Chapter 2 — Ratio and Proportion

What is a ratio?
A ratio compares two quantities of the same kind by showing how many times one contains or is contained in the other. We usually write a ratio as a:b, read as "a to b". It can also be written as the fraction a/b or in words "a for every b". Ratios give relative information: knowing the ratio 3:5 tells us the relationship between the two parts but not their exact sizes unless total or one part is known.

Forming correct ratios.
To form a ratio, pick the two quantities you want to compare and put them in the correct order. Order matters: boys:girls is not the same as girls:boys. Before comparing make sure quantities are measured in the same units. If units differ (metres and centimetres), convert so both are comparable. A ratio may represent parts of a whole (e.g., red:blue in a bag of beads) or rates (e.g., speed as km:hour).

Simplifying ratios.
Simplify a ratio the same way as a fraction: divide both terms by their greatest common divisor until no common factor remains. For example 20:30 reduces to 2:3. Simplest form makes comparison easy and avoids mistakes when scaling ratios later. Also be comfortable with ratios involving fractions or decimals; convert to common form before simplifying.

Reading ratios in context.
A ratio 2:5 could mean two parts of sugar to five parts of flour; if the total amount is fixed, these parts show proportions of the whole. If units are given as rates, such as rupees per kilogram, read them as a ratio rupees:kilogram. Practice converting words like "for every", "out of", and "as many as" into ratio notation. Understanding this basic meaning is the first step to solving proportion and scaling problems in later topics.

Understand the language first.
Word problems describe relationships using everyday language. To write the correct ratio, read the sentence slowly and underline the two quantities being compared. Words like "for every", "per", "is to", or "out of" signal a ratio. Decide which quantity is first and which is second; the phrase order usually tells you this. For example, "apples to oranges" means apples appear before oranges in the ratio.

Convert units when necessary.
If the quantities use different units, convert them to the same unit before forming the ratio. For instance, if one quantity is in metres and the other in centimetres, convert metres into centimetres or vice versa. Also convert percentages and fractions into parts over 100 or a fraction of the whole so you can create a correct pair of numbers to compare.

Form the ratio and simplify.
After identifying and converting, write the ratio in the required order, using colon notation or as a fraction. Then simplify by dividing by the greatest common divisor. If the question asks for one term to the whole (e.g. boys to total students), be careful: "boys to total" places total as the second term, not girls.

Check with examples and practice.
Convert several word sentences to ratio form and verify by making small examples. For instance, "3 apples for every 2 oranges" is 3:2; if you had 6 apples and 4 oranges that also fits. Practise creating ratios from sentences with implied totals, rates, and comparisons. Writing clear steps — identify, convert, order, simplify — reduces errors under exam conditions.

Understanding equivalent ratios.
Equivalent ratios express the same relationship using different numbers. Two ratios a:b and c:d are equivalent when they represent the same fraction a/b = c/d. You can think of equivalent ratios as different ways of showing the same part–part relationship. For instance 2:3, 4:6 and 6:9 all say the first quantity is two-thirds of the second when simplified.

How to make equivalent ratios.
Create an equivalent ratio by multiplying or dividing both terms by the same non-zero number. Multiplying by a whole number scales both parts up; dividing by a common factor scales both down. You may also multiply by fractions to produce decimals, but it is often simplest to keep whole-number multipliers when working with counts of objects.

Testing for equivalence.
Two principal tests are used. Simplify both ratios to their lowest terms and compare, or use cross-multiplication: a:b and c:d are equivalent if and only if a×d = b×c. Cross-multiplication is quick and avoids mistakes when numbers are large or not obviously reducible. Apply this test when comparing rates or checking answers in an exam.

Use in problem solving.
Equivalent ratios are used to scale recipes, convert map distances, and compare prices. To find a missing term use multiplication or division: to get a second term of 20 from 3:4, multiply both parts by 5 to make 15:20. Practice generating several equivalent ratios for the same relation to recognise patterns and become fluent at scaling up and down without recalculating from scratch.

What is proportion?
A proportion states that two ratios are equal. It is written as a:b = c:d and read "a is to b as c is to d." This equality allows us to compare different pairs of quantities and solve for an unknown when one term is missing. Proportion expresses a balance between the two sides and is used widely in arithmetic and applied problems.

The fundamental property — cross-multiplication.
The central tool for working with proportions is cross-multiplication. From a/b = c/d we obtain a×d = b×c. This equality of cross-products is true for all valid proportions and is the quickest way to check whether two ratios form a proportion. When one of the four terms is unknown, cross-multiplication produces an equation that can be solved directly for the unknown term.

Solving for an unknown.
If a:b = c:d and one term, say d, is unknown, rearrange using the fundamental property: d = (b×c)/a provided a ≠ 0. Always show the steps: write the proportion, cross-multiply, isolate the unknown, compute and then simplify. Replace back into the original to verify correctness. Work carefully with fractions and decimals; clear denominators early if needed to avoid fractional algebra mistakes.

Practical advice and checks.
Use cross-multiplication in word problems involving scaling, price comparisons, and rate conversions. Estimate the answer roughly before final calculation to catch gross errors. Remember to keep units consistent and annotate which term corresponds to which physical quantity. Clear notation reduces order errors and makes checking straightforward. Practice several varied examples to become comfortable forming proportions from words and solving by cross-multiplication.

Systematic approach.
Begin by reading the problem carefully and identifying the two pairs that form the proportions. Decide which quantity corresponds to which and write down the proportion in proper order. If the question involves a missing number, assign a symbol such as x to the unknown. Use cross-multiplication to obtain an equation and then solve for x. Finally, check by substituting the value back into the original proportion to ensure both sides are equal.

Types of situations.
Proportion problems commonly involve scaling (e.g., maps, models), sharing in a given ratio, rates (cost per unit, speed), and conversion of units. Translate words like "for every", "per", "is to" into ratio form and be careful when the question compares one part to the whole (e.g., girls to total students) — this changes the denominator. Use clear steps on paper to earn method marks in exams.

Handling totals and parts.
When sharing a total in a given ratio, convert the ratio to fractions of the whole by summing ratio parts. If A:B = m:n and total T is to be divided, A receives (m/(m+n))×T and B receives (n/(m+n))×T. For problems giving rates (like price per kg), compute unit price by dividing then scale to required quantity. When multiple ratios are chained, reduce to a common base before calculating actual amounts.

Checking answers and estimation.
After finding a solution, estimate roughly to check reasonableness. If scaling up nearly doubles one quantity, expect the other side of the proportion to almost double too. Also substitute the computed value back into the proportion to confirm both sides match. Practice with varied word problems increases speed and reduces translation errors under exam conditions.

Definition and meaning.
Continued proportion connects three or more quantities so that each neighbouring pair has the same ratio. For three numbers a, b, c in continued proportion we write a:b = b:c. The middle term b is the mean proportional between a and c. Continued proportion extends to any length: a:b = b:c = c:d = ... forming a chain of equal ratios.

How to find the mean proportional.
From a:b = b:c we get b² = ac by cross-multiplying, so b = √(ac). This formula shows the mean proportional is the square root of the product of the extremes. In class 8 problems the product often comes out to a perfect square, producing an integer mean proportional; otherwise the answer may be an irrational number which you can leave in root form if permitted.

Using continued proportion in sequences.
When several consecutive equal ratios are given, the terms form a geometric progression with common ratio r. For example, if a, b, c, d are in continued proportion, then b = ar, c = ar², d = ar³ and so on. This view makes it easier to generate missing terms once you find the common multiplier between adjacent terms.

Problem solving tips.
Write the equalities and use algebra to find unknowns. If only extremes are known, compute the mean proportional directly. For longer chains, determine the common ratio by comparing known neighbouring terms or by solving equations produced by cross-multiplying. Sketch small number examples to see the pattern and avoid ordering mistakes. Practice both exact root computation and manipulation of terms in geometric progression style.

Core idea.
Direct proportion means that two quantities change in the same ratio: when one doubles, the other doubles; when one halves, the other halves. If y is directly proportional to x, write y ∝ x or y = kx where k is constant. This constant k is the ratio y/x which stays the same for all matching pairs of values.

Finding the constant and solving problems.
Given one pair (x1, y1) compute k = y1/x1. For any new x2, find y2 = kx2. This method is very useful in money and measurement problems: unit price is cost per unit, speed is distance per time. Alternatively use proportions x1:y1 = x2:y2 and cross-multiply to solve for the unknown. Direct proportion problems are straightforward if you identify that the relation is linear and passes through the origin.

Graphical meaning.
On a graph with x on the horizontal axis and y on the vertical axis, direct proportionality produces a straight line passing through the origin. The slope of this line equals the constant k. Different values of k give lines with different steepness. Observing such a line helps understand how fast y grows with x: a steeper line means a larger k and faster growth.

Applications and exam tips.
Typical class examples include price per kg, distance for constant speed, and converting quantities at fixed rates. When solving, clearly state the constant k and show multiplication steps. Use units in answers (Rs/kg, km/hr) and check by plugging back into the relation. If more than two quantities are involved, combine ratios correctly before applying direct proportion rules.

Definition and examples.
In inverse proportion describes pairs where one quantity grows while the other decreases so that their product stays constant. If y is inversely proportional to x, write y ∝ 1/x or equivalently xy = k for some constant k. Common examples include time taken to finish a job and number of workers (assuming each worker works at same rate), and speed and time for a fixed distance.

How to work problems.
Given one pair (x1, y1) compute k = x1y1. For a new x2, find y2 = k/x2. Alternatively use the equality x1y1 = x2y2 directly. This form of relation is different from direct proportion: you cannot use y = kx here. It often helps to reason qualitatively first: doubling x should halve y in exact inverse proportion situations, which gives a quick check on answers.

Visual and algebraic insight.
Graphs of inverse proportion are rectangular hyperbolas in the positive quadrant: as x increases, y decreases but never becomes zero. Algebraically, solving involves manipulating products rather than ratios. Ensure units are consistent and remember that inverse proportion problems usually stem from a fixed total workload or fixed product quantity in context.

Practical warnings and checks.
When applying to real contexts, check assumptions: inverse proportion holds when each unit of one quantity contributes equally (e.g., workers with equal efficiency). If workers have differing speed or there are idle times, the simple model fails. Always compute k and then use substitution to verify results. Estimation helps catch gross mistakes: if x doubles, y should roughly halve in inverse proportion.

Essence of the method.
The unitary method finds the value of a single unit and then scales to the required number. It is useful when a problem gives a total for many identical units and asks for the value of one unit or a different number of units. This approach clarifies intermediate steps and makes mental estimation simple.

Steps to use unitary method.
First, find the value of one unit by dividing the given total by the number of units. Second, multiply the unit value by the number of units required to get the answer. For example, if 8 pens cost Rs.240, one pen costs 240 ÷ 8 = 30; so 5 pens cost 5×30 = 150. Always keep track of units to avoid confusion between quantity, cost and rate.

When to use and when not to.
The unitary method is best when dealing with direct proportion problems and simple conversions. It is often longer than writing a proportion but more intuitive for many students. For inverse proportion, adapt the idea: find total constant like total work or product and then divide appropriately. For chained ratios, compute unit values only after combining ratios into a consistent set of parts.

Practical tips.
Use the unitary method for price, speed, and coverage problems where unit rates are natural. Show the intermediate unit value in exam answers to gain method credit. When working with decimals, maintain sufficient precision until the final step. Practice both mental and written approaches to become quick with this very handy technique.

Ratios and similar figures.
When two shapes are similar, corresponding sides are in the same ratio. This scale factor k shows how one shape is stretched or shrunk relative to another. Knowing k for side lengths lets you find ratios of perimeters and areas quickly: perimeters scale by k while areas scale by k². These facts follow because perimeter is a sum of lengths and area involves two dimensions.

Perimeter scaling explained.
If corresponding side lengths of two similar polygons are in the ratio a:b, then every side in the larger shape is b/a times the smaller. The perimeter, being the sum of corresponding sides, also has ratio a:b. This is because adding corresponding sides preserves the common multiplicative factor. For example, squares with sides 3 cm and 6 cm have perimeters 12 cm and 24 cm: ratio 1:2, same as side ratio.

Area scaling explained.
Area involves multiplying two linear measures (for rectangles, length × breadth). If each linear measure scales by k, area scales by k×k = k². Thus when side ratio is 1:2, area ratio is 1²:2² = 1:4. This is important when working with scale models: a small change in linear size produces a larger change in area. Use this to compute missing sides or areas when similarity is given.

Using the idea in problems.
Identify if figures are similar; if so, find side ratio k and apply it to perimeters and areas. Remember: volumes (for later classes) would scale by k³. In class 8 questions, sketch similar figures, label corresponding sides, compute k, and then apply k and k² as needed. Always check units and simplify final ratios to their lowest terms for clarity in answers.

Mixing in given ratio.
Mixture problems ask how much of each component is present when two or more are combined in a certain ratio or how to mix given quantities to achieve a required ratio or concentration. Treat the given ratio as parts of the whole. If milk and water are mixed in ratio 7:3 and total is 50 litres, milk fraction is 7/(7+3) and water fraction is 3/(7+3). Multiply these fractions by the total to find actual quantities.

Basic alligation idea.
Alligation is a simple arithmetic method to determine how to mix two different concentrations to get a desired concentration. For class 8, stick to the basic idea: difference of concentrations tells how many parts of each to mix relative to the target. A more reliable approach is to form proportion equations: let x and y be amounts of each component, and write fraction equations equating the final concentration to the weighted average of the parts, then solve. This method avoids memorising rules and works in all cases.

Solving mixture questions stepwise.
1) Express required ratio or concentration as fractions of the total. 2) Set up equations for amounts or concentrations using these fractions. 3) Solve for unknown quantities using cross-multiplication or substitution. Always check that the sum of parts equals the given total and that concentrations are within realistic bounds (e.g., percentages between 0% and 100%).

Applications and tips.
Mixture problems occur in food recipes, chemical solutions, and blending commodities. Represent mixtures visually with bar or pie models to aid understanding. Practise converting ratios into fractions of totals and forming the correct proportion equations. In exams present clear working: state fractions from the ratio, multiply by the total, and show final simplified answers with units.

Unit conversion as ratio reasoning.
Converting units is an application of ratio and proportion. A conversion factor states an equality between two units, for example 1 m = 100 cm gives the ratio m:cm = 1:100. To convert from one unit to another multiply or divide by this factor according to the direction of conversion. View the conversion factor as a ratio that can be used to change units while keeping the measured quantity the same.

Multiplying or dividing correctly.
When converting a larger unit to a smaller unit multiply (e.g., km to m multiply by 1000); when converting a smaller unit to a larger one divide (e.g., cm to m divide by 100). Always write the intermediate step and cancel units if converting with chained factors. For compound conversions (hours to seconds), chain the ratios (1 hour = 60 minutes, 1 minute = 60 seconds) and multiply the factors together to get the direct factor (1 hour = 3600 seconds).

Compound and rate conversions.
For compound units like km/h to m/s apply the conversion to both distance and time parts: 1 km/h = (1000 m)/(3600 s) = 5/18 m/s. Hence multiply km/h by 5/18 to get m/s. Keep track of the units at each step to avoid reversing the factor. If a question mixes units in ratio form (e.g., rupees per kg), convert the unit in the denominator before comparing prices or forming proportions.

Practical examples and checks.
Practice common conversions: km↔m, m↔cm, hours↔seconds, litres↔millilitres. For exam problems write the conversion factor clearly and show multiplication or division steps. Estimate the order of magnitude of the answer to catch mistakes (e.g., converting 2.75 km to metres should give a number in the thousands). Mastery of conversions makes many proportion problems straightforward.

Combining multiple ratios.
Often problems give ratios between pairs of quantities and ask for a combined ratio of three or more terms. The key is to make the shared quantity the same in every ratio. For example, if A:B = 2:3 and B:C = 4:5, make the B part equal in both by scaling the ratios: multiply 2:3 by 4 to get 8:12 and 4:5 by 3 to get 12:15. Then combine into A:B:C = 8:12:15. This reduction to a common reference is the main technique.

Step-by-step method.
1) List the given ratios. 2) Identify the common term(s). 3) Find a common value for each shared term (use lcm if necessary). 4) Scale each ratio to make shared terms equal. 5) Write the final combined ratio including all terms and simplify. If no direct common term exists, introduce variables and form equations using proportions to solve for relative values.

Applications in sharing and mixtures.
This method is used when dividing money among several people with different pairwise relations, or combining ingredients where pairwise ratios are given. After obtaining A:B:C:... convert to actual amounts by multiplying each term by a common factor found from total quantity or other constraints. Always check the combined ratio by reconstructing the original pairwise ratios to ensure consistency.

Common errors and checks.
Watch the order of terms and ensure correct scaling: mixing up which term is common leads to wrong results. After computing the final ratio, simplify only if it does not change the meaning regarding the shared terms. Test with simple numbers or verify by substituting back into one of the original pairwise ratios to confirm correctness. Practice several chained-ratio examples to gain fluency.