Chapter 2 — Ratio and Proportion

What is a ratio?
Ratio is a way to compare two quantities of the same kind. When we say the ratio of A to B is 3:2, it means that for every 3 units of A there are 2 units of B. A ratio does not give exact amounts unless one of the quantities is known; instead it shows a rule or relationship between the quantities. Ratios are used when the items compared are of the same type, for example litres with litres, rupees with rupees, or number of students with number of teachers.

Different forms of a ratio
A ratio can be written in several equivalent forms. The colon form uses a colon: 3:2. The fraction form writes it as 3/2. And in words we write '3 to 2'. All three forms express the same idea. We must be careful about order: the ratio 3:2 is not the same as 2:3. The first part always refers to the first quantity named and the second part to the second quantity.

How to read and use ratios
Think of a ratio as instructions to build groups. If a toy mixture requires paint and thinner in ratio 5:1, this means make groups where paint has five parts and thinner one part. To make many groups, repeat the parts. If you want actual amounts, you must know the size of a part or the total. For example, if the total mixture is 12 litres and the ratio is 2:1, you find total parts 3 and each part equals 4 litres. Then allocate 8 litres and 4 litres accordingly.

Visualising ratios
Using bars, blocks or number lines helps children see the relation. Draw a bar split into parts: if ratio is 3:2, draw a bar with 5 equal parts, shade 3 for one quantity and 2 for the other. This makes it easy to convert the ratio to actual values when the size of one part is known. Working with simple whole-number examples first builds the idea clearly before moving to fractions or decimals.

Why simplify a ratio?
Simplifying a ratio makes the numbers smaller while keeping the same relationship. Smaller numbers are easier to work with, compare and use in calculations. For example, 18:24 looks larger than 3:4, but they show the same relationship. Simplifying also helps when sharing or scaling quantities because it shows the basic pattern of parts.

How to simplify
To simplify a ratio, find a common factor of both numbers and divide both by it. If you can do this repeatedly until no common factor greater than one remains, the ratio is in its simplest form. The best factor to use is the greatest common divisor (GCD), because dividing by the GCD gives the final simplest form in one step. For example, to simplify 28:42, the GCD of 28 and 42 is 14. Dividing both by 14 gives 2:3. If students cannot find the GCD quickly, they can use repeated division by small primes (2,3,5,7) until no longer possible.

Methods to find GCD
One practical method is prime factorisation: write each number as a product of primes and then take the common primes. For 18 and 24: 18 = 2×3×3, 24 = 2×2×2×3. The common primes are 2 and 3, multiply them to get GCD = 6. Dividing both by 6 gives 3:4. Another simple method is trial division: keep dividing both numbers by the smallest common factor you spot until they share no more factors.

Checking and practising
After simplifying, check that the two numbers share no factor other than 1. Practise with different pairs so students become quick at spotting factors. Also practice simplifying ratios arising from real situations like recipes, money sharing, and classroom counts. Remind students that simplifying changes appearance but not the relationship; 50:75 and 2:3 are the same ratio in different forms.

Understanding equivalent ratios
Equivalent ratios look different but show the same relationship between two quantities. For example, 2:3 and 6:9 are equivalent because both tell us that for every 2 units of the first quantity there are 3 units of the second. Equivalent ratios are found by multiplying or dividing both parts of a ratio by the same non-zero number. This is useful when comparing situations that use different units or scales.

How to make equivalent ratios
Start from a base ratio and multiply both parts by the same factor. For instance, from 3:5 multiply by 2 to get 6:10, multiply by 3 to get 9:15, and so on. Conversely, if both parts share a common factor you can divide by it to reduce to a simpler equivalent ratio. Working with tables helps to list many equivalents at once. A ratio table has rows showing k×a and k×b for k = 1, 2, 3,.. This visual method makes it easy to find a matching pair when one part is given in a problem.

Using equivalence to solve problems
Equivalent ratios let you scale quantities up or down. If a recipe uses 2 cups of milk for 3 cups of flour and you want to make more, use equivalent ratios to find how much of each ingredient. If you know one part of an equivalent ratio, search the table or compute the appropriate factor. For classroom questions, verify equivalence by cross-multiplying or simplifying both ratios and checking if results match.

Common mistakes and tips
Remember not to multiply just one part; both parts must be changed by the same factor. Order matters: 2:3 multiplied by 2 becomes 4:6, not 6:4. When working with larger numbers always check by simplifying the result or using cross-product test a×d = b×c to avoid errors. Practice with different factors and real-life examples like scaling maps, adjusting recipes or enlarging drawings to get comfortable creating and using equivalent ratios.

What is the unitary method?
The unitary method is a simple two-step process used to find the value of one unit and then scale it to find the value of many units. It is especially useful when a known number of items has a total value and you need the value of any other number of items. The idea is to first find the value for 1 unit by dividing, and then multiply to find the desired number of units.

Step-by-step procedure
Step 1: Find the value of one unit by dividing the total value by the number of units. Step 2: Multiply that unit value by the required number of units to get the result. For example, if 8 notebooks cost Rs 240, the cost of one notebook is 240 ÷ 8 = 30. Thus 5 notebooks cost 5 × 30 = Rs 150. This method works equally well for measures of weight, distance, or time when quantities are directly proportional.

Connection with ratios
Unitary method is closely linked to ratios because a ratio gives parts and unitary method assigns an actual value to one part. In ratio language, if a quantity is divided in ratio a:b and the total is known, the unitary method helps find the value of one part and then the share corresponding to a or b. This makes the unitary method a useful bridge between abstract ratios and concrete numbers.

Using unitary method in word problems
Identify the known total and the number of equal parts it represents. Always check units (rupees, kg, litres) and keep work organized: write down one-unit value clearly, then perform multiplication. Use the unitary method for pricing problems, scaling quantities in recipes, converting map scales and finding per-item rates. For practice, try both direct use and use combined with ratio tables to strengthen understanding.

Understanding proportion
Proportion means two ratios are equal. When we write a:b = c:d we say the ratio of a to b is equal to the ratio of c to d. Proportion shows that two different pairs of quantities have the same relationship. This idea is useful when comparing rates, solving for unknowns and working with scale drawings or maps.

Cross-multiplication test
A quick test for proportion is cross-multiplication. For a:b = c:d, multiply the outer terms a and d, and the inner terms b and c. If a×d = b×c, the two ratios are in proportion. This method is simple and reliable when numbers are whole. It also helps to find a missing term: if one of a, b, c, d is unknown, rearrange the cross-product equation to solve for it.

Solving problems using proportion
To solve a proportion problem, identify the equal ratios, set them equal and use cross-multiplication. For example, if 4 pens cost Rs 20, and we want the cost of x pens for a different number, set 4:20 = x:? or use unitary method. In school problems you often see questions like 5:8 = 15:x; use cross-product to get 5×x = 8×15 and solve x. Keep steps neat so reversing is easy to check.

Visual and practical use
Use drawings or ratio tables to represent proportions visually. For scale maps, the proportion between map distance and real distance stays constant. Proportions also appear in mixing problems where different substances must be added in a fixed ratio. Encourage students to check their answers by substituting back into the original ratio and verifying equality. Practise problems where both ratios vary so students learn when proportions apply and how to manipulate them accurately.

Definition and examples
Direct proportion describes a relationship where two quantities increase or decrease together, keeping the same ratio. If x is directly proportional to y, we write x ∝ y, which means x/y is constant. A simple example is cost and number of items at a fixed price: if one pen costs Rs 5, then two cost Rs 10, three cost Rs 15, and so on. The cost and number of pens are directly proportional.

Finding the constant of proportionality
To work with direct proportion, find the constant k such that x = k×y. From an example, if 4 books cost Rs 120, then cost per book k = 120 ÷ 4 = 30. Then cost for any number y is x = 30×y. Using this constant helps solve many problems quickly. It is the same as using the unitary method, but stated as a fixed multiplier.

Solving word problems
Read the problem to identify the two directly proportional quantities. Either set up a proportion x1:y1 = x2:y2 or compute the constant k. Use cross-multiplication or multiplication by k to find the missing value. For example, if 5 workers make 100 toys in one day, then 10 workers (double) make 200 toys. Practice with price, distance-speed-time (when time is constant), and recipe scaling problems to see direct proportion in many contexts.

Using tables and checks
Ratio tables are useful for direct proportion: list multiples and find matching rows. Always check the result by confirming the ratio remains constant. For numerical accuracy, keep units consistent and round only at the end. Encourage students to explain why quantities are directly proportional in the problem so they apply the right method and avoid confusion with inverse situations.

Basic idea of inverse proportion
Inverse proportion is a relationship where one quantity increases and the other decreases so that their product remains constant. If x and y are inversely proportional, we write x ∝ 1/y or x×y = constant. This often appears in work-time problems: more workers mean fewer days needed to finish the same job, assuming all workers are equally efficient.

Simple examples and reasoning
Suppose 4 workers finish a job in 6 days. The total work can be thought of as 4×6 = 24 worker-days. If more workers are assigned, say 8 workers, they will share the same total work, so days = 24 ÷ 8 = 3 days. Here the product (workers × days) stays the same. Another example: if one tap fills a tank in 10 hours, two identical taps will fill it in 5 hours because the filling rate doubles and time halves.

Solving using inverse proportion
Set up the equality x1×y1 = x2×y2 for two situations and solve for the unknown. Identify which pairs are inversely related. Write the given pairs, compute the constant product if helpful, and then rearrange to find the missing value. Keep work neat to avoid mixing up with direct proportion problems. For young learners, practise with whole numbers and small integer answers to build confidence.

Limitations and when not to use
Inverse proportion applies only when the total quantity being shared remains fixed and when all units behave similarly (workers have same efficiency). It does not apply to problems where adding more of one thing changes the total or efficiency. Discuss real-life examples to show when inverse proportion is appropriate and when it is not. This helps students choose the correct approach in mixed problems.

Purpose and approach
Many problems ask to split a total amount among people or parts in a given ratio. The process is systematic: first find the total number of parts, then determine how much each part equals, and finally multiply the part value by each ratio number to get every share. This method works for money, sweets, time, or any divisible quantity.

Step-by-step method
Step 1: Add the numbers of the ratio to get the total parts. Step 2: Divide the total quantity by the total parts to find the value of one part. Step 3: Multiply that value by each ratio term to obtain individual shares. Example: share Rs 360 in ratio 2:3:1. Total parts = 6. One part = 360 ÷ 6 = 60. Shares are 120, 180 and 60. Always write each step to avoid mistakes.

Using mixed ratios and more than two parts
When there are three or more parts, the method is the same. If the ratio includes fractions or decimals first convert them to whole numbers by multiplying through by a common factor. Remember to simplify the ratio if possible before dividing. If the ratio is given in simplest form, one can directly find the value of a part; otherwise clear denominators first to work with whole-number parts.

Checking your answer
After calculating shares, add them to ensure the sum equals the original total. Also check the ratio between the shares by simplifying them to confirm they match the given ratio. Use bar models to visualise the splits: draw a long bar divided into equal parts and write the value of each part below. This helps students see why the method works and makes checking straightforward.

What is a ratio table?
A ratio table is a two-column (or more) chart showing pairs that follow the same ratio by multiplying both parts by the same multiplier. It is a simple visual tool that helps us see how numbers grow or shrink together. Start with the simplest form of the ratio and then list multiples. For example, for ratio 2:5 the table rows are (2,5), (4,10), (6,15), (8,20) and so on.

How to use the table
To solve a problem, build the table and look for a row where one column matches the given value. Then read the other column to find the unknown. If the exact value does not appear, you can use fractional multipliers too. Ratio tables are very helpful when students prefer mental multiplication and avoid formal algebra. They also help in understanding scaling in steps.

Benefits and strategies
Ratio tables make it easy to compare different multiples and find matching pairs. They are especially useful for unit price problems, adjusting recipes, and map scale questions. To make the table effective, start with k = 1 row to see the unit relationship. Then increase k progressively until the needed value is found. For larger numbers use division first to find the multiplier quickly: if second term is 150 and base second term is 10, multiplier k = 150 ÷ 10 = 15.

Practical tips and checks
Keep the table neat, label columns clearly, and include a row for k = 1 to identify one unit value. Check answers by simplifying the resulting pair to ensure the original ratio remains. Use the table together with unitary method: find the one-unit row then scale. Practice building tables quickly to gain fluency, and use them to teach proportional reasoning before introducing formal algebraic methods.

Translate words into maths
Word problems ask students to read, understand and convert a real-life situation into mathematical statements using ratios or proportions. Begin by underlining key numbers and identifying what is being compared. Decide whether the problem needs dividing a total in a ratio, scaling using the unitary method, or solving a proportion. Write the known ratio clearly and mark the unknown with a letter or box.

Strategy to solve
1) Read carefully and identify quantities and their relation. 2) Choose a method: direct proportion, inverse proportion, unitary method, ratio table or divide-in-ratio. 3) Set up the mathematical equation (for example a:b = c:d or T ÷ (a+b) to find one part). 4) Solve step by step, showing working. 5) Check the answer by substituting back and confirming that totals match or ratios hold true.

Types of routine problems
Common problems include sharing money in a ratio, mixing ingredients in given proportions, scaling recipes or converting map distances to actual distances. Some involve finding unit rates such as price per kg. Others ask to compare two ratios or find how many times greater one quantity is than another. Clear diagrams or bar models often make the setup easier and reduce errors.

Pitfalls and tips
Avoid mixing units; convert them to same unit before applying ratios. When fractions or decimals appear, clear denominators or multiply to remove decimals for easy calculation. Always verify final answers against the context of the problem to ensure reasonableness. Encourage students to write short sentences explaining each step; this practice improves understanding and helps spot mistakes during checking.

Why compare ratios?
Often we need to know which of two ratios is larger or whether two ratios are equal. Comparing ratios is similar to comparing fractions. Use cross-multiplication to compare without converting to decimals. This keeps calculations exact and avoids rounding errors.

Cross-multiplication explained
For two ratios a:b and c:d, compare a/b and c/d by computing a×d and b×c. If a×d > b×c, then a/b > c/d. If a×d = b×c, the ratios are equal; if a×d < b×c, then a/b < c/d. This method works because cross-multiplying eliminates the denominators and compares whole numbers directly. It is the same check used to test proportions where equality is expected.

Step-by-step use in problems
To use cross-multiplication: (1) Write both ratios as fractions. (2) Multiply the numerator of the first by the denominator of the second (outer product). (3) Multiply the denominator of the first by the numerator of the second (inner product). (4) Compare the two products to decide which ratio is larger or whether they are equal. When one term is unknown in a proportion, set up the equation a×d = b×c and solve for the unknown by simple division.

Worked application and checks
Example: compare 5:8 and 3:5. Compute 5×5 = 25 and 8×3 = 24, so 5:8 is larger. For solving 7:9 = x:18 use 7×18 = 9×x ⇒ x = (7×18)÷9 = 14. After finding x, always substitute back to check the ratio equality to avoid arithmetic slips. For comparison problems involving prices, speeds or densities, cross-multiplication gives a quick and accurate answer without decimals.

Tips, common mistakes and practice
Be careful to keep the order of terms correct when forming products; swapping numbers changes the result. Do not divide before multiplying, as this can cause rounding errors; multiply first then divide when solving for an unknown. When numbers are large, simplify by cancelling common factors before multiplying to keep calculations easy. Regular practice with varied examples (whole numbers, fractions, decimals) will help students become confident using cross-multiplication in exams and real life.

Ratios need not be whole numbers
A ratio can include fractions or decimals, for example 1/2 : 3/4 or 0.75 : 1.5. The methods used earlier still apply but you may first convert all parts to a common form to make simplifying and calculation easier. The key is to clear fractions or decimals so that you work with whole numbers when possible.

Working with fractional parts
If the ratio contains fractions, convert them to a common denominator and compare numerators. For 1/2 : 3/4, write both with denominator 4: 2/4 : 3/4, so the ratio of numerators is 2:3. If decimals appear, multiply both terms by a power of ten that removes decimals. For 0.5 : 1.25 multiply both by 100 to get 50:125, then simplify to 2:5. These steps avoid dealing with awkward fractional parts in later calculation.

When using unitary method
After clearing fractions or decimals and simplifying, use the unitary method as usual: find one part and scale. For example, if a mixture is 1/2 : 3/4 and the total quantity is known, first convert to 2:3 then divide the total into 5 parts. Show each transformation in steps so reasoning is clear and teachers can follow the work.

Care with rounding
If calculations produce decimals in the final answer, round only when the question asks or when necessary for practical units (for example money to two decimal places). Practise many examples converting between fractions and decimals inside ratios to build fluency. Using visual aids like fraction bars helps students see how fractional parts combine to form whole-number ratio parts.

Bringing ideas together
This final topic encourages students to use all the tools learned in the unit. Review the meaning of ratio, simplification, equivalent ratios, unitary method, proportion and the difference between direct and inverse proportion. Practise recognising which method suits a problem: whether to divide a total into parts, use unitary method, construct a ratio table, or set up a proportion and use cross-multiplication.

Structured practice
Begin with short numerical drills: simplify ratios, find equivalent ratios, and compute unit values. Move to slightly longer problems: divide quantities in given ratios, scale recipes and solve simple price or distance problems. Finally attempt mixed word problems that require choosing an approach. Encourage use of bar models and tables for visual reasoning. Organise practice in sets: five quick questions on simplification, five using unitary method, and five word problems to mix methods.

Checking and self-assessment
Teach students to check answers by reversing steps: add shares to confirm they equal the total, verify proportions by cross-multiplication, and ensure ratios simplify to the expected form. Encourage mental estimation before full calculation to see if the answer is reasonable. Keep a short checklist: identify quantities, choose method, compute one-unit value if needed, solve, and check.

Exam preparation tips
Practice writing neat steps and labelling units. Many exam questions award marks for correct method even if final arithmetic has small errors. Time yourself on mixed sets to build speed. Finally, solve a few past-style questions that combine ratios with fractions or decimals, and one or two inverse proportion questions, so you are ready for the variety that appears in tests.