Chapter 12 — Ratio and Proportion

Definition and basic idea
A ratio is a way to compare two amounts of the same kind. When we want to say how many times one quantity is in relation to another, we use a ratio. For example, if there are 6 red marbles and 3 green marbles, we say the ratio of red to green is 6:3. This tells us the relative sizes — not the exact amounts alone — but how they compare.

Different notations
We can write a ratio using a colon (6:3), the word 'to' (6 to 3), or as a fraction (6/3). All three are ways to express the same relation. Which form to use depends on the problem: the colon is common for counting objects, the fraction form helps if we need to divide or simplify, and the word 'to' is useful in spoken explanations.

Order and units matter
Always compare quantities of the same unit: litres to litres, rupees to rupees, or apples to apples. The order of terms matters in a ratio: 2:5 is different from 5:2. Read a ratio aloud to avoid mistakes: 2:5 is 'two to five' and means the first quantity is smaller.

Why ratios are useful
Ratios help in everyday choices: making mixtures, dividing money, comparing speeds, or understanding recipes. With ratios we can scale quantities up or down while keeping the same relationship. Learning ratios builds a foundation for fractions, percentages and algebra later in school.

Everyday examples and thinking
If a recipe needs 2 cups of flour and 1 cup of sugar, the ratio flour to sugar is 2:1. If a map shows 1 cm for 2 km, that is a ratio of map length to real distance. Practise by looking for two related quantities and writing their ratio; this helps the idea become natural.

Forms of writing
Ratios can be written as a:b, a to b, or a/b. Each form has a use. The colon form a:b is compact and used frequently in answers. The 'to' form is used when speaking, and the fraction form a/b is helpful when we want to simplify or calculate. For example, 8:2, 8 to 2 and 8/2 all represent the same comparison.

Choosing the right units
Always ensure both quantities are measured in the same unit before forming a ratio. If one length is in metres and the other in centimetres, convert one so both use the same unit. For example, 2 m and 150 cm become 200 cm and 150 cm, so the ratio is 200:150. After forming the ratio, simplify if needed.

Reading order and clarity
When reading a ratio aloud, say the first number, then 'to', then the second number. This helps avoid swapping terms which changes meaning. Also label the ratio when necessary, for instance ‘boys to girls 10 to 15’ rather than leaving it ambiguous. Labels are important in word problems to keep track of what each term stands for.

Writing in words and symbols
If you convert a ratio to fraction form, you can perform division and get decimal values which help in comparison. The fraction a/b equals the value of one part of the second quantity; for example, 6/3 = 2 tells you the first is twice the second. Use fraction division carefully and write simple steps in your working so marks are clear in exams.

Practice and conversion tips
Practice converting between forms: take several real life pairs (e.g., 3 apples and 5 oranges) and write the ratio in all three forms. When units differ, practise converting units first. This habit prevents errors in homework and tests.

Understanding equivalent ratios
Equivalent ratios show the same relationship between quantities although the numbers are different. If we multiply or divide both parts of a ratio by the same nonzero number, the new ratio keeps the same comparison. For example, 2:3 is equivalent to 4:6, 6:9, and 8:12. These are all the same relationship, just scaled up by multipliers 2, 3 and 4.

How to make them
To make an equivalent ratio multiply both terms by k (k ≠ 0): a:b = (a×k):(b×k). To reduce to smaller equivalents, divide both terms by the same factor. When dividing, always use a common factor of both numbers so the result remains whole numbers if that is required by the problem.

Why equivalent ratios help
Equivalent ratios let us compare values more easily by choosing convenient numbers. When solving problems, we often scale ratios until one term matches a given number, then read off the corresponding partner. This method is helpful in recipe scaling, sharing money, or matching units in map problems.

Checking equivalence
One way to check is to reduce both ratios to simplest form. If their simplest forms match, the ratios are equivalent. Another method is cross-multiplication: a:b and c:d are equivalent if a×d = b×c. Both checks are useful in tests. For whole-number problems prefer simplification; for quick checks use cross-products.

Practice patterns
Create ratio tables to list equivalent pairs and observe patterns: from 3:4 get 6:8, 9:12, 12:16, and so on. This strengthens quick recognition and mental arithmetic. Also practise creating equivalents where k is fractional, leading to fractional partners when necessary.

Meaning and importance
The simplest form of a ratio is the version where both numbers share no common factor other than 1. This is similar to reducing a fraction to lowest terms. Giving ratios in simplest form makes answers neat and helps comparison. For example, 12:18 in simplest form is 2:3 after dividing both by their greatest common divisor (GCD) 6.

Steps to simplify
1. Find a common factor of both numbers. Start with small primes such as 2, 3, 5, and so on. 2. Divide both numbers by that common factor and repeat until no further division is possible. 3. The result is the simplest form. For many Class 6 problems repeated trial by small primes works fine.

Using GCD
If you know the GCD of the two numbers, divide each term by the GCD to get simplest form in one step. For two numbers like 45 and 60, GCD = 15, so 45:60 reduces immediately to 3:4. Learning to spot common factors helps speed up exams. If one number is 1, the ratio is already simplest. If a term is 0 and the other positive, ratios like a:0 indicate all to one side; treat such cases carefully in context.

Practical checks
After simplifying, multiply back by the factor to verify you get the original numbers. Also ensure you maintain the order of terms: simplifying 8:12 gives 2:3, not 3:2. In sharing problems, always present simplest form when the question asks. Don’t simplify if the question asks for a certain total or asks explicitly not to reduce.

Common mistakes to avoid
Do not divide by different numbers for each term. Do not forget to convert units beforehand. Avoid treating negative numbers or zero incorrectly in real sharing contexts. Practice many examples to make simplification routine.

Purpose
Often we must compare two ratios to know which one represents a larger proportion. Comparing ratios helps in choosing the bigger share, better rate, or faster speed. For instance, which ratio gives more sugar per cup: 3:5 or 4:7? We need reliable methods to decide.

Method 1: Convert to decimals or fractions
Write each ratio as a fraction and divide: a:b becomes a/b. Compute decimals or compare common denominators. For example, 3:4 = 3/4 = 0.75 and 4:5 = 0.8 so 4:5 is larger. This method needs simple division but is straightforward for understanding.

Method 2: Cross-multiplication
Cross-multiplication avoids division. To compare a:b and c:d compute a×d and b×c. If a×d > b×c then a/b > c/d. This method works with whole numbers and gives an immediate answer without decimals. For example, compare 2:3 and 3:5: 2×5 = 10, 3×3 = 9, so 2:3 > 3:5.

Method 3: Bring to same first term or same second term
Scale equivalent ratios so one term matches, then compare the other. For example to compare 3:7 and 4:9, make first terms equal: multiply 3:7 by 4 to get 12:28 and 4:9 by 3 to get 12:27. Now 12:28 < 12:27 so 3:7 < 4:9. This is like making common denominators in fractions.

Practical checks and common errors
Always keep the order consistent and units the same. Convert both to simplest form to make quick mental checks. Avoid comparing 2:5 with 5:2 without noticing order. Practise several examples to build speed and confidence.

Definition and meaning
A proportion states that two ratios are equal. When we say a:b = c:d we mean the ratio of a to b is the same as the ratio of c to d. Proportion is a useful idea because it links four numbers together and allows us to find missing values or check if two pairs match the same relation.

Testing a proportion
The common test for proportion is cross-multiplication: a:b = c:d is true if and only if a×d = b×c. This is easy to use and works well with whole numbers. For example, to test whether 4:9 equals 8:18 compute 4×18 = 72 and 9×8 = 72; they match so the two ratios are in proportion.

Using proportion to find missing numbers
If three terms are known we can find the fourth using cross-multiplication. For a:b = c:d with d missing, compute d = (b×c)/a provided a ≠ 0. Make sure to check that the division gives the required form (whole number if the problem requires). After finding the unknown, substitute back and cross-check the equality.

Applications
Proportions are common in map reading, scaling models, mixing solutions, and sharing money proportionally. For example, if maps show 1:50,000 and another map shows 2:100,000, these are in proportion because the ratios reduce to the same simplest form. In recipes, proportions ensure the same taste when scaled up.

Common mistakes and tips
Keep the order of terms consistent; swapping terms changes the meaning. Always use same units before forming ratios. Practice many examples by both reducing to simplest form and using cross-products to become fluent in checking proportions quickly.

Situations with one unknown
In many problems one of the four numbers in a proportion is unknown. Using cross-multiplication we can find this missing term easily. For a:b = c:d, if any one of a, b, c or d is unknown, rearrange the cross-product a×d = b×c to solve for that unknown.

How to set up
Write the proportion carefully with each term in its correct place. If the unknown is in the second position, for example a:x = c:d, then x = (a×d)/c. If the unknown is the first term, x:b = c:d then x = (b×c)/d. Label the terms A, B, C, D if that helps to avoid confusion.

Whole number and fractional results
Solve the multiplication before division. If the division does not give a whole number, the answer may be fractional; check the context. In dividing things like people or whole items, fractional results may need interpretation (rounding or changing the problem). For measurements and money, fractional answers are acceptable if stated properly.

Checking and simplifying
After finding the missing term, substitute back and verify by cross-multiplication. Simplify the found value if possible and ensure the proportion remains correct in simplest terms. If you found the unknown by using an equivalent ratio table, check the scaling factor used is consistent.

Tips and common errors
Keep the order of the ratio fixed. Do not swap numerator and denominator accidentally. Use parentheses when writing expressions to avoid algebra mistakes: x = (a×d)/c. Practice many examples with the unknown in different positions so the method becomes automatic.

Concept
The unitary method finds the value of one unit first and then uses that to find the value of many units. It works when the relation between quantity and value is proportional. By finding the value of a single part (one unit), we can multiply to find any number of parts. This method is practical and often faster in daily problems.

Step-by-step approach
Step 1: Find the total number of equal parts or units in the given ratio or problem. Step 2: Divide the given total by the number of parts to find one part. Step 3: Multiply the value of one part by the required number of parts to find the desired value. For example, if 6 bottles cost Rs 180, cost of 1 bottle = 180 ÷ 6 = 30, so cost of 4 bottles = 30 × 4 = 120.

Applications
The unitary method is useful in scaling recipes, sharing money, finding price per item, and converting rates like km per hour or cost per kg. It is the practical face of ratios and proportions and links directly to the fraction form of ratios: one part is like dividing by the denominator in a/b and then multiplying by another numerator when needed.

Checking and careful use
Always check units and ensure exact division when a whole number is required. If the one-part value is fractional, decide whether the context allows fractions (e.g., weight or money) or requires whole numbers (people). Use the unitary method along with ratio tables for quick mental calculations and to avoid repeated multiplication steps.

Common mistakes to avoid
Do not forget to divide by the total number of parts before multiplying; many students reverse the steps. Label the unit clearly to avoid confusion when multiple ratios appear. Practise with several types of questions to build confidence.

Meaning
Direct proportion describes two quantities that change together in the same way. If one quantity doubles, the directly proportional quantity also doubles. We write this relation as y ∝ x or y = kx where k is constant. This constant gives the rate or cost per unit depending on the context.

Identifying direct proportion
To test if two quantities are in direct proportion, divide one by the other for several pairs of values. If the ratio y/x is the same each time, the quantities are directly proportional and that common value is k. For instance, if 2 kg of apples cost Rs 80 and 5 kg cost Rs 200, cost per kg is constant at Rs 40 so cost is directly proportional to weight.

Solving problems
When y = kx and k is known, find missing values by multiplication or division. To find k use k = y/x from given values. Then calculate unknowns: y2 = k×x2 or x2 = y2/k. In Class 6 problems these operations are simple and usually involve small whole numbers or decimals.

Applications
Direct proportion appears in money (cost and quantity), speed and distance when time is fixed, and material required per unit. If a photocopy machine copies 30 pages in 2 minutes, it will copy 60 pages in 4 minutes; pages copied are directly proportional to time.

Difference from other relations
Direct proportion is different from simple proportion statements because it relates variables continuously; proportion a:b = c:d is an equality of two ratios, while direct proportion y = kx shows how one variable depends on another with constant k. Both ideas are connected and used together in many problems.

Understanding the problem
Start by reading the question carefully and identifying what two quantities are compared. Decide which quantity is the first term and which is the second; write the ratio in that order. Underline numbers and labels so you do not confuse parts such as 'apples to oranges' or 'boys to girls'.

Plan the method
Decide whether to use unitary method, cross-multiplication, a ratio table, or simplification. For sharing problems use one-part value; for finding missing terms in equality of ratios use cross-multiplication; for scaling use direct proportion. Visual models like bars or pie charts often make the plan clearer.

Working step-by-step
1. Write the ratio and simplify if it helps. 2. If sharing a total, compute one part = total ÷ sum of parts. 3. If scaling from one pair to another, use equivalent ratios or cross-multiplication. 4. Work out numbers cleanly, show division and multiplication steps, and keep units. 5. Check that your shares add up to totals and that ratios hold after calculation.

Example problem types
Sharing money or sweets in a ratio, scaling a recipe, finding distance from a map scale, or mixing chemicals in lab practice fit this topic. Some problems combine steps: first scale, then share, or first convert units then apply ratio. Keep calm and solve systematically.

Checking answer
After solving, substitute back into original ratio or use cross-multiplication to ensure the relation is satisfied. Check units and whether the answer should be whole numbers. If the result seems odd, re-read the question and verify which quantity corresponds to each ratio term.

Use in cooking and mixtures
Recipes and mixtures rely on fixed ratios so the taste or properties remain consistent when quantities change. For example, a salad dressing that uses oil:vinegar = 3:1 will taste the same whether you make a small or large amount as long as you keep the same proportion of ingredients.

Scaling recipes up or down
To scale, find the multiplier k that changes the recipe size. If the original calls for 2 cups sugar and 5 cups flour and you need to use 10 cups flour, then k = 10/5 = 2 so multiply sugar by 2 to get 4 cups. Use the same multiplier for all ingredients so the ratio remains 2:5.

Finding one ingredient given another
If one ingredient amount is given and the ratio is known, find the value of one part by dividing given amount by that ingredient’s ratio number. Then multiply by other ratio numbers to get amounts of other ingredients. Keep units consistent; convert grams to kilograms or ml to litres if needed before scaling.

Practical cautions
When scaling recipes, rounding sometimes becomes necessary: small fractional amounts (like 0.3 teaspoon) can be adjusted sensibly. For construction mixtures, accurate measurement matters for strength, so keep precision. When mixing liquids, volume units add in simple cases but be careful with temperature or density changes in scientific contexts.

Checking work
After scaling, check the total and taste or property by making a small test batch if possible. Use ratio tables to make the scaling quick and to avoid repeated multiplication. Practise problems that involve both scaling and sharing to master the mixture applications of ratios.

Sharing money and objects
When a total must be divided among people in a given ratio, each person’s share is proportional to their part of the ratio. The unitary method gives an easy procedure: add the ratio parts to find total parts, divide the total amount by this sum to get one part, then multiply by each person’s part to find their share.

Worked procedure
Example: Share Rs 360 in ratio 2:4. Sum of parts = 6. One part = 360 ÷ 6 = 60. First gets 2×60 = 120, second gets 4×60 = 240. This method works for any number of people and for sharing items, money, time or distance when proportional distribution is needed.

Special situations
If items are indivisible (like whole toys), either the initial ratio must yield whole number shares or the problem will require rounding or an agreed rule (e.g., keep some items aside). If someone’s ratio part is zero they receive nothing. Ratios should always be non-negative in practical sharing problems.

Applications beyond money
Profits among business partners, dividing work hours, or sharing time on a computer between students can use ratios. In these contexts also ensure units and total make sense: hours add to total available time, profits add to total profit, etc. Write units clearly to avoid mistakes.

Checking answers and fairness
Always check that shares sum to the total and that proportions match the original ratio by comparing share/part values. If any share looks too small or too big, re-check the calculations. Practice with varied totals and parts so you can perform these divisions quickly and accurately in exams.

What is a ratio table?
A ratio table is a simple grid that lists equivalent pairs of numbers found by multiplying the basic ratio by whole multiples. It is a helpful tool to see patterns and to find matching values quickly without doing cross-multiplication each time. Students can build tables to fit particular given numbers or totals.

Building a table
Start with a simplest form ratio a:b. Create rows for k = 1,2,3,... and compute (a×k):(b×k). For ratio 2:3 the table rows are 2:3, 4:6, 6:9, 8:12, 10:15 and so on. If a problem gives one of the numbers, scan the table to find where it appears and read off the partner value. This avoids algebra and is reliable for many exam questions.

Solving problems using tables
If a question states that in a ratio 5:7 the larger number is 35, students can find k = 35/7 = 5 and then compute the smaller number 5×5 = 25. For totals, add corresponding row values and find which row gives the required total. Tables are also useful to spot whether an exact integer solution exists.

Advantages and practice tips
Tables strengthen mental multiplication and pattern recognition and reduce calculation errors. They are especially useful in competitive time-limited tests. Practice making tables for several base ratios and use them to solve matching and sharing problems quickly. Also practise finding fractional or non-integer partners when no exact multiple appears; then use cross-multiplication to find exact fractional answers.

Tools you have learned
By now you know how to write ratios in different forms, simplify them, create equivalent ratios, use cross-multiplication to check proportions, and apply the unitary method to find one part. You have also used ratio tables, applied ratios to recipes and mixtures, and divided totals among people. These tools work together; choose the right one for each problem.

General strategy for solving ratio problems
1. Read carefully and label what each number means. 2. Convert units if needed and write the ratio in the correct order. 3. Simplify the ratio if that makes work easier. 4. Choose a method: unitary method for sharing and totals, cross-multiplication for missing terms, ratio tables for matching values, and direct proportion for scaling. 5. Do arithmetic step-by-step and keep answers in simplest form unless told otherwise.

Checking your answers
Always verify: substitute your values back into the original relation and confirm the ratio equality using cross-multiplication or by comparing fractions. Check that shares add up to the total and that you respected units. Use a different method (for example use a ratio table to confirm a cross-multiplication result) to be sure your answer is correct.

Exam tips and practice
Show clear steps even if mental math gave the answer — examiners give marks for method. Label units and parts, and write the final ratio in simplest form. Practice many problems of each type so you can recognise which method to use quickly. Work on multiplication and division speed to save time during tests.

Final advice
Work with real-life examples: recipes, sharing, speeds and maps. Visual models like bars or pie charts help understanding. With steady practice, ratio and proportion become easy tools you can use across maths and daily life.