Chapter 2 — Ratio and Proportion

What is a ratio? A ratio compares two quantities of the same kind by division. When we write 5 : 3, we mean that for every 5 units of the first quantity there are 3 units of the second. Ratios tell us how large one quantity is relative to another and do not depend on the unit as long as both quantities use the same unit.

Reading and writing ratios. A ratio a : b can be read as ‘a to b’. It can also be written as the fraction a/b when comparing a with b. If the quantities are equal, the ratio is 1 : 1. If one quantity is zero, the ratio may be 0 : b (provided b ≠ 0), but a : 0 is undefined as a comparison by division.

Types of ratios. Ratios may compare parts of a whole (part-to-part) or part to whole (part-to-whole). For example, in a bag with 4 red and 6 blue beads, the part-to-part ratio of red to blue is 4 : 6, while red to total is 4 : 10.

Key ideas to remember. Ratios are simplified by dividing both terms by a common factor. Ratios are multiplicative: multiplying or dividing both terms by the same positive number gives an equivalent ratio. Ratios are used to maintain relative amounts when scaling or sharing.

Why simplify? Simplifying a ratio makes it easier to understand and compare. When you reduce a ratio to its simplest form you remove any common factors in both parts. This gives the clearest statement of how two quantities relate. Working with the simplest form also helps when checking if two ratios are equal.

How to simplify. To simplify a ratio a : b, find the greatest common divisor (GCD) of a and b and divide both terms by that number. For example for 42 : 56 the GCD is 14; divide both terms by 14 to get 3 : 4. If you cannot find a common factor greater than 1, the ratio is already in simplest form.

Finding equivalent ratios. Equivalent ratios are formed by multiplying or dividing both terms by the same positive integer. For example 2 : 3, 4 : 6 and 6 : 9 are all equivalent because each term of 2 : 3 is multiplied by 2 and 3 respectively to get the others. Equivalent ratios represent the same proportional relationship but with different units. This is helpful when you must compare ratios with different scales or when you match a required unit size (for example, making the second term a specified number).

Techniques to use. Use prime factorisation or the Euclidean algorithm to find the GCD quickly for larger numbers. When you need an equivalent ratio with a given term, divide or multiply by the factor that transforms the known term to the required one. Always perform the same operation on both terms to keep the ratio equivalent. If working with fractions, convert the ratio to a fraction a/b and simplify the fraction; after simplifying you may convert back to a : b format.

Common classroom checks. After simplifying, multiply the simplified ratio by the GCD to check you retrieve the original ratio. To compare two ratios, simplify both and see if the simplified forms match. Practice with numbers that have multiple common factors to get comfortable with the reduction process; with experience you will spot factors quickly and simplify in your head.

Idea behind comparison. To decide which of two ratios is larger or whether they are equal, a reliable way is to write each ratio as a fraction and compare those fractions. Ratios a : b and c : d correspond to fractions a/b and c/d. Comparing fractions can be done by finding a common denominator, converting to decimals, or by cross-multiplication.

Cross-multiplication method. Cross-multiplication is quick and exact: compare ad and bc. If ad > bc then a/b > c/d; if ad < bc then a/b < c/d; if equality holds ad = bc then the ratios are equal. This works provided b and d are not zero. Cross-multiplication is often faster than finding common denominators, especially with larger numbers.

Common denominator method. Another method is to convert both fractions to the same denominator and compare the numerators. For instance, to compare 2/3 and 3/5, write both with denominator 15: 10/15 and 9/15, so 2/3 is larger. This method helps when denominators are small or share factors, as students can mentally scale numbers to a common base.

Using decimal form. Dividing numerator by denominator gives decimal values which are easy to compare. For example 7/12 ≈ 0.583 and 5/8 = 0.625, so 5 : 8 is larger. Beware of rounding errors when decimals are similar; prefer cross-multiplication for exact checks.

Practical examples. Comparing concentrations, speeds, densities or prices per unit uses these methods. For price comparison, compute price per unit as a fraction and compare. For concentration problems, compare solute/solution fractions. Always keep the same units for a fair comparison and use the method that best fits the numbers given to reduce calculation errors.

What is a proportion? A proportion states that two ratios are equal. It is written as a : b = c : d or as a/b = c/d. Proportion is the algebraic form of saying that two comparisons represent the same relationship. Proportion problems often ask us to find a missing term when three terms are known.

Terminology and positions. In the proportion a : b = c : d, the numbers a and d are called the extremes and b and c are called the means. This naming is useful when you talk about cross-products and check if terms are in the correct places before solving. Always ensure the order of terms is correct when setting up a proportion from a word problem.

Cross-multiplication rule. Cross-multiplication is the fundamental tool for solving proportions. From a/b = c/d we get ad = bc. This equality of cross-products lets you find the unknown term by simple algebraic steps. For example, to find x in a : b = c : x rearrange to x = (b c)/a provided a ≠ 0. Cross-multiplication also serves as a quick test for whether two ratios are equivalent: check if ad and bc are equal.

Solving with fractions and care with units. Sometimes it is easier to convert ratios to fractions and solve with algebra. Keep units consistent: ratios must compare same types of quantities (length with length, cost with cost). If a word problem mixes units, convert them first so the proportion is meaningful. After finding a value, substitute back into the original proportion to verify it satisfies the equality and that values are practical (for instance, whole numbers where required).

Applications and checks. Proportion models many everyday situations: recipe scaling, converting currencies at a fixed rate, calculating speeds and distances, and splitting amounts. When solving, always simplify the final ratio if asked, and check by cross-multiplying to ensure ad = bc holds. Practice with varied word problems helps to recognise whether a proportion is the correct approach and to avoid mistaken assumptions such as treating a non-linear relation as proportional.

Concept and use. The unitary method is a simple two-step approach: first find the value for one unit, then use that to find the value for any number of units. This method connects directly with ratio because it establishes a standard ‘per one’ quantity. It is especially handy when quantities are given for several units and you need a different number of units, or when comparing prices, speeds, or amounts per item.

Step-by-step explanation. Step 1: Divide the given total by the given number of units to obtain the amount for one unit. Step 2: Multiply the unit amount by the required number of units. For example, if 9 metre cloth costs Rs 270, cost per metre is 270 ÷ 9 = Rs 30; for 5 metres cost = 30×5 = Rs 150. The method works with whole numbers, fractions and decimals, so be comfortable with all forms.

When to prefer unitary method. Unitary method is clear in word problems where the phrase ‘per’ or ‘for each’ appears implicitly or explicitly, such as ‘‘per litre’, ‘for one student’, or ‘for one hour’. It also helps when converting measures: for example, converting cost per dozen to cost per piece or vice versa. For proportional scaling problems it gives an immediate and intuitive answer and can be faster than setting up a full proportion equation.

Handling fractions and checks. If dividing gives a fraction, keep it as a fraction or convert to a decimal depending on which is simpler for subsequent multiplication. After calculating the required amount, always multiply back to check you recover the original total when appropriate. For discrete contexts like people or whole items, consider whether the fractional part makes sense; if not, explain whether rounding is needed and why.

Practice tips. Try varied examples: cost per item, distance per hour, pages per minute, and work done per worker. Practise switching between unitary and proportional approaches; with experience you will be able to choose the faster method by looking at the numbers and the wording of the question.

Definition. Two quantities are in direct proportion when they change in the same ratio: as one doubles, the other doubles; as one halves, the other halves. Mathematically, y is directly proportional to x if y = kx for some constant k called the constant of proportionality. This means the ratio y/x remains fixed.

Identifying direct proportion. Look for words that indicate sameness of scale: ‘for every’, ‘per’, ‘at this rate’, or examples like ‘‘cost of apples at a fixed price’’, ‘‘distance covered at constant speed’’. If two pairs of values keep the ratio y1/x1 = y2/x2 then the relationship is direct proportion. Always test with known values to be sure the relation is linear and does not have a fixed add-on.

Solving direct proportion problems. If y1 corresponds to x1 and we need y2 for x2, use y2 = (y1/x1)×x2 or write y = kx and find k = y1/x1 then compute. Alternatively unitary method finds value for one unit and scales. For example, if 5 kg sugar costs Rs 200, cost per kg = 40 and cost for 8 kg = 40×8 = Rs 320. The algebraic form y1/x1 = y2/x2 is often simplest for two known pairs.

Graphical view and properties. The graph of y against x is a straight line through the origin with slope k. This means the line passes through (0,0), so zero x gives zero y. If the graph does not pass through the origin, the relationship is not pure direct proportion. Keep this check in mind for real world problems: fixed charges or base values break direct proportionality.

Applications and cautions. Direct proportion appears in many contexts: cost with quantity, distance with time at constant speed, weight with mass, and brightness with number of bulbs at fixed wattage. Always ensure the condition of constancy (fixed rate or speed) holds. If there are additional fixed components (for instance a flat fee), use a linear equation rather than direct proportion.

Definition and idea. Two quantities are in inverse proportion when their product is constant: as one increases, the other decreases in such a way that x×y = k for some constant k. This means if x is multiplied by a factor, y must be divided by the same factor to keep the product unchanged. Inverse proportion models situations where more of one resource shortens the required amount of the other, under fixed total work or fixed total effect.

Recognising inverse proportion. Look for phrases like ‘‘if more workers are employed, time taken decreases’’, ‘‘more taps fill the tank faster’’ or ‘‘if speed increases, time for fixed distance decreases’’. These ideas express that one quantity varies inversely with another. Test given pairs: if x1y1 = x2y2 then the relationship is inverse for those pairs; otherwise it is not.

Solving inverse proportion problems. Use the relation x1y1 = x2y2. When a constant amount of work is done, set product equal for both situations. For example, if 6 workers take 10 days to finish, work = 6×10 = 60 worker-days. For 15 workers days needed x = 60/15 = 4 days. Alternatively, express y = k/x by finding k from one pair and then computing the unknown. Always check units and note if efficiency per worker remains constant—only then does inverse proportion apply.

Graph and properties. The graph of y against x for inverse proportion is a smooth curve (rectangular hyperbola) that does not pass through the origin. As x approaches zero, y grows large; as x grows, y approaches zero. This distinguishes it clearly from direct proportion. Use this visual idea for checking answers and understanding how changes affect the other quantity.

Applications and limits. Typical applications include workers and time, number of machines and time to finish work, speed and time for a fixed distance, and pipes filling a tank together (with care when combined rates are involved). Note that some real situations have limits (e.g., there is a minimum possible time), so inverse proportionality may be an approximation; check assumptions before applying.

Problem type. Often we must split an amount into parts that are proportional to given numbers. If amount A is to be divided in ratio a : b, then first find the total parts a + b, the value of one part is A/(a+b), and each share is (a or b) × (A/(a+b)). This extends to more than two parts: for ratio a : b : c, total parts = a + b + c and each share is its part times A/(sum).

Step-by-step method. Step 1: Add the parts to get the total number of parts. Step 2: Divide the total amount by total parts to get single part value. Step 3: Multiply single part by required number of parts for each share. Use fractions if division does not give an integer and interpret context for rounding.

Applications and checks. This method is used to share money, divide time, distribute ingredients or divide angles. Check by adding all shares to see if they equal the original amount. Also check proportionality by dividing pairs of shares to see if the ratio matches the given one.

Special note. If one part must be integer (like people), ensure the total allows integer shares or state fractional results as necessary. When more than two parts are present, simplify the ratio first if possible to make computation easier.

Practical examples and tips. When numbers are large, simplify the ratio before computing to reduce arithmetic. If a share is given and you must find the whole, divide the given share by its part count to find one part and then multiply by total parts. Always state units and round only after reasoning whether rounding changes meaning (money, people, ingredients behave differently).

What mixture problems ask. Mixture problems involve combining two or more substances with different concentrations or amounts and finding the composition or quantities in the final mixture. In Class 7 we focus on simple mixing where components are combined in certain ratios, and we track how much of each part is present in the final mix. The key ideas are conservation of mass (or volume when liquids mix without change) and proportion of parts.

Working with ratios directly. If a mixture is to be made in ratio m : n of components A and B, and the total required volume/weight is T, then total parts are m + n and one part equals T/(m + n). The amounts needed are m×(T/(m + n)) for A and n×(T/(m + n)) for B. This method avoids percentages when the ratio is given explicitly and is quick for recipes and model problems.

Mixing concentrations (percentages). When concentrations are given (for example A is 10% sugar and B is 30% sugar) and equal volumes are mixed, the final percentage is the weighted average based on volumes: (p1V1 + p2V2)/(V1 + V2). If volumes are equal, the average simplifies to (p1 + p2)/2. For differing volumes use the general weighted average formula. Always convert percentages to fractions when you need exact amounts of solute: p% means p/100 of the mass or volume.

Practical steps and checks. Keep units consistent (litres with litres, kg with kg). After computing, verify the total amount equals the sum of parts and that computed concentration matches the intended value: amount of solute divided by total mixture times 100. For classroom problems practise both forming a mixture to achieve a required concentration and finding the concentration of a given combination.

Common classroom examples. Mixing syrups and water, combining alloys, or blending paints with given ratios are typical tasks. When required amounts must be whole numbers, choose totals that produce integer parts or accept fractional answers with clear units. For more advanced study, the alligation method gives a shortcut for finding ratios when target concentration is known, but in Class 7 practising ration-based splitting and weighted averaging builds a clear foundation.

Definition and significance. Continued proportion describes a chain of numbers where each successive pair keeps the same ratio: a : b = b : c = c : d = ... . This is a natural extension of proportion and ties directly to geometric progression. Understanding continued proportion helps find geometric means and describes sequences where each term is obtained by multiplying the previous by a common ratio.

Basic property for three terms. For three numbers a, b, c to be in continued proportion we must have a : b = b : c. Rearranging gives b^2 = ac. This formula is a quick test: compute the product ac and see if its square root equals b. If ac is not a perfect square, b may be an irrational number; in school problems usually numbers are chosen so b is an integer.

Extending to more terms. For four numbers a, b, c, d in continued proportion, the common ratio r satisfies b = ar, c = ar^2, d = ar^3. In general the nth term equals a × r^(n-1). The constant r can be found as b/a (or c/b etc.) provided a ≠ 0. This connects continued proportion to geometric progression and helps when constructing sequences or solving problems that ask for intermediate geometric means.

Finding terms and checks. To find the middle term between two given numbers when they are required to be in continued proportion, use the geometric mean: b = sqrt(ac) for three-term case. For more terms, if you know a and d for a sequence of equal multiplicative steps, find r = (d/a)^(1/3) for four terms and then compute intermediate values provided the root yields a rational number. Always check by multiplying adjacent terms ratio to confirm equality.

Classroom practice and examples. Students often meet questions like ‘find a number between X and Y in continued proportion’ or ‘are these numbers in continued proportion?’ Practice includes integer and non-integer results; discuss when square roots are whole numbers and when decimals are acceptable. Continued proportion gives an early bridge to geometric sequences and the concept of geometric mean, both useful in higher classes.

Role of ratio in geometry. Ratios compare lengths of sides, angles (by measure), areas and volumes in geometric shapes. The concept is most powerful when applied to similar figures: shapes that have the same form but different sizes. In similar figures corresponding sides are proportional, which means their ratios are equal. Understanding this helps solve many geometric problems involving scaling, area and volume comparisons.

Similar figures and scale factor. If two figures are similar with scale factor k, every length in the larger figure equals k times the corresponding length in the smaller one. For example, if triangles are similar and one has side 4 cm while the corresponding side in the other is 6 cm, the scale factor is 6/4 = 3/2. All corresponding sides have this same scale. Use the scale factor to find missing lengths easily by multiplying or dividing by k.

Areas and volumes. When shapes are similar, areas scale as the square of the scale factor and volumes (or capacities) scale as the cube. That is, area ratio = k^2 and volume ratio = k^3. For example, if side ratio is 2 : 3, area ratio is 4 : 9 and volume ratio is 8 : 27. This is because area units are square units and volume units are cubic, so the power of k corresponds to the dimension.

Applications and checks. Use ratios to work with models, maps, scale drawings and prototypes. For model-making, convert real dimensions to scaled ones using the scale factor. When working with areas and volumes, remember to square or cube the factor appropriately. Verify results by checking corresponding side ratios and ensuring area/volume ratios match the expected power of the scale factor.

Practical classroom examples. Problems include finding missing sides of similar triangles, converting model measurements to real-life sizes, and comparing areas of similar polygons. Draw figures clearly, mark corresponding sides, and label the scale factor. Ensure students understand units: length in cm, area in cm² and volume in cm³ so the interpretation of the power of k is meaningful and accurate.

Percent means per hundred. Percentage expresses a part out of 100. Writing p% means p out of every 100 parts. Because percent has 100 as a standard denominator, it is an easy way to compare and communicate proportions. For example, 25% means 25 out of 100, which equals 1/4 or 0.25.

Conversions between percent, fraction and decimal. To convert a percent to a fraction, divide by 100 and simplify: p% = p/100. To convert a percent to decimal form, divide by 100 as well: p% = p/100 = 0.p when written in decimal. To turn a fraction into a percent, compute (a/b)×100% and simplify. Practice these conversions because many problems require switching forms to compute easily.

Finding percentage of a quantity. If a quantity A is to be reduced to p% of its value, compute (p/100)×A. For example, 20% of 250 is (20/100)×250 = 50. Percentage is used widely: marks (percent of total), discounts, profit/loss rates, concentrations in solutions and population statistics. Always state the base (the ‘whole’) clearly since percentage is relative to that base.

Percent increase and decrease. To find percent increase from value X to Y, compute [(Y − X)/X]×100%. For decrease, use [(X − Y)/X]×100%. Remember that percent changes are relative to the original amount X, not to the final amount. For successive percentage changes multiply factors: a p% increase corresponds to ×(1 + p/100) and a q% decrease corresponds to ×(1 − q/100).

Using percent with ratio and proportion. Percent problems can be solved with ratio techniques by treating 100 as the total parts. Splitting an amount according to percentages is the same as dividing in ratio p : (100 − p). For mixtures and concentrations, convert percent to fraction to compute exact amounts of solute and solvent and verify totals. Practice with real-life examples like discounts, tax calculations and examination percentages.

Reading and understanding. The first step in any word problem is careful reading: underline the important numbers and identify what is asked. Decide which quantities are being compared and whether the relationship is a ratio, direct proportion or inverse proportion. Translate the phrases into mathematical statements: ‘‘for every’’, ‘‘per’’, ‘‘in the ratio’’, ‘‘if x then y’’ help determine the correct model.

Choosing the method. Decide whether the unitary method, proportion (cross-multiplication), direct or inverse proportion best fits the situation. For splitting amounts use division in ratios; for scaling use direct proportion; for work-time-worker problems use inverse proportion. If unsure, set up variables and form equations guided by the text; explicit equations reduce mistakes from misreading.

Setting up equations. Assign symbols for unknowns and express given relations as ratio or proportion equations. For proportion use a/b = c/d and apply cross-multiplication: ad = bc. For unitary approach compute value for one unit then scale. For inverse relations use the product constant xy = k. Keep track of units and ensure terms correspond correctly (e.g., cost per kg, hours per worker).

Solving and checking. Solve algebraically, simplify fractions, and interpret the result in context. Check by substituting the answer into the original statement and verifying that numbers match. Also check whether answers are reasonable by estimation: if sharing money among people yields a fraction of a rupee, decide whether rounding or giving paise is appropriate.

Practice with varied examples. Word problems often combine ideas: a mixture with ratio then scaling to a total, or a proportion followed by percentage change. Practice translating sentences to equations and solve step by step. Mark which problems require integers and which allow fractions. Use diagrams or bar models for complex situations to visualise relationships before algebraic work.

Why checking matters. Even when you get an answer, checking confirms it is correct and sensible. Mistakes in arithmetic, wrong setup, or unit errors are common; simple checks catch most errors. Estimation is a quick way to see whether the exact answer is in the right range before finalising.

Methods of checking. For proportions, substitute the found value back and check the equality of ratios or cross-products. When a quantity has been divided in a ratio, add shares to see if they match the original total. For percent problems compute the original value using reverse operations to check whether the forward calculation is consistent.

Estimation techniques. Round numbers to one or two significant digits and perform the calculation mentally to get a rough answer. Use benchmarks: 50% halves a number, 25% is a quarter, doubling or halving gives quick checks. If the exact value differs greatly from the estimate, review the steps. Estimation is also useful to choose which method is faster or less error-prone for a given question.

Quick arithmetic checks. Use multiplication to reverse division steps. For example, if one part was found as A/(a+b), multiply this by (a+b) to confirm you get A. For proportional scaling, check that ratios remain equal by division or cross-multiplication. If results should be integers (people, whole items), check divisibility and explain rounding if needed.

Real-life sense checks. Consider units and realism: time cannot be negative and people counts must be whole. For percentage increases, an increase more than 100% means the final amount is more than double—check whether this is reasonable in context. Training in estimation and routine checks prevents small mistakes costing marks in exams and builds confidence in problem solving.

Aim of revision. Revision brings together all parts of the unit so students can recognise which method applies quickly: simplifying ratios, creating equivalent ratios, setting up proportions, using unitary method, applying direct or inverse proportion, dividing amounts, mixing components, and handling percents. Mixed practice helps in deciding the right approach for a given problem and improves speed and accuracy.

Organising revision. Start with short questions from each topic to refresh procedures: simplify ratios, find missing terms in a proportion, split amounts in ratios, and calculate percentages. Then move to mixed questions that combine ideas, such as a percentage of a share divided in a ratio, or scaling a recipe and comparing times for different numbers of workers. Time yourself for some sets to build exam readiness.

Exam technique and presentation. Always write clear steps: show how you formed the ratio or equation, perform cross-multiplication when used, and write units. Simplify ratios early to reduce arithmetic. For multi-step problems, write intermediate values neatly to avoid confusion and make checking easier. If a question allows fractional amounts, indicate whether rounding is needed and explain your choice.

Common mixed problem types. Expect map scale and distance problems, cost and price per unit, work and time with changing numbers of workers, mixture concentration problems, and splitting sums in ratios. Practice converting a written situation into the correct proportional model; that skill is often the hardest part and improves with varied problems.

Final revision tips. Review mistakes after each practice session to find recurring errors: forgetting units, mixing up part-to-part and part-to-whole, or assuming direct proportion when inverse is correct. Make a small formula sheet of ad = bc, y = kx, xy = k and unitary method steps and use it while practising until the methods become automatic.