Ratio and Proportion | ICSE Class 7 Maths Notes

This note covers ratios, equivalent ratios, proportion, the unitary method, division in a given ratio, percentages, conversions between fractions and decimals, profit and loss for a single transaction, simple interest for complete years, and speed, distance and time.

What is a ratio, and why does its order matter?

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

In the ratio a : b, the letters a and b represent the first and second quantities, expressed in the same unit. It can be written as the fraction a/b, where the slash means division and b is not zero.

A fraction has a numerator, the number above the division line, and a denominator, the number below it. Writing a ratio as a fraction lets us use fraction methods to simplify it or compare it with another ratio.

How do we keep the comparison in the correct order?

Read the quantities in the order asked. The first quantity belongs in the first term and the second quantity in the second term. A comparison of books in one group to books in another group must keep those groups in that order.

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

Answer: The ratio is 8 : 20. Divide both terms by 4 to obtain 2 : 5. This compares the first group with the second group, preserving the order of the question.

The lowest form of a ratio has no common factor other than 1. A factor divides a whole number exactly; a common factor divides both terms exactly. Simplification changes the numbers used to express the ratio but preserves the comparison.

The ratios 5 : 4 and 4 : 5 are different. Their fractions are 5/4 and 4/5, so reversing the terms changes the value. Do not put the smaller number first merely because it looks easier to simplify.

Note: A ratio is a comparison by division. It does not tell us how much larger one quantity is by subtraction. Keep the meanings of difference and ratio separate.

How do we express quantities in the same unit?

Before forming a ratio of measured quantities, express them in the same unit. A unit is the agreed measure used for a quantity, such as a metre for length. The numerical comparison becomes misleading if one term uses a larger unit than the other.

For lengths, m means metre and cm means centimetre, with 1 m = 100 cm. For money, ₹ means rupees, and 1 rupee = 100 paise. The sign = means “is equal to”. The signs +, −, × and ÷ mean addition, subtraction, multiplication and division respectively.

How should a conversion be recorded?

1. Identify the two quantities and the order requested.
2. Choose a common unit for both quantities.
3. Convert the quantity that uses the other unit.
4. Write the ratio and divide both terms by a common factor.

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

Answer: ₹8 = 800 paise. The required ratio is 800 : 80 = 10 : 1. Both terms now refer to paise, so the comparison correctly shows the relative amounts.

Writing 8 : 80 directly would compare the displayed numerals without accounting for the different units. The conversion is part of the mathematical reasoning, not an optional step after simplification. Once the units match, the ratio can be expressed using numbers alone.

Worked example 3. Two heights are 1.50 m and 75 cm. Find their ratio in that order, using 1 m = 100 cm.

Answer: 1.50 m = 150 cm. The ratio is 150 : 75 = 2 : 1. The first height is twice the second height.

Record the converted value beside its unit before simplifying. This makes it possible to check whether a mistake came from the conversion or the division. Keep the original order throughout: conversion changes the unit of measurement, not which quantity comes first.

How do equivalent ratios help us compare quantities?

Equivalent ratios are ratios whose corresponding fractions are equal. They express the same comparison using different terms. The word equivalent refers to equality of value, rather than to the terms being identical as written.

Property: Equivalent ratios preserve the comparison

Multiplying both terms of a ratio by the same non-zero number gives an equivalent ratio. Dividing both terms by the same non-zero number also preserves its value. This follows from the corresponding rule for equivalent fractions.

For example, 4 : 7 and 20 : 35 are equivalent because multiplying both terms of 4 : 7 by 5 gives 20 : 35. Equally, dividing both terms of 20 : 35 by 5 returns 4 : 7.

How can fractions show that ratios differ?

Write each ratio as a fraction and simplify. If the fractions are equal, the ratios are equivalent. Alternatively, express the fractions with a common denominator. Such fractions are called like fractions; their numerators can then be compared directly.

Worked example 4. Decide whether 12 : 18 and 28 : 56 are equal.

Answer: 12 : 18 simplifies to 2 : 3, while 28 : 56 simplifies to 1 : 2. The fractions 2/3 and 1/2 are different, so the original ratios are not equal.

The simplification method compares values rather than appearances. Large terms do not by themselves make a ratio larger. What matters is the quotient obtained when the first term is divided by the second term.

Use one operation on both terms when generating an equivalent ratio. Changing just one term changes the fraction. When checking an answer, trace the multiplier or divisor used on each term and confirm that it is the same.

Note: Simplifying each ratio separately is often convenient. If the simplified fractions still look different, compare them using a common denominator rather than assuming that different-looking terms prove inequality.

What is a proportion, and how do we find a missing term?

Definition: A proportion states that two ratios are equal. Four quantities are in proportion when the ratio of the first to the second equals the ratio of the third to the fourth.

The notation a : b :: c : d means a : b = c : d. Here a, b, c and d represent the four quantities in order; b and d must not be zero. The symbol :: means that the ratios form a proportion.

Property: Cross products in a proportion are equal

For a/b = c/d, the cross products are a × d and b × c. A product is the result of multiplication. These products are equal when the fractions form a proportion.

Thus 6/8 = 9/12 can be checked using 6 × 12 = 8 × 9. Both products are 72. Keep the fractions in their original order while choosing the diagonally opposite numerator and denominator for each multiplication.

Worked example 5. Find the missing number in x : 8 = 12 : 32, where x represents the missing first term.

Answer: 12 : 32 simplifies to 3 : 8, so x = 3. Checking by cross products gives 3 × 32 = 8 × 12 = 96.

Why is the order of all four terms important?

The numbers 3, 8, 24 and 64 are in proportion in that order: 24 : 64 reduces to 3 : 8. The order 3, 8, 64 and 24 does not make a proportion because it reverses just the second ratio.

For a missing term, first see whether an equivalent-ratio step gives it directly. If not, write the fractions and use their equal cross products. Finally substitute the answer into the original proportion to check the relationship that was actually asked for.

How does the unitary method lead to a general expression?

The unitary method finds the value for one unit first and then the value for the required number of units. In a cost problem, the unit value is the cost of one item or one unit of mass.

Use the same rate throughout the calculation. A rate relates one quantity to another, such as the cost per kilogram. Here kg means kilogram, a unit of mass; “per kilogram” means for each kilogram.

How can the method be written with letters?

Let C be the total cost of n units, where n is positive, and let m be the number of units required. At the same cost per unit, one unit costs C/n and m units cost (C/n) × m.

This general expression follows the same two operations as the verbal method. First divide by the number of units already priced. Then multiply by the number wanted. Define what one unit represents before using the expression.

Worked example 6. Twenty tons of iron cost ₹600000. Find the cost of 560 kg at the same rate. Use 1 ton = 1000 kg.

Answer: Twenty tons = 20000 kg. One kilogram costs ₹600000 ÷ 20000 = ₹30. Therefore 560 kg costs ₹30 × 560 = ₹16800.

The intermediate value ₹30 is a price for one kilogram, not the answer to the entire question. The required cost comes after multiplying by 560. Writing each stage with its quantity and unit helps distinguish the rate from the final total.

How can the result be checked?

Read the question again and identify the quantity being requested. Check that the answer has the matching unit and that the same rate has been used. A calculation may contain correct arithmetic but answer for one unit instead of the required quantity.

Unitary reasoning also connects to percentages and interest: establish the value associated with a known number of units, reduce it to one unit, and build up to the requested number.

How do we divide a total in a given ratio?

A ratio between two parts is different from the fraction that either part forms of the total. To divide a total in a ratio, add the ratio terms to find the total number of equal shares. Then find the value of one share.

The ratio terms count shares, rather than giving the final sizes of the parts. Multiplying each term by the value of one share produces the two required quantities. Their sum should give the original total, and their ratio should match the stated ratio.

How do we move from parts to the whole?

1. Add the two ratio terms to find the total number of shares.
2. Divide the given total by that number to find one share.
3. Multiply one share by each ratio term separately.
4. Check both the sum of the parts and their simplified ratio.

Worked example 7. Divide a line segment 56 cm long into two parts in the ratio 2 : 5.

Answer: There are 2 + 5 = 7 shares, in total. One share is 56 ÷ 7 = 8 cm. The two lengths are 2 × 8 = 16 cm and 5 × 8 = 40 cm.

The first part forms 2/7 of the entire length and the second forms 5/7. The fraction 2/5 compares the first part with the second part; it does not describe the first part as a fraction of the whole length.

For the check, 16 + 40 = 56 cm, and 16 : 40 simplifies to 2 : 5. Both checks matter. Getting the correct total by itself does not establish that the division has the required ratio.

Note: In a part-to-part ratio, the denominator of the corresponding fraction represents the second part. For a part-to-whole fraction, the denominator represents the combined total. Read those labels before calculating.

How are percentages related to fractions and decimals?

A percentage expresses a comparison per hundred. The symbol % means per cent, or per hundred. This gives a common basis for comparisons: the fraction is expressed in terms of a denominator of 100.

Property: Per cent means hundredths

For a number p representing a percentage value, p% means p/100. To write a fraction as a percentage, multiply its value by 100 and attach %. To reverse the conversion, divide the percentage number by 100.

A decimal is a number written using decimal place value. The same conversions apply to decimals: multiply by 100 to find the percentage number, or divide the percentage number by 100 to obtain the decimal.

How can we check a conversion in both directions?

Worked example 8. Write 0.05 as a percentage and convert that percentage back to a decimal.

Answer: 0.05 × 100 = 5, so the percentage is 5%. Reversing the conversion gives 5/100 = 0.05. The decimal and percentage express the same value.

Do not attach a percentage sign to a decimal without adjusting the number. The sign itself means division by 100. The number written before it therefore changes even though the value being represented stays the same.

A percentage can exceed 100 when the compared quantity exceeds the reference quantity. To find what percentage ₹9000 is of ₹4500, use 9000/4500 × 100 = 200%. The reference quantity is ₹4500, so it belongs in the denominator.

Read the words “of” and “what percentage of” carefully. They identify the whole used as the basis of comparison. Changing that reference changes the percentage even if the same two quantities are involved.

How do we calculate a percentage of a quantity?

To find a stated percentage of a quantity, replace the percentage with its fraction over 100 and multiply by the quantity. The result has the same unit as the original quantity. The percentage describes a proportion; the result gives an actual amount.

To find what percentage a part forms of a whole, divide the part by the whole and multiply by 100. Identify the whole before writing the fraction. A total number of people, a distance or an amount of money can provide that reference.

How does the percentage calculation work?

Worked example 9. Find 40% of 250 km, where km means kilometre, a unit of distance.

Answer: 40% means 40/100. Thus 40% of 250 km is (40/100) × 250 km = 100 km. The answer is a distance, so km remains attached to the result.

The same reasoning can begin with one per cent: divide the whole by 100, then multiply by the percentage number. This is a unitary approach to percentage calculation. It explains the operations rather than treating them as an unrelated formula.

Worked example 10. A class has 50 children, of whom 20 are boys and the rest are girls. Find the percentage of girls.

Answer: The number of girls is 50 − 20 = 30, after subtracting the boys from the total. Girls therefore form 30/50 of the class. Their percentage is (30/50) × 100 = 60%.

What if the given number is not the required part?

In the class example, the question supplies the number of boys but asks about girls. First calculate the required part, then compare it with the whole. Using 20/50 would answer a different question about the boys.

Label the part and whole in words whenever the context includes several quantities. This habit is useful in profit and loss too, where the relevant comparison is with the cost price rather than whichever price appears nearest to the percentage.

How do percentages describe profit and loss?

The cost price, written CP, is the price paid to buy an item. The selling price, written SP, is the price for which it is sold. Compare those two amounts for a single transaction.

A profit occurs when the selling price exceeds the cost price; it equals SP − CP. A loss occurs when the cost price exceeds the selling price; it equals CP − SP. Equal prices mean no profit or loss.

Which price is the basis of the percentage?

Profit per cent = (Profit/CP) × 100. Similarly, Loss per cent = (Loss/CP) × 100. Both percentages use the cost price as the denominator. Here the cost price is positive, and the profit or loss is the difference between the two prices.

Worked example 11. A bicycle is bought for ₹1800 and sold at a profit of 12%. Find its selling price.

Answer: Profit = (12/100) × ₹1800 = ₹216. Therefore SP = CP + Profit = ₹1800 + ₹216 = ₹2016.

The percentage tells us how much is gained relative to the buying price. It is not itself the selling price. First calculate the gain in money, then add it to the cost price to find the amount received on selling.

Worked example 12. An article is sold for ₹329 at a loss of 6%. Find its cost price.

Answer: The selling price is 100% − 6% = 94% of the cost price. Therefore CP = ₹329 × 100/94 = ₹350. The loss is ₹350 − ₹329 = ₹21.

When a loss percentage and selling price are given, subtracting that percentage of the selling price uses the wrong reference amount. Instead, express the selling price as the remaining percentage of the cost price, then recover the whole.

Check that a profit gives a selling price above the cost price, and a loss gives a selling price below it. Then check the percentage using the original cost price. This verifies both the direction of the change and its size.

How is simple interest calculated for complete years?

The principal is the original money borrowed. Interest is the extra money paid for using that money over a given time. The time period is how long the money is borrowed, and the rate of interest determines the interest payable.

The rate of interest is generally given in per cent per year. The expression per annum means per year. For these simple-interest calculations, take the time period in complete years and apply the stated annual rate to the principal.

What do the letters in the formula represent?

Let P represent principal, R the numerical annual percentage rate, T the time in years, I simple interest, and A the amount, which is principal plus interest. Then I = P × R × T / 100 and A = P + I.

The formula follows unitary reasoning. For one year, interest is R/100 of the principal. For T complete years at that simple-interest rate, multiply the one-year interest by T. Keep all money values in the same unit.

Worked example 13. Find the simple interest on ₹30000 for 3 years at 15% per annum.

Answer: P = ₹30000, R = 15 and T = 3. Therefore I = 30000 × 15 × 3 / 100 = ₹13500. This is the interest, not the principal plus interest.

How do we calculate the total amount?

Worked example 14. Find the amount on ₹3000 for 2 complete years at 11% per annum simple interest.

Answer: I = 3000 × 11 × 2 / 100 = ₹660. Then A = P + I = ₹3000 + ₹660 = ₹3660.

Before substituting, identify whether the question asks for interest or amount. The multiplication-and-division calculation gives the interest. Add the principal only when the total amount is required. Distinguishing the two quantities prevents an otherwise correct calculation from giving the wrong requested result.

How are speed, distance and time connected?

Speed is distance travelled divided by time taken. It describes the distance covered per unit of time. For a journey measured in kilometres and hours, km/h means kilometres per hour; an hour is the time unit used in that rate.

Let s represent speed, d distance travelled and t time taken. Then s = d/t, with positive time. For travel at a uniform speed, d = s × t and t = d/s, with positive speed for the last calculation.

How can we compare two speeds?

Worked example 15. A scooter travels 120 km in 3 hours and a train travels 120 km in 2 hours. Find the ratio of their speeds, scooter to train.

Answer: The scooter's speed is 120 ÷ 3 = 40 km/h. The train's speed is 120 ÷ 2 = 60 km/h. Their ratio is 40 : 60 = 2 : 3.

The distances in this example are equal, but the times are different. Comparing distances alone would miss the speed comparison. Find the distance per hour for each journey, then compare those rates in the order requested.

How does uniform speed support the unitary method?

Uniform speed means that the speed stays the same. With this condition, the distance travelled in one hour can be used to calculate the time for a different distance. Keep the distance and time units consistent throughout.

Worked example 16. A train covers 130 km in 2 hours. At the same uniform speed, how long will it take to cover 780 km?

Answer: Its speed is 130 ÷ 2 = 65 km/h. The required time is 780 ÷ 65 = 12 hours. Checking gives 65 × 12 = 780 km.

State the uniform-speed condition when extending a rate from one journey to another. A known distance and time establish a rate for the given journey; applying that rate elsewhere requires the stated assumption that it remains the same.

Glossary

• Ratio — A comparison of two numbers or quantities by division, with the order of the quantities preserved.
• Equivalent ratios — Ratios whose corresponding fractions have equal values and therefore express the same comparison.
• Proportion — An equality between two ratios, involving four quantities written in a particular order.
• Cross products — Products formed by multiplying each fraction's numerator by the other fraction's denominator.
• Unitary method — A method that first finds the value of one unit, then the value of the required units.
• Percentage — A comparison per hundred, represented by a fraction with a denominator of one hundred.
• Cost price — The price paid to buy an item, used as the basis for profit and loss percentages.
• Selling price — The price received when an item is sold, compared with its cost price to identify profit or loss.
• Profit — The amount by which an item's selling price exceeds its cost price in a transaction.
• Loss — The amount by which an item's cost price exceeds its selling price in a transaction.
• Principal — The original sum of money borrowed, on which simple interest is calculated.
• Simple interest — Interest calculated using the principal, the stated rate per year and the time in years.
• Amount — The total money payable, obtained by adding the interest to the principal.
• Speed — The distance travelled divided by the time taken, expressed as distance per unit of time.
• Uniform speed — Speed that stays the same, allowing the same distance-per-time rate to be used throughout.

Common errors and misconceptions

• Misconception: The ratio of ₹8 to 80 paise is found by comparing 8 with 80 directly. Correct: Convert ₹8 into 800 paise first; the ratio is 10 : 1.
• Misconception: Reversing the terms leaves any ratio unchanged. Correct: Order matters. The ratios 5 : 4 and 4 : 5 express different comparisons.
• Misconception: A part-to-part ratio gives each part as a fraction of the whole. Correct: Add the ratio terms first; a division in 2 : 5 gives fractions 2/7 and 5/7 of the whole.
• Misconception: The unitary method ends when the value of one unit is found. Correct: Use that value to calculate the required number of units and state the requested total.
• Misconception: A decimal becomes a percentage by attaching the percentage sign. Correct: Multiply by 100 to find the percentage number; 0.05 is 5%.
• Misconception: Profit and loss percentages use the selling price as their base. Correct: Divide the profit or loss by the cost price before multiplying by 100.
• Misconception: Simple interest and amount are the same quantity. Correct: Interest is the extra money; amount is the principal plus that interest.
• Misconception: Equal distances imply equal speeds. Correct: Speed also depends on time; compare distance divided by time for each journey, using the same units.

Exam-style questions with model answers

Q1. Find the ratio of ₹8 to 80 paise in lowest form. Use 1 rupee = 100 paise. [2 marks]

1. Express both amounts in paise: ₹8 equals 800 paise, while the second amount is already 80 paise.
2. The required ratio is 800 : 80 = 10 : 1, after dividing both terms by 80.

Q2. Find x in x : 8 = 12 : 32, where x is the missing number, and verify your answer. [3 marks]

1. Simplify the known ratio by dividing both terms by 4: 12 : 32 = 3 : 8.
2. The missing ratio has the same second term, 8, so its first term must be 3. Hence x = 3.
3. Verify with cross products: 3 × 32 = 96 and 8 × 12 = 96. Their equality confirms the proportion.

Q3. Twenty tons of iron cost ₹600000. Find the cost of 560 kg at the same rate. Use 1 ton = 1000 kg. [3 marks]

1. Convert the known mass into the requested unit: 20 tons = 20 × 1000 = 20000 kg.
2. Find the cost of one kilogram by dividing the given total cost by its mass: ₹600000 ÷ 20000 = ₹30.
3. Multiply the unit cost by the required mass: ₹30 × 560 = ₹16800.
4. Therefore 560 kg costs ₹16800 at the same rate of ₹30 per kilogram; the answer is a cost in rupees.

Q4. A line segment 56 cm long is divided into two parts in the ratio 2 : 5. Find both lengths and check the result. [5 marks]

1. The ratio 2 : 5 describes two groups of equal shares. Add its terms to obtain 2 + 5 = 7 shares in the whole line segment.
2. Find one share by dividing the total length by the number of shares: 56 ÷ 7 = 8 cm.
3. The first part contains two shares, so its length is 2 × 8 = 16 cm.
4. The second part contains five shares, so its length is 5 × 8 = 40 cm.
5. Check the total, 16 + 40 = 56 cm, and the ratio, 16 : 40 = 2 : 5. Both match the given conditions.

Q5. A class contains 50 children: 20 boys and the rest girls. Find the percentage of girls. [3 marks]

1. The required group is the girls, so subtract the number of boys from the class total: 50 − 20 = 30 girls.
2. Use all 50 children as the whole. The fraction of the class who are girls is therefore 30/50.
3. Convert that fraction to a percentage: (30/50) × 100 = 60%. Thus girls make up 60% of the class.

Q6. A bicycle is bought for ₹1800 and sold at a profit of 12%. Find the profit in rupees and the selling price. [4 marks]

1. The cost price is ₹1800. The profit percentage is calculated using this buying price as its reference amount.
2. Write 12% as 12/100 and multiply by the cost price: Profit = (12/100) × ₹1800 = ₹216.
3. Add the profit to the buying price: Selling price = ₹1800 + ₹216 = ₹2016.
4. The bicycle is therefore sold for ₹2016, giving a profit of ₹216. Its selling price exceeds its cost price, as required for a profit.

Q7. Find the simple interest and total amount on ₹3000 for 2 complete years at 11% per annum. Explain the distinction between interest and amount. [6 marks]

1. The principal is the original sum, ₹3000. This is the money on which the simple interest must be calculated.
2. The annual rate is 11%, meaning eleven per hundred for each year, and the stated time is 2 complete years.
3. Use Simple interest = Principal × numerical annual percentage rate × time in years / 100.
4. Substitute the given values: Simple interest = 3000 × 11 × 2 / 100 = ₹660.
5. The total amount is principal plus interest, so Amount = ₹3000 + ₹660 = ₹3660.
6. Interest is the extra ₹660 for using the money; the amount of ₹3660 includes both the original principal and that interest.

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

1. Use speed = distance travelled divided by time taken. Since both distances are in kilometres and both times in hours, the speeds have matching units.
2. The scooter's speed is 120 ÷ 3 = 40 kilometres per hour.
3. The train's speed is 120 ÷ 2 = 60 kilometres per hour.
4. In the requested order, the speed ratio is 40 : 60 = 2 : 3. The scooter's speed forms the first term.

Key takeaways

• A ratio compares quantities by division; preserve the requested order and convert measurements into the same unit before simplifying.
• Equivalent ratios represent equal fractions, and a proportion states that two ratios have the same value.
• The unitary method first finds a value for one unit, then uses it to find the value for the required quantity.
• To divide a total in a ratio, add the ratio terms, find one share and multiply by each term.
• A percentage means per hundred; convert it into a fraction over one hundred before calculating a percentage of a quantity.
• Profit and loss compare selling price with cost price, while both percentage calculations use cost price as their reference.
• Simple interest uses principal, annual percentage rate and time; the total amount includes both the principal and the calculated interest.
• Speed is distance divided by time; use matching units and the stated uniform-speed condition when applying a rate to another journey.

Test yourself

What operation defines a ratio, and why is order important?

A ratio uses division. Reversing its terms changes which quantity is divided by which, so it can change the comparison.

Are 4 : 7 and 20 : 35 equivalent?

Yes. Dividing both terms of 20 : 35 by 5 gives 4 : 7, so their corresponding fractions are equal.

What condition do cross products satisfy when two fractions are in proportion?

The cross products are equal: each numerator multiplied by the other fraction's denominator gives the same result.

What are the two main stages of the unitary method?

First find the value for one unit. Then use it to calculate the value for the required number of units.

Convert 0.05 to a percentage.

Multiply 0.05 by 100 to obtain the percentage number 5. Therefore the equivalent percentage is 5%.

Find 40% of 250 km.

Write 40% as 40/100 and multiply by 250 km. The resulting distance is 100 km.

Which price is the denominator when finding profit or loss per cent?

The cost price is the denominator. Divide the profit or loss by the cost price, then multiply by 100.

How do interest and amount differ?

Interest is the extra money paid for using the principal. Amount is the total of principal and interest.