion

To test four quantities in a stated order, form the two ratios and simplify them. If they reduce to the same ratio, the quantities are in proportion in that order. If they do not, they are not in proportion.

Worked example 6. Are 3, 8, 24 and 64 in proportion in that order?

Answer: Compare 3 : 8 with 24 : 64. Dividing 24 and 64 by 8 gives 3 : 8. The ratios are equal, so 3 : 8 = 24 : 64 is a proportion.

The order remains essential. The sequence 3, 8, 64, 24 would compare 3 : 8 with 64 : 24. Reversing the second pair changes its ratio, so this reordered sequence does not express the same proportion.

How do you show that ratios are unequal?

Worked example 7. Decide whether 12 : 18 and 28 : 56 form a proportion.

Answer: Dividing the first pair by 6 gives 2 : 3. Dividing the second pair by 28 gives 1 : 2. These ratios differ, so the original ratios do not form a proportion.

A complete explanation identifies both simplified ratios before giving the conclusion. Saying that the numbers are different is insufficient: equivalent ratios can have different terms. What matters is the comparison those terms express.

How can you find a missing term in a proportion?

A missing term can be found by making two ratios equivalent. Use the known ratio to determine how its terms relate, then match the term already given in the incomplete ratio. Preserve the position of each term.

Let x represent the missing number. This letter is a placeholder for a value to be found. In x : 8 = 12 : 32, x is the first term of the first ratio; it is not a multiplication sign.

How does scaling reveal the missing number?

Worked example 8. Find x in x : 8 = 12 : 32, where x is the missing number.

Answer: The second term 32 becomes 8 when divided by 4. Divide the matching first term by 4 as well: 12 ÷ 4 = 3. Therefore, x = 3.

This method uses the same operation on both terms of the known ratio. Dividing one term by 4 while leaving the other unchanged would produce a different ratio and would not solve the proportion.

How do you check the result?

Replace the letter with the number found. The statement becomes 3 : 8 = 12 : 32. Since multiplying both 3 and 8 by 4 gives 12 and 32, the equality is correct.

This final replacement is called substitution: putting a known value in place of a letter. It checks the original question directly. A missing-number answer should satisfy the given proportion, not just a rearranged comparison with a different order.

How do you divide a quantity in a given ratio?

To divide a total in a ratio, interpret the terms as numbers of equal parts. Add the terms to find the total number of parts. Divide the actual total by that number to find the value of one part.

Then multiply the value of one part by each ratio term. The resulting shares must add to the original total. Their ratio must also match the requested ratio. These two checks test different features of the answer.

How do ratio parts relate to actual lengths?

Worked example 9. A line segment 56 cm long is divided into two parts in the ratio 2 : 5. Find both lengths.

Answer: Total ratio parts = 2 + 5 = 7. One part is 56 ÷ 7 = 8 cm. The lengths are 2 × 8 = 16 cm and 5 × 8 = 40 cm.

A line segment is a straight portion of a line with two endpoints. Here, the original segment is separated into two shorter segments. The terms 2 and 5 describe their relative lengths, not their lengths in centimetres.

The checks are 16 + 40 = 56 cm and 16 : 40 = 2 : 5. The first restores the total length; the second restores the intended comparison. Both are satisfied.

What if one share is given instead of the total?

Books with green and brown covers are in the ratio 2 : 3. If 18 books have green covers, the 2 parts correspond to 18 books. One part represents 18 ÷ 2 = 9 books, so the brown-covered books number 3 × 9 = 27.

The distinction is the starting information. A total is divided by the sum of the ratio terms. A known share is divided by its own ratio term. Match the given amount to the correct number of parts before calculating.

How does the unitary method solve everyday problems?

Definition: The unitary method first finds the value of one unit and then the value of the required number of units.

For these problems, use direct variation: the related quantities change in the same ratio while the rate remains fixed. For purchases, the rate is the cost of one unit of the item. The same cost per unit must apply throughout the calculation.

How do you work from a cost to a unit rate?

The abbreviation kg means kilogram, a unit of mass. In the following problem, a ton is taken as 1,000 kg. Convert the given mass into kilograms before calculating the cost of one kilogram.

Worked example 10. If 20 tons of iron cost ₹600,000, find the cost of 560 kg at the same rate. Use 1 ton = 1,000 kg.

Answer: 20 tons = 20,000 kg. Cost of 1 kg = ₹600,000 ÷ 20,000 = ₹30. Cost of 560 kg = ₹30 × 560 = ₹16,800.

The one-unit step is the centre of the method. It connects the given amount with the required amount. It also makes clear why division is followed by multiplication in this example.

1. Identify the quantity whose single unit is needed.
2. Express the given and required quantities in compatible units.
3. Divide the given total value by the given number of units.
4. Multiply the value of one unit by the required number of units.

How can you check the interpretation?

The answer ₹30 is a cost per kilogram, while ₹16,800 is a total cost. Naming each result prevents confusion between a rate and an amount. A calculation can be numerically correct yet answer the wrong part of the problem.

The condition at the same rate matters. The original purchase tells us the cost of the required mass only if the cost per kilogram is unchanged. State this condition when using the unitary method for a purchase.

What does percentage mean as a fraction out of 100?

Per cent means “per hundred”. The symbol % denotes per cent. A percentage expresses a comparison using 100 as the denominator. Thus, 50% means 50/100, which is the same fraction as 1/2.

The denominator identifies the number of equal parts used for the comparison. With percentage, the reference is a hundred equal parts. It does not mean that the original collection must actually contain exactly 100 objects.

How do you express a simple fraction as a percentage?

An equivalent fraction with denominator 100 makes the percentage visible. The numerator then gives the number per hundred. For example, 1/2 = 50/100 = 50%, and 3/4 = 75/100 = 75%.

Worked example 11. A class has 50 children, of whom 20 are boys and the rest are girls. Express the girls as a fraction of the class and as a percentage.

Answer: Girls = 50 − 20 = 30. Their fraction is 30/50. Multiplying numerator and denominator by 2 gives 60/100, so the girls form 60% of the class.

The fraction uses all 50 children as the whole. It does not compare girls with boys. The meaning of the denominator remains just as important in percentage as it is in any other part-to-whole comparison.

What do half and the whole mean in percentage?

50% of a quantity means half of that quantity. 100% means the whole quantity. These statements describe a fraction of a chosen whole; the actual amount depends on the size of that whole.

When reading a percentage, identify what it is a percentage of. In the class example, the whole is the class of 50 children, and the group being measured is the girls. This keeps the numerical calculation connected to the question.

How are speed, distance and time related?

Distance is the length travelled. Time is the duration of the journey. Speed expresses the distance travelled per unit of time. Divide the distance travelled by the time taken to calculate the speed for a journey.

The abbreviation km means kilometre. The unit km/h means kilometres per hour. A uniform speed stays constant: equal time intervals correspond to equal distances. Keep this condition when using a known speed to predict another journey.

How can you compare two speeds?

Worked example 12. A scooter travels 120 km in 3 hours and a train travels 120 km in 2 hours. Find the ratio of their speeds.

Answer: Scooter speed = 120 ÷ 3 = 40 km/h. Train speed = 120 ÷ 2 = 60 km/h. Scooter speed to train speed = 40 : 60 = 2 : 3.

Although the distances are equal, the times are different. Comparing the distances alone would therefore not compare the speeds. Find the distance per hour for each journey, then form the ratio in the requested order.

Which calculation finds distance or time?

A train takes 2 hours to cover a given distance of 130 km. Its speed is 130 ÷ 2 = 65 km/h. At this same uniform speed, covering a given distance of 780 km takes 780 ÷ 65 = 12 hours.

This is another use of the unitary method. The distance covered in one hour links the original journey to the required journey. At a fixed speed, distance and time are directly related: increasing the time by a given factor increases the distance by that factor.

Write units beside intermediate answers. Dividing distance by time gives a speed; dividing distance by speed gives a time. The unit attached to the answer helps confirm that the calculation addresses the quantity requested.

Glossary

• Ratio — A comparison of two numbers or quantities by division, written with a colon between its terms.
• Term of a ratio — Either of the two numbers written on opposite sides of the colon in a ratio.
• Equivalent ratios — Ratios whose corresponding fractions have equal values and express the same comparison.
• Lowest form — The form of a ratio whose whole-number terms share no common factor other than one.
• Numerator — The number above a fraction's dividing line, or before the slash when written horizontally.
• Denominator — The number below a fraction's dividing line, or after the slash when written horizontally.
• Proportion — An equality of two ratios, comparing the first quantity with the second and the third with the fourth.
• Part-to-whole ratio — A ratio comparing one part of a collection with the complete collection.
• Unitary method — A method that finds the value of one unit before finding the value of the required units.
• Direct variation — A relationship in which two quantities change in the same ratio while their rate remains fixed.
• Per cent — Per hundred, expressing a quantity as a fraction with one hundred as its denominator.
• Speed — The distance travelled per unit of time, calculated by dividing distance by time.
• Uniform speed — A speed that stays constant, giving equal distances travelled during equal intervals of time.

Common errors and misconceptions

• Misconception: A ratio is found by subtracting one quantity from another. Correct: Subtraction gives a difference; a ratio compares quantities by division.
• Misconception: The ratio 5 : 4 means the same as 4 : 5. Correct: The order identifies which quantity is being compared with which. Reversing it changes the comparison.
• Misconception: ₹8 to 80 paise gives 8 : 80. Correct: Convert ₹8 to 800 paise first. The required ratio is 800 : 80 = 10 : 1.
• Misconception: Dividing just one term simplifies a ratio. Correct: Divide both terms by the same common factor to preserve their relationship.
• Misconception: Shaded to unshaded means shaded to the whole figure. Correct: The whole includes both groups. With 15 shaded and 33 unshaded squares, the comparisons use 33 and 48 respectively.
• Misconception: A total divided in the ratio 2 : 5 should be divided by 5 to find one part. Correct: The total contains 2 + 5 = 7 equal parts.
• Misconception: Equal journey distances imply equal speeds. Correct: Time matters too. Calculate distance divided by time for each journey before comparing speeds.

Exam-style questions with model answers

Q1. Explain why the ratios 5 : 4 and 4 : 5 are different. [2 marks]

1. The ratio 5 : 4 compares the first quantity with the second and has fraction value 5/4.
2. The ratio 4 : 5 reverses the comparison and has value 4/5. These values are unequal, so order matters.

Q2. Find the ratio of ₹8 to 80 paise in lowest form. Use 1 rupee = 100 paise. [3 marks]

1. Express both amounts in paise before comparing them. The first amount is ₹8, which equals 8 × 100 = 800 paise.
2. Keep the order specified in the question. The first amount to the second amount is therefore 800 : 80.
3. Divide both terms by their common factor 80. This gives 10 : 1, so the first amount is ten times the second amount.

Q3. A grid contains 15 shaded squares and 33 unshaded squares, all equal in size. Find the shaded-to-unshaded ratio and the fraction of the whole that is shaded. Explain the different denominators. [4 marks]

1. The shaded-to-unshaded ratio is 15 : 33. Dividing both terms by 3 gives the simplest ratio 5 : 11.
2. The whole grid contains both shaded and unshaded squares, so the total number of squares is 15 + 33 = 48.
3. The fraction of all squares that is shaded is therefore 15/48. Dividing its numerator and denominator by 3 gives 5/16.
4. The first comparison uses the unshaded group as its second quantity. The second uses the whole grid, so the denominators describe different quantities.

Q4. Find the missing number x in x : 8 = 12 : 32 and check your answer. [3 marks]

1. The known second term 32 must become 8. Since 32 ÷ 4 = 8, divide both terms of the known ratio by 4.
2. The first term becomes 12 ÷ 4 = 3. Therefore the missing number, represented by x, is 3.
3. Substitute 3 into the original statement. It gives 3 : 8 = 12 : 32. Multiplying both terms of 3 : 8 by 4 confirms the equality.

Q5. A line segment 56 cm long is divided in the ratio 2 : 5. Find both lengths and check the total and the ratio. [5 marks]

1. The terms 2 and 5 represent equal parts assigned to the two lengths. The complete segment therefore contains 2 + 5 = 7 equal parts.
2. Find the length of one equal part by dividing the full length by the total number of parts: 56 ÷ 7 = 8 cm.
3. The first length has 2 parts, so it measures 2 × 8 = 16 cm. The second has 5 parts, so it measures 5 × 8 = 40 cm.
4. Check the combined length: 16 + 40 = 56 cm. The two calculated lengths use the whole of the original line segment.
5. Check the comparison: 16 : 40 reduces to 2 : 5 after dividing both terms by 8. Both required conditions are satisfied.

Q6. Twenty tons of iron cost ₹600,000. Find the cost of 560 kg at the same rate, using 1 ton = 1,000 kg. Show the unitary method. [4 marks]

1. The given mass is measured in tons, but the required mass is measured in kilograms. Express both masses in kilograms before finding a unit cost.
2. Convert the given mass using the stated conversion: 20 × 1,000 = 20,000 kg. This is the mass that costs ₹600,000.
3. Divide the total cost by the number of kilograms. The cost of one kilogram is ₹600,000 ÷ 20,000 = ₹30.
4. The rate remains unchanged, so multiply the cost of one kilogram by the required number of kilograms: ₹30 × 560 = ₹16,800.
5. Therefore, 560 kg of iron costs ₹16,800. The method first found the value of one unit and then the value of the required units.

Q7. A class contains 50 children, of whom 20 are boys and all the others are girls. Find the fraction and percentage of girls. [3 marks]

1. Find the number of girls by subtracting the boys from the complete class: 50 − 20 = 30 girls.
2. The girls form 30/50 of the class. This is a part-to-whole comparison, because the denominator includes all the children.
3. Multiply numerator and denominator by 2 to obtain 60/100. Per cent means per hundred, so 60/100 is 60%. Therefore, 60% of the class are girls.

Q8. A scooter travels 120 km in 3 hours and a train travels 120 km in 2 hours. Calculate their speeds in km/h and the scooter-to-train speed ratio. [4 marks]

1. Speed is distance travelled divided by time taken. Using kilometres and hours expresses both calculated speeds in kilometres per hour.
2. The scooter's speed is 120 ÷ 3 = 40 km/h, using the scooter's own distance and time.
3. The train's speed is 120 ÷ 2 = 60 km/h. Equal distances do not give equal speeds when the journey times differ.
4. Keep the requested scooter-to-train order: 40 : 60. Dividing both terms by 20 gives the speed ratio 2 : 3.

Key takeaways

• A ratio compares quantities by division, while subtraction finds the difference between the amounts.
• Read the order carefully: the first named quantity supplies the first term of the ratio.
• Express measurements of the same kind in a common unit before forming and simplifying their ratio.
• Equivalent ratios preserve the comparison by multiplying or dividing both terms by the same suitable number.
• A proportion states that two ratios are equal; compare their simplified forms to check the statement.
• A part-to-part comparison and a fraction of the whole use different second quantities or denominators.
• The unitary method finds one unit first, then the required amount, using the stated fixed rate.
• Per cent means per hundred, while speed compares the distance travelled with the time taken.

Test yourself

Which operation defines a ratio: subtraction or division?

Division defines a ratio; subtraction finds the difference between two quantities.

Why is the ratio of ₹8 to 80 paise not 8 : 80? Use 1 rupee = 100 paise.

The units differ. Convert ₹8 to 800 paise, giving 800 : 80 = 10 : 1.

Are 3, 8, 24 and 64 in proportion in that order?

Yes. The ratio 24 : 64 simplifies to 3 : 8, so the two ratios are equal.

A grid has 15 shaded and 33 unshaded equal squares. What denominator gives the shaded fraction of the whole?

Use 48, because the whole contains 15 + 33 squares. The fraction is 15/48.

A total is divided in the ratio 2 : 5. How many equal ratio parts does it contain?

It contains 7 equal parts, found by adding the two ratio terms, 2 and 5.

What is the first value found in the unitary method?

Find the value of one unit before finding the value of the required number of units.

What do 50% and 100% of a quantity mean?

Fifty per cent means half of the quantity; one hundred per cent means the whole quantity.

A scooter covers 120 km in 3 hours. What is its speed in km/h?

Divide distance by time: 120 ÷ 3 = 40 km/h, meaning forty kilometres per hour.