Overview
This chapter introduces the fundamentals of Ratio and Proportion for Class 6. Students learn how to compare two or more quantities using ratios, express ratios in simplest form, recognize equivalent ratios and convert ratios to fractions. The chapter then defines proportion as the equality of two ratios and teaches the basic property of proportions (cross‑product equality) to find missing terms. Emphasis is placed on practical applications — dividing quantities in a given ratio, solving simple word problems, and using the unitary method for proportionate reasoning. Mastery of these ideas builds number sense, prepares students for percent, scaling and algebra, and develops problem‑solving skills useful in everyday contexts such as recipes, sharing, maps and models.
Learning Objectives
- Define ratio and express it in three forms: fraction, colon notation and words
- Identify equivalent ratios and write a given ratio in its simplest form
- Simplify given ratios to lowest terms using highest common factor (HCF)
- Compare two ratios to determine which is greater or whether they are equal
- Express a given ratio as a fraction and convert a fraction into ratio form
- Use the concept of proportion to test whether four numbers form a proportion
- Solve for a missing term in a proportion using cross-multiplication
- Apply ratio and proportion to solve word problems on sharing, mixtures and scaling
Topics in this chapter
10 topics · tap a topic title to jump straight to it.
Introduction to Ratio
What is a ratio? A ratio is a way to compare two (or more) quantities of the same kind by showing how many times one quantity is of the other. We write a ratio of a to b as a:b (read "a to b"). A ratio is dimensionless — the units must be the same for both quantities before comparing.
Notation and meaning:
- a:b can be interpreted as the fraction a/b.
- Order matters: a:b is generally different from b:a.
- A ratio can be simplified by dividing both terms by their greatest common divisor (GCD). For example, 8:6 = 4:3.
Equivalent ratios and scaling: If you multiply or divide both terms by the same nonzero number k, the new ratio is equivalent: a:b = (ka):(kb). Conversely, if a:b = m:n, then there exists k such that a = km and b = kn.
Part-to-whole and conversions: When you want the part-to-whole form, convert a:b to the fraction a/(a+b). That fraction can be changed to a decimal or percentage by standard methods (e.g., multiply by 100 to get percent).
Simple example (explained): There are 8 apples and 6 oranges. The ratio apples:oranges = 8:6. Simplify by dividing both by 2: 8:6 = 4:3. As a fraction, apples are 4/(4+3)=4/7 of the fruits (≈57.14%).
Key ideas to remember: Always compare like with like (same units), simplify ratios using GCD, and use equivalent ratios to scale up or down (for recipes, mixtures, maps, etc.).
- 8 apples and 6 oranges → ratio apples:oranges = 8:6 = 4:3. Part-to-whole: apples = 4/(4+3) = 4/7 of the fruits.
- Classroom example: 12 boys and 8 girls → ratio boys:girls = 12:8 = 3:2.
- Recipe: If sugar:flour = 2:5, for every 2 cups sugar use 5 cups flour. To make double, use 4:10 (equivalent).
- Map scale (ratio): 1:100,000 means 1 unit on map represents 100,000 units in real life.
- Paint mixture: Red:Blue = 3:1 means 3 parts red for every 1 part blue (e.g., 3 litres red and 1 litre blue).
- Ratio notation: a:b (read 'a to b').
- Ratio as fraction: a:b = a/b.
- Equivalent ratios: a:b = (k·a):(k·b) for any nonzero k.
- If a:b = m:n then a = k·m and b = k·n for some k (scale factor).
- Simplify a:b by dividing both terms by GCD(a,b). Example: 8:6 → divide by 2 → 4:3.
- Part-to-whole: a in a:b is a/(a+b) of the whole. Convert to percent by ×100%.
Notation and Representation
What is a ratio? A ratio compares two quantities by showing how many times one quantity is of the other. It tells us the relative sizes of two quantities.
How to write a ratio (notation):
- With a colon: a:b (read as "a to b").
- As a fraction: a/b (which shows how many times a is of b).
- In words: "a to b" or "a for every b".
Representation and simplification:
- Two ratios are equivalent if one can be obtained from the other by multiplying or dividing both terms by the same non‑zero number. Example: 2:3 = 4:6 = 6:9.
- To write a ratio in its simplest form, divide both terms by their greatest common divisor (GCD). Example: 12:8 → divide by 4 → 3:2.
- Unit ratio / unitary method: If a ratio is a:b, the amount corresponding to one unit of b is a/b. This helps find parts for any amount.
Connection with proportion:
If two ratios are equal, we call them a proportion. For example, a:b = c:d is written as a:b :: c:d. In such a case cross multiplication gives ad = bc. Proportion helps solve many practical problems (scaling, map distances, recipes).
Visual models commonly used:
- Bar (strip) models: divide a rectangle into (a + b) equal parts and shade a parts for a and b parts for b.
- Number line: mark equal divisions showing each part of the ratio.
- Pie charts or area models: divide a circle into parts proportional to the ratio terms.
Key idea for students: A ratio gives a relative comparison. You can write it in different forms (colon, fraction, words), simplify it, find equivalent ratios, and represent it visually to make comparisons clear.
- Apples to oranges: 12 apples and 8 oranges → ratio = 12:8 = (divide by 4) = 3:2. As a fraction 12/8 = 3/2.
- Recipe: For every 2 cups of flour and 1 cup of sugar → flour:sugar = 2:1. If you use 6 cups of flour, sugar needed = 3 cups (multiply both terms by 3).
- Classroom: Boys:girls = 15:10 → simplify (divide by 5) → 3:2. For 30 boys, girls would be 20 to keep the same ratio.
- Paint mix: Paint A to Paint B = 2:5. To make a larger batch keeping same shade, multiply both by same factor (e.g., ×3 → 6:15).
- Equivalent ratios: 4:7 = 8:14 = 12:21 (all multiplied by 2 and 3 respectively).
- Ratio notation: a:b or a/b or "a to b".
- Equivalent ratios: a:b = (ka):(kb) for any nonzero k.
- Simplest form: divide both terms by GCD(a, b) to get lowest terms.
- Proportion: a:b = c:d ⇔ a/b = c/d ⇔ ad = bc (cross multiplication).
- Unit value (unitary method): if a corresponds to b, then one unit of b = a/b. To find value for k units of b, multiply by k.
Equivalent Ratios and Simplification
Ratio compares two quantities of the same kind and is written as a:b or a/b. It tells how many times one quantity is of the other.
Equivalent ratios are different ratios that express the same relationship between quantities. For example, 2:3, 4:6 and 6:9 are equivalent because each pair has the same relative size.
How to get equivalent ratios: Multiply or divide both terms of a ratio by the same nonzero number. If a:b is a ratio and k is a nonzero number, then (ka):(kb) is equivalent to a:b.
Simplification (reducing a ratio to lowest terms): To simplify a ratio, divide both terms by their greatest common divisor (GCD). The simplified form has no common factor other than 1.
Test for equivalence: Two ratios a:b and c:d are equivalent if a/b = c/d (when b and d are not zero). Equivalently, cross-multiply: a × d = b × c.
Why this matters: Working with equivalent ratios and simplified ratios makes comparison, calculation and real-life tasks (recipes, maps, mixtures) easier and less error-prone.
- Simple numeric example: 18:24. GCD(18,24)=6, so divide both by 6 → 3:4. Thus 18:24 is equivalent to 3:4.
- Scaling up: 3:5 → multiply both by 2 → 6:10; multiply by 3 → 9:15. All are equivalent to 3:5.
- Cross-multiplication check: Are 2:7 and 6:21 equivalent? 2×21 = 42 and 7×6 = 42, so yes, they are equivalent.
- Recipe example (real-life): A pancake recipe needs 2 cups flour : 1 cup milk. For a larger batch, multiply by 3 → 6 cups flour : 3 cups milk — equivalent ratio keeps the same taste.
- Classroom example: If in one class the ratio of boys to girls is 4:5, and another class has 8:10, these are equivalent (both simplify to 4:5) so proportions of boys to girls are the same.
- Ratio notation: a:b or a/b (b ≠ 0).
- Scaling (equivalent ratio): a:b = (k·a):(k·b) for any nonzero k.
- Equivalence test: a:b is equivalent to c:d if a/b = c/d (or if a·d = b·c).
- Simplification: If g = gcd(a,b) then simplified ratio = (a/g):(b/g).
- Lowest terms: a and b are in lowest terms when gcd(a,b) = 1.
Comparing Ratios
What is comparing ratios? Comparing ratios means deciding which of two (or more) ratios is greater, or whether they are equal. A ratio a:b can be treated as the fraction a/b, so comparing ratios is like comparing fractions or unit rates.
Common methods
- Simplify: Reduce each ratio to its simplest form by dividing both terms by their greatest common divisor (GCD). Then compare the simplified pairs.
- Convert to fractions or unit rates: Treat a:b as the fraction a/b (amount of first per one of the second). Compare the fractions or convert to decimals.
- Cross-multiplication: For two ratios a:b and c:d, they are equal if ad = bc. To see which is larger, compare ad and bc: if ad > bc then a:b > c:d.
- Scaling: Multiply or divide both terms of a ratio by the same number to get comparable terms (for example, make both ratios have the same second term) and then compare the first terms.
Why it works: Ratios describe relative sizes. Converting to a common basis (unit rate or same second term) removes the effect of different scales so you can compare the actual relative sizes directly. Cross-multiplication is a quick arithmetic way to compare the two fractions a/b and c/d without dividing.
- Example 1 (Equality): Compare 3:4 and 6:8. Simplify 6:8 by dividing both terms by 2 to get 3:4. So 3:4 = 6:8 (they are equal).
- Example 2 (Cross-multiplication): Which is larger, 5:8 or 4:7? Compute cross-products: 5×7 = 35 and 4×8 = 32. Since 35 > 32, 5:8 > 4:7.
- Example 3 (Unit rate / price): Apples cost ₹150 for 3 kg and ₹200 for 4 kg. Compare price per kg. First: 150/3 = ₹50 per kg. Second: 200/4 = ₹50 per kg. Both are equal in price (50 = 50).
- Example 4 (Scaling to same second term): Compare 2:3 and 5:7. Make second terms equal (LCM of 3 and 7 = 21). Scale 2:3 by 7 → 14:21. Scale 5:7 by 3 → 15:21. Compare first terms 14 and 15 → 15>14, so 5:7 > 2:3.
- Ratio to fraction: a:b = a/b (compare as fractions)
- Equality condition: a:b = c:d iff a × d = b × c (cross-multiplication)
- Unit rate (per one): a:b → a/b (amount of first per one of second)
- Scaling equivalent ratios: a:b = (a×k):(b×k) for any nonzero k
- Simplification: divide both terms by gcd(a,b) to get simplest form
Proportion
What is Proportion?
Proportion is a statement that two ratios are equal. If two ratios a:b and c:d have the same value, we write a:b = c:d and say “a is to b as c is to d”.
Terms: In a:b = c:d, a and d are called extremes, b and c are called means.
How to check a proportion (Cross-multiplication): For a:b = c:d, multiply extremes and means. If ad = bc, then the two ratios are equal. This method also helps to find a missing term.
Constant of Proportionality (Direct Proportion): When a and b are directly proportional (a:b = constant), we write a/b = k. That means a = k·b. In a proportion a:b = c:d, if the common value is k then a = k·b and c = k·d.
Solving a missing term (unitary method): To find an unknown in a proportion, either use cross-multiplication or find the value for one unit and scale (unitary method). Example: If 5:x = 3:9, then 5·9 = 3·x so x = 45/3 = 15.
- Simple numeric: Check if 3:4 and 6:8 are in proportion. Cross-multiply: 3×8 = 24 and 4×6 = 24 → equal, so they are proportional.
- Find a missing term: 5:7 = x:21. Using cross-multiplication 5×21 = 7×x → 105 = 7x → x = 15.
- Cost and quantity (real life): If 4 apples cost Rs. 60, cost is proportional to number of apples. Cost per apple = 60/4 = Rs.15, so 7 apples cost 7×15 = Rs.105.
- Recipe proportions: If a cake recipe uses 2 cups of flour for 3 cups of sugar, to make twice the cake use 4 cups flour and 6 cups sugar — ratios remain equal (2:3 = 4:6).
- Map scale: If 1 cm on a map represents 50 km, then 5 cm represents 5×50 = 250 km — distance is directly proportional to map length.
- Distance and time at constant speed: If a car travels 40 km in 1 hour, then distance and time are proportional. In 3 hours it travels 3×40 = 120 km.
- Definition: a:b = c:d (two ratios are equal)
- Cross-multiplication test: a:b = c:d ⇔ a·d = b·c
- Constant of proportionality: if a:b = k then a = k·b and k = a/b
- Solving for a missing term: from a:b = c:d → d = (b·c)/a, c = (a·d)/b, b = (a·c)/d, etc.
- Unitary method: Find value for one unit then multiply. Example: If 4 items cost 200, 1 item costs 200/4 = 50, so 7 items cost 7×50 = 350.
Properties of Proportion (Means and Extremes)
What is a proportion? A proportion is an equality of two ratios. If a:b = c:d, we say the two ratios a:b and c:d are in proportion.
Means and extremes — definitions: In the proportion a:b = c:d, the terms a and d are called the extremes, and b and c are called the means.
Fundamental property (Product of Means = Product of Extremes): For any proportion a:b = c:d (where b and d are not zero), the cross products are equal:
- a × d = b × c
This is the most important property and is often called cross-multiplication. It allows us to test whether two ratios are in proportion and to find a missing term.
Other useful properties of proportions:
- If a:b = c:d then a/b = c/d (fractions are equal).
- If a:b = c:d then b:a = d:c (reciprocals are equal).
- Multiplying or dividing all four terms by the same non-zero number k preserves the proportion: (ka):(kb) = (kc):(kd).
- If a:b = c:d then (a + c):(b + d) = a:b = c:d (sum of antecedents to sum of consequents gives same ratio).
- If a:b = c:d then (a - c):(b - d) = a:b = c:d (difference of corresponding terms gives same ratio), provided subtraction gives valid (non-zero) terms.
- From a:b = c:d we also get (a + b):b = (c + d):d and (a - b):b = (c - d):d where terms are valid.
How to use the properties: Use cross-multiplication to check or find missing numbers. Example: If 3:4 = x:12, then 3×12 = 4×x, so x = 9.
Why it works (intuitive idea): If two ratios are equal, they represent the same multiplier k: a = kb and c = kd. Then a×d = (kb)×(kd)/k? More directly, a×d = (kb)×d = k(bd) and b×c = b×(kd) = k(bd), so a×d = b×c.
- Numeric check: 2:3 = 4:6. Means are 3 and 4, extremes are 2 and 6. Product of means = 3×4 = 12 and product of extremes = 2×6 = 12, so the proportion holds.
- Find missing term: If 5:8 = x:32, use cross-multiplication: 5×32 = 8×x → 160 = 8x → x = 20.
- Sum property: Given 3:5 = 6:10, (3+6):(5+10) = 9:15 = 3:5 (both simplify to 3:5).
- Real-life (recipe): A recipe needs sugar:flour = 2:5. For 10 cups of flour, sugar = (2/5)×10 = 4 cups. Here ratios remain in proportion.
- Real-life (map scale): If 1 cm on a map represents 5 km, then 3 cm represents 15 km. Ratios 1:5 = 3:15, and 1×15 = 5×3 (15 = 15).
- Definition: a:b = c:d (a, b, c, d are terms of proportion)
- Means: b and c; Extremes: a and d
- Fundamental property (cross-multiplication): a × d = b × c
- Equality as fractions: a/b = c/d
- Reciprocal property: b/a = d/c
- Scaling: (ka):(kb) = (kc):(kd) for any nonzero k
Division of a Quantity in a Given Ratio
What it means
To divide a quantity in a given ratio means to split that quantity into parts that are proportional to the numbers in the ratio. If the ratio is a:b (or a:b:c for three parts), the parts should be in the same proportion as a to b (or a to b to c).
Step-by-step method
- Write down the ratio parts and add them to find the total number of equal parts. (E.g., for 2:3 the total is 2+3 = 5.)
- Divide the whole quantity by this total to find the value of one part (unit value). (Unit = Quantity ÷ Total parts.)
- Multiply the unit value by each ratio number to get each share. (Share for a = a × Unit, for b = b × Unit, etc.)
- Check by adding all shares — the sum must equal the original quantity.
Worked example (two parts)
Divide 90 in the ratio 4:5.
Total parts = 4 + 5 = 9. Unit = 90 ÷ 9 = 10. Shares: 4 × 10 = 40 and 5 × 10 = 50. Check: 40 + 50 = 90.
Worked example (three parts)
Divide 180 in the ratio 2:3:5.
Total parts = 2 + 3 + 5 = 10. Unit = 180 ÷ 10 = 18. Shares: 2×18 = 36, 3×18 = 54, 5×18 = 90. Check: 36 + 54 + 90 = 180.
Notes
• The ratio need not be simplified first; the method works for any positive integers. If you simplify a ratio (divide all parts by the same factor), the shares will be the same proportions but you must use the simplified total correctly.
• Always check that the parts add up to the original quantity.
- Divide ₹480 in the ratio 3:5. Total parts = 3+5 = 8; unit = 480 ÷ 8 = 60; shares = 3×60 = ₹180 and 5×60 = ₹300.
- A recipe needs ingredients in the ratio 2:1:1. If you have 800 g total, total parts = 4; unit = 200 g; parts = 400 g, 200 g, 200 g.
- Three friends share prize money of ₹1200 in ratio 4:3:5. Total parts = 12; unit = 100; shares = ₹400, ₹300, ₹500.
- Split 1 hour (60 minutes) between two tasks in ratio 2:1. Total parts = 3; unit = 20 minutes; tasks get 40 minutes and 20 minutes.
- For two parts a:b and quantity Q: First find total parts = a + b. Unit = Q ÷ (a + b). Shares = a × Unit and b × Unit.
- For three parts a:b:c and quantity Q: Total parts = a + b + c. Unit = Q ÷ (a + b + c). Shares = a×Unit, b×Unit, c×Unit.
- General: For ratio r1:r2:...:rn and quantity Q: Total parts = Σ ri. Unit = Q ÷ (Σ ri). Share i = ri × Unit.
- Check: Sum of shares = Q (because Σ(ri × Unit) = Unit × Σ ri = Q).
Unitary Method and Simple Applications
What is the Unitary Method?
The unitary method is a way to find the value of a single unit from a known value of multiple units, and then use that single-unit value to find the value of any other number of units. It is often used when quantities are in direct proportion (i.e., doubling one quantity doubles the other).
Steps (basic idea)
- Find the value of 1 unit by dividing the given total by the number of units.
- Multiply the value of 1 unit by the required number of units to get the answer.
When to use it
Use the unitary method for problems involving prices, work done, distance and speed (when time is fixed), ingredients in a recipe, and any situation where one quantity changes directly in proportion to another.
Tips
- Always identify the unit you want to find (the '1' in unitary).
- Keep units consistent (e.g., hours, kilometres, rupees).
- If quantities are inversely proportional (one increases while the other decreases), use inverse reasoning: find per-unit value and then adjust accordingly by division where needed.
- Example 1 — Price of one apple and many apples: If 6 apples cost Rs 30, find the price of 1 apple and of 15 apples. Solution: Price of 1 apple = 30 ÷ 6 = Rs 5. Price of 15 apples = 5 × 15 = Rs 75.
- Example 2 — Scaling a recipe: A recipe needs 4 cups of flour to make 8 cookies. How much flour is needed for 20 cookies? Solution: Flour per cookie = 4 ÷ 8 = 0.5 cup. For 20 cookies = 0.5 × 20 = 10 cups.
- Example 3 — Distance and speed (direct proportion for fixed time): A car travels 60 km in 1 hour. How far will it travel in 3 hours at the same speed? Solution: Distance per hour = 60 km. For 3 hours = 60 × 3 = 180 km.
- Example 4 — Wages problem: 5 workers earn Rs 240 in one day. How much does 1 worker earn in one day? How much do 8 workers earn in one day? Solution: Earning per worker = 240 ÷ 5 = Rs 48. For 8 workers = 48 × 8 = Rs 384.
- Value of 1 unit = Given total value ÷ Number of units
- Value for n units = (Value of 1 unit) × n
- If A units correspond to B value, then value per unit = B ÷ A and for x units value = (B ÷ A) × x
- Direct proportion linear relation: y = kx (k is value for 1 unit, slope through origin)
- For scaling quantities in the same ratio: new quantity = original quantity × (new factor ÷ original factor)
- When quantities are inversely proportional (e.g., time and speed for fixed work): If x1·y1 = x2·y2, use appropriate inverse relation (unitary reasoning after converting to 'per unit' form)
Word Problems and Real-life Applications
What is a ratio? A ratio compares two quantities of the same kind by division. If there are a and b of something, the ratio of a to b is written as a:b and means a/b.
What is a proportion? A proportion is a statement that two ratios are equal. It is written as a:b = c:d or a:b :: c:d and means a/b = c/d.
How to approach word problems using ratio and proportion
- Read the problem carefully and identify what quantities are being compared.
- Write the ratio(s) given in the problem and simplify if possible.
- Decide whether you should use direct proportion, the unitary method (find one unit then multiply), or set up a proportion (use cross-multiplication).
- Solve step by step and check your answer to see if it makes sense in the context of the problem.
Common methods
- Unitary method: Find the value of one part and then multiply to get the required number of parts. Useful when you know total for several equal parts and need one part or vice versa.
- Proportion and cross-multiplication: If a/b = c/d, then a×d = b×c. Use this to find an unknown when three numbers are known.
- Scaling ratios: To find quantities in the same ratio for a different total, first add the parts of the ratio to get total parts, find value of one part, then multiply by each ratio part.
Tips for word problems
- Use diagrams (tape diagrams or pie slices) for parts-of-a-whole problems.
- Label what each part of the ratio represents (e.g., boys:girls = 3:2 means for every 3 boys there are 2 girls).
- Check units (litres, rupees, kilograms, etc.) so answers are in the correct unit.
- Example 1 — Sharing money: A prize of Rs 480 is to be shared among A, B and C in ratio 3:2:1. Find each share. Solution: Total parts = 3+2+1 = 6. One part = 480 ÷ 6 = 80. A = 3×80 = 240, B = 2×80 = 160, C = 1×80 = 80.
- Example 2 — Parts of a mixture: A juice mixture has water and concentrate in ratio 4:1. If total mixture is 25 litres, how much concentrate is used? Solution: Total parts = 4+1 = 5. One part = 25 ÷ 5 = 5 litres. Concentrate = 1×5 = 5 litres.
- Example 3 — Scale a recipe: A recipe needs sugar and flour in ratio 1:3. If you want to use 600 g of flour, how much sugar is needed? Solution: For every 3 parts flour, sugar = 1 part. One part = 600 ÷ 3 = 200 g. Sugar needed = 200 g.
- Example 4 — Cost and unitary method: 5 pens cost Rs 150. How many pens can you buy for Rs 210? Solution: Cost of 1 pen = 150 ÷ 5 = Rs 30. Number of pens for Rs 210 = 210 ÷ 30 = 7 pens.
- Example 5 — Class strength: In a class, ratio of boys to girls is 7:5. If there are 48 students in all, find number of boys and girls. Solution: Total parts = 7+5 = 12. One part = 48 ÷ 12 = 4. Boys = 7×4 = 28, Girls = 5×4 = 20.
- Example 6 — Proportion (missing term): If 4 notebooks cost Rs 96, how much will 7 notebooks cost? Solution using proportion: 4 : 96 = 7 : x ⇒ 4x = 96×7 ⇒ x = (96×7)÷4 = 24×7 = Rs 168.
- Ratio of a to b: a:b = a/b (b ≠ 0)
- Simplest form: divide both terms by their greatest common divisor (gcd)
- Proportion: a:b = c:d ⇒ a/b = c/d
- Cross-multiplication: a:b = c:d ⇒ a×d = b×c
- Unitary method: value of one part = total value ÷ number of parts; required value = (value of one part) × number of required parts
- To scale a ratio to a new total: one part = new total ÷ (sum of ratio parts); then multiply each ratio part by one part
Practice Exercises and Problem-solving Strategies
Practice with ratio and proportion problems builds understanding of parts, scaling and direct relationships. Effective problem-solving uses clear steps, shortcuts (like the unitary method and cross‑multiplication), and visual models (bar/tape diagrams, tables or graphs) to organize information and check answers.
Step-by-step problem-solving strategy
- Read and identify: Determine what quantities are in ratio form, what is asked (each part, total, missing term).
- Assign parts: Write the ratio clearly (a:b or a:b:c). If total is given, find sum of parts.
- Simplify: Reduce ratios to simplest form if useful.
- Choose method: Use unitary method (find one part), cross‑multiplication for equations of proportion, or scaling factor for enlargement/reduction.
- Solve and check: Compute values, then check by reassembling totals or comparing the ratios.
- Use diagrams: Draw tape/bar diagrams, ratio tables or number lines to make relationships visual, especially in word problems.
Common problem types and short tips
- Divide a total in a given ratio: Add ratio parts; each part = (total)/(sum of parts) × part size.
- Find a missing term in proportion: Use cross‑multiplication: if a:b = c:d then ad = bc.
- Scaling/recipes/maps: Use scale factor = new/old or unitary method to compute each ingredient or distance.
- Mixing/sharing: Convert to parts and apply totals; draw bars for clarity.
- Checking proportionality: Reduce ratios to simplest form or check cross products equal.
Practice approach: Start with simple numeric ratio problems, move to word problems (recipes, sharing, maps), then mixed problems (combine ratios or change totals). Time yourself on sets of problems to build fluency, and always draw a quick diagram for word problems.
- Divide 56 in the ratio 3:4. Sum of parts = 3+4=7. One part = 56/7=8. So parts = 3×8=24 and 4×8=32.
- Find x if 5:x = 15:9. Cross‑multiply: 5×9 = 15×x ⇒ 45 = 15x ⇒ x = 3.
- Recipe problem: Sugar:Flour = 2:5 and total required = 350 g. Sum parts = 7. Sugar = (2/7)×350 = 100 g, Flour = 250 g.
- Map scale: 1 cm = 5 km. If distance on map = 7 cm, real distance = 7×5 = 35 km.
- Mixture: Alcohol:Water = 3:2, total 25 L. Sum parts = 5. Alcohol = (3/5)×25 = 15 L, Water = 10 L.
- Are 4:6 and 10:15 in proportion? Simplify 4:6 = 2:3 and 10:15 = 2:3, so yes they are proportional (or check 4×15 = 6×10 = 60).
- Ratio of a and b written a:b (compare sizes; order matters).
- Convert ratio to fraction: a:b = a/(a+b) of the total for the first part (when dividing total among parts).
- If a:b = c:d (proportion) then cross‑multiplication gives ad = bc.
- Unitary method: one part = total ÷ (sum of ratio parts); required part = (one part) × (part's share).
- Scaling factor: new value = old value × scale (e.g., to enlarge a ratio by k, multiply each part by k).
- To check proportionality: reduce both ratios to simplest form or verify cross products are equal.
Key Concepts
- Ratio
- A comparison of two quantities of the same kind showing how many times one is of the other.
- Proportion
- An equation stating that two ratios are equal.
- Antecedent
- The first term (left-hand quantity) of a ratio.
- Consequent
- The second term (right-hand quantity) of a ratio.
- Colon notation
- A way to write a ratio using a colon ":" between two numbers.
- Fraction form (of a ratio)
- A ratio written as a fraction with antecedent as numerator and consequent as denominator.
- Equivalent ratios
- Two or more ratios that express the same relationship or simplify to the same simplest form.
- Simplest form (of a ratio)
- A ratio whose terms have no common factor other than 1.
- Unit ratio
- A ratio whose antecedent (first term) is 1.
- Unitary method
- A method to find the value of one unit first, then use it to find the value of required units.
- Continued proportion
- A sequence of three or more terms where consecutive ratios are equal.
- Directly proportional (Direct proportion)
- Two quantities are directly proportional if they increase or decrease in the same ratio so their quotient is constant.
- Inverse proportion
- Two quantities are inversely proportional if one increases while the other decreases so their product is constant.
- Means (in a proportion)
- The middle two terms (second and third) in a proportion a:b = c:d are called means.
- Extremes (in a proportion)
- The first and last terms of a proportion a:b = c:d are called extremes.
- Cross multiplication
- A method stating that in a proportion a:b = c:d the product of extremes equals product of means (a×d = b×c).
- Part-to-whole ratio
- A ratio that compares a part of a quantity to the whole quantity.
- Rate
- A ratio that compares two quantities having different units (often expressed using 'per').
- Percentage
- A ratio expressed as a fraction of 100 (parts per hundred).
- Sharing in a given ratio
- Dividing a total quantity into parts such that the parts are in the specified ratio.
End-of-Chapter Trial Paper & Test Questions
Topic-wise questions to test your understanding of every concept in this chapter.
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The ratio of 15 to 25 in simplest form is: / 15 से 25 का अनुपात सरलतम रूप में है: (a) 15:25 / 15:25 (b) 3:5 / 3:5 (c) 5:3 / 5:3 (d) 1:2 / 1:2
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(b) 3:5 / 3:5 — HCF(15, 25) = 5. Divide both terms by 5: 15÷5 = 3, 25÷5 = 5. So 15:25 = 3:5. / HCF(15, 25) = 5। दोनों पदों को 5 से भाग दें: 15÷5 = 3, 25÷5 = 5। अतः 15:25 = 3:5।
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Which pair of ratios forms a proportion? / निम्न में से कौन-सा अनुपात युगल समानुपात बनाता है? (a) 2:3 and 4:5 / 2:3 और 4:5 (b) 3:4 and 9:12 / 3:4 और 9:12 (c) 1:2 and 3:5 / 1:2 और 3:5 (d) 5:8 and 3:4 / 5:8 और 3:4
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(b) 3:4 and 9:12 / 3:4 और 9:12 — Cross-multiply: 3 × 12 = 36 and 4 × 9 = 36. Since products are equal, 3:4 = 9:12 (both simplify to 3:4). / क्रॉस गुणा: 3 × 12 = 36 और 4 × 9 = 36। गुणनफल बराबर होने पर, 3:4 = 9:12 (दोनों 3:4 में सरल होते हैं)।
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If 5:x = 15:9, then x is: / यदि 5:x = 15:9, तो x का मान है: (a) 3 / 3 (b) 45 / 45 (c) 5 / 5 (d) 27 / 27
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(a) 3 / 3 — Cross-multiply: 5 × 9 = 15 × x → 45 = 15x → x = 3. / क्रॉस गुणा: 5 × 9 = 15 × x → 45 = 15x → x = 3।
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Divide ₹480 in the ratio 5:3. The larger share is ₹ ______. / ₹480 को 5:3 के अनुपात में बाँटें। बड़ा हिस्सा ₹ ______ है।
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₹300 / ₹300 — Total parts = 5 + 3 = 8. One part = 480 ÷ 8 = ₹60. Larger share (5 parts) = 5 × 60 = ₹300. / कुल भाग = 5 + 3 = 8। एक भाग = 480 ÷ 8 = ₹60। बड़ा हिस्सा (5 भाग) = 5 × 60 = ₹300।
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If 8 notebooks cost ₹200, then 5 notebooks cost ₹ ______. (Use unitary method.) / यदि 8 नोटबुक का मूल्य ₹200 है, तो 5 नोटबुक का मूल्य ₹ ______ है। (एकात्मक विधि प्रयोग करें।)
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₹125 / ₹125 — Cost of 1 notebook = 200 ÷ 8 = ₹25. Cost of 5 notebooks = 25 × 5 = ₹125. / 1 नोटबुक का मूल्य = 200 ÷ 8 = ₹25। 5 नोटबुक का मूल्य = 25 × 5 = ₹125।
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True or False: The ratio 4:6 and 6:9 are equivalent. / सत्य या असत्य: 4:6 और 6:9 समतुल्य अनुपात हैं।
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True / सत्य — Simplify 4:6 (÷2) = 2:3 and 6:9 (÷3) = 2:3. Both simplify to 2:3, so they are equivalent. Cross-check: 4×9 = 36 = 6×6. / 4:6 (÷2) = 2:3 और 6:9 (÷3) = 2:3। दोनों 2:3 में सरल होते हैं, अतः समतुल्य हैं। जाँच: 4×9 = 36 = 6×6।
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In a class of 48 students, the ratio of boys to girls is 5:3. How many girls are in the class? / 48 छात्रों की कक्षा में लड़कों और लड़कियों का अनुपात 5:3 है। कक्षा में कितनी लड़कियाँ हैं?
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18 girls / 18 लड़कियाँ — Total parts = 5 + 3 = 8. One part = 48 ÷ 8 = 6. Girls (3 parts) = 3 × 6 = 18. / कुल भाग = 5 + 3 = 8। एक भाग = 48 ÷ 8 = 6। लड़कियाँ (3 भाग) = 3 × 6 = 18।
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A recipe for 4 people needs 2 cups of sugar and 6 cups of flour. What is the ratio of sugar to flour in simplest form, and how much sugar is needed for 10 people? / 4 लोगों की रेसिपी में 2 कप चीनी और 6 कप मैदा चाहिए। चीनी और मैदा का सरलतम अनुपात क्या है, और 10 लोगों के लिए कितनी चीनी चाहिए?
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Ratio: 1:3; Sugar for 10 people = 5 cups / अनुपात: 1:3; 10 लोगों के लिए चीनी = 5 कप — Sugar:Flour = 2:6 = 1:3 (÷2). For 10 people (scale factor = 10/4 = 2.5): sugar = 2 × 2.5 = 5 cups. Or: 1 cup per 2 people, so 10 people need 5 cups. / चीनी:मैदा = 2:6 = 1:3। 10 लोगों के लिए (गुणक = 10/4 = 2.5): चीनी = 2 × 2.5 = 5 कप।
Related Laws & Principles
Explore allFoundational laws & principles behind this chapter. Each one opens a full page — what it says, why it matters, five practice questions and the mistakes to avoid.