Proportional Reasoning-2 | CBSE Class 8 Maths Notes

This note covers proportional relationships, map scales, ratios with several terms, division of a whole in a given ratio, triangle angles, pie charts, direct and inverse proportions, and work done together.

What makes two ratios proportional?

A ratio compares quantities in a stated order. Its terms are the numbers used in that comparison. The notation 2 : 1 reads “two to one”. In one mixture for idli batter, it means two cups of rice for every one cup of urad dal. The proportions used can have regional variations.

A proportional relationship is a relationship in which related quantities change by the same factor. A factor is the number by which a quantity is multiplied. Multiplying both terms of 2 : 1 by the same factor preserves the relative amounts.

How does cross-multiplication check a proportion?

Let a, b, c and d represent the four quantities in two ratios a : b and c : d. The colon separates terms within a ratio. The double-colon notation a : b :: c : d means that the two ratios are proportional.

Definition: Two ratios a : b and c : d are proportional when a × d = b × c. The symbol × means multiplication, and = means “is equal to”. A product is the result of multiplication. This check is called cross-multiplication because it compares the products across the two ratios.

Worked example 1. Viswanath mixes 6 cups of rice with 3 cups of urad dal. Puneet mixes 4 cups of rice with 2 cups of urad dal. Check whether the ratios are proportional.

Answer: The cross-products are 6 × 2 = 12 and 3 × 4 = 12. They are equal, so 6 : 3 and 4 : 2 are proportional. Both mixtures have rice and urad dal in the ratio 2 : 1.

Keep the order of the ingredients unchanged while comparing them. Here rice comes first and urad dal second. It is likely that the idlis would taste the same if cooked in the same way and if all the other ingredients are proportional too.

How does a ratio describe a map scale?

A Representative Fraction (RF) expresses the ratio of a distance on a map to the corresponding actual distance on the ground. The two distances must be read in the same unit. Maps usually show a scale containing 1 and a much larger number.

The abbreviation cm means centimetres, while km means kilometres. With RF 1 : 60,00,000, a map distance of 1 cm represents a geographical distance of 60,00,000 cm. This is 60 km. The distance represented is geographical distance, not road distance.

Worked example 2. Interpret a map scale of 1 : 60,00,000 for a distance of 1 cm on the map.

Answer: Map distance : ground distance = 1 : 60,00,000. Therefore, 1 cm on the map corresponds to 60,00,000 cm on the ground, or 60 km. Keep both quantities in centimetres when first applying the ratio.

How can a scale be used carefully?

Measure the map distance between the required locations with a ruler. Use the given ratio to obtain the corresponding ground distance, then convert units if needed. Measurements from maps with different scales should give approximately the same geographical distance between the same locations.

The map marks cities including Bengaluru, Chennai and Mangaluru. It shows RF 1 : 60,00,000 and also carries the label “Map not to scale”.

Reference: NCERT Class 8, page 56, unnumbered map

The classroom map activity uses a scale of 1 : 50. Positions of the teacher’s desk, blackboard, fans and lights are marked according to that scale. Symbols can represent objects such as tables and chairs, but their positions still need to follow the chosen scale.

How do ratios with more than two terms work?

A multi-term ratio compares several quantities in a fixed order. Each term tells the amount associated with one ingredient or category. To maintain a proportional relationship, multiply every term by the same factor, including terms that are fractions or decimals.

Viswanath’s spice mixture uses 8 spoons of coriander seeds, 4 red chillies, 2 spoons of toor dal and 1 spoon of fenugreek, also called methi, seeds. In that order, its ratio is 8 : 4 : 2 : 1.

Worked example 3. Puneet has 2 red chillies and wants a mixture similar to Viswanath’s. Find the quantities of the remaining ingredients.

Answer: Two chillies are half of four chillies, so halve each quantity. Use 4 spoons of coriander seeds, 2 red chillies, 1 spoon of toor dal and half a spoon of fenugreek seeds. The new ratio is 4 : 2 : 1 : 0.5.

Why must corresponding ingredients stay together?

The chilli quantity is a count, while the other quantities are measured in spoons. Retain these labels when interpreting the ratio. Halving the chilli count but leaving all the spoonfuls unchanged would not preserve the original relationship between the ingredients.

One part means the amount represented by one unit of a ratio. The symbol + means addition.

Worked example 4. Yasmin mixes red, blue and white paint in the ratio 2 : 3 : 5. She has 10 litres of white paint. Find the other quantities and the total volume.

Answer: Five parts represent 10 litres, so one part represents 10 ÷ 5 = 2 litres. The symbol ÷ means division. Red paint requires 2 × 2 = 4 litres, and blue requires 3 × 2 = 6 litres. Total volume = 4 + 6 + 10 = 20 litres.

White paint gives the value of five parts, not the value of the entire mixture. Finding one part links the ratio to the actual quantities.

How is a whole divided in a given ratio?

When the whole quantity is known, first add the terms of the ratio. This sum is the total number of parts. Divide the whole by that sum to find one part, then multiply by each ratio term to obtain the separate shares.

Result: Each share depends on its fraction of the total parts

Let x denote the whole quantity, and let a, b and c denote the three terms of its division ratio. Then first share = x × a/(a + b + c). The slash means division; the brackets keep the sum together as the divisor, the number being divided by.

The second and third shares use b and c respectively in place of a before the slash. This method extends to ratios with more terms. A share depends on its own term divided by the sum of all the terms, rather than by another individual term.

Worked example 5. Divide 110 units of concrete into cement, sand and gravel in the ratio 1 : 1.5 : 3.

Answer: Total parts = 1 + 1.5 + 3 = 5.5. One part = 110 ÷ 5.5 = 20 units. Cement = 20 units, sand = 1.5 × 20 = 30 units, and gravel = 3 × 20 = 60 units. Their sum is 110 units.

Cement concrete is a mixture of cement, sand and gravel. Its component ratio varies with the strength needed. The calculation treats the quantities in the common units given in the problem. Keep the ingredients in the order attached to the ratio.

Worked example 6. Make 50 millilitres, abbreviated ml, of purple paint using red : blue : white = 2 : 3 : 5.

Answer: Total parts = 10, so one part = 50 ÷ 10 = 5 ml. Use 10 ml of red, 15 ml of blue and 25 ml of white paint. These add to the required 50 ml.

Compare this with Yasmin’s problem: there, one ingredient was known; here, the whole mixture is known. Identify which quantity the question gives before deciding whether to divide by one ratio term or by their sum.

How can ratios divide angles and practice time?

The same division method works for a total angle or a total duration. What changes is the whole being divided. In a triangle, the angles add to 180 degrees. The symbol ° denotes degrees, the unit used here to measure angles.

Worked example 7. Find the angles of a triangle whose angles are in the ratio 1 : 3 : 5.

Answer: Total parts = 1 + 3 + 5 = 9. One part = 180° ÷ 9 = 20°. The three angles are 1 × 20° = 20°, 3 × 20° = 60° and 5 × 20° = 100°.

The triangle has vertices, or corners, labelled A, B and C. The angles at these vertices are labelled 20°, 60° and 100° respectively.

Reference: NCERT Class 8, page 60, unnumbered figure

The letters A, B and C here name corners, rather than quantities in a formula. The ratio describes angles. It does not say that the triangle’s side lengths have the same ratio. Read the type of measurement before using the given terms.

How does the method apply to a practice session?

A cricket practice session lasts 150 minutes. Its times for warm-up/cool-down, batting, bowling and fielding are in the ratio 3 : 4 : 3 : 5. The ratio concerns durations, so the answer must give minutes for each activity.

The terms add to 15. One part therefore represents 150 ÷ 15 = 10 minutes. The four allocations are 30, 40, 30 and 50 minutes, in the stated order. Checking their sum gives 150 minutes, the full session.

Equal terms produce equal shares of the same whole: warm-up/cool-down and bowling both receive 30 minutes. A larger ratio term receives a larger allocation. These checks help confirm that the calculation and the category labels agree.

How are the angles of a pie chart calculated?

A pie chart represents proportions of a whole using slices of a circle. A radius joins the centre to the circle; its plural is radii. A slice, also called a sector, is the part between two radii and the circle’s curved edge. Its angle at the centre must be proportional to the quantity it represents.

The total angle around a circle’s centre is 360°. For the grade distribution below, A, B, C, D and E are grade labels. The 40 students form the whole. Divide 360° in the ratio of their grade counts.

How does simplifying the ratio help?

The highest common factor (HCF) is the greatest whole number that divides every term exactly. Divide all terms by their HCF to put the ratio in simplest form. The HCF of 12, 10, 8, 6 and 4 is 2.

Dividing every term by 2 changes 12 : 10 : 8 : 6 : 4 to 6 : 5 : 4 : 3 : 2. The relative shares stay the same. The simplified terms add to 20, and 360° ÷ 20 = 18° per part.

Worked example 8. Calculate the angle for grade A when 12 of the 40 students have that grade.

Answer: Grade A represents 12/40 of the class, so its angle is (12/40) × 360° = 108°. Equivalently, the simplified ratio gives (6/20) × 360° = 108°. Both calculations describe the same share.

Use the remaining terms to obtain 90°, 72°, 54° and 36°. These angles total 360°, so together they account for the complete circle. Reducing the ratio simplifies the arithmetic without changing the number of students represented by any grade.

How do you construct and read a pie chart?

After calculating the angles, draw the sectors consecutively around the centre. Each new radius forms the boundary between neighbouring sectors. Label the sectors so the quantities they represent are clear.

What is the construction sequence for the grade chart?

In this construction, A names the centre of the circle. B, C, D, E and F name points on its circumference, the curved boundary. A pair of letters such as AB names the line segment joining those two points.

1. Draw a circle with centre A and mark radius AB as the starting line.
2. At A, measure 108° from AB in the anti-clockwise direction, opposite to a clock’s hands. Draw radius AC for the grade A sector.
3. Measure 90° from AC and draw AD. This creates the next sector, representing grade B.
4. Measure 72° from AD and draw AE for grade C. Continue with 54° and then 36° to complete the circle.
5. Colour and label the slices appropriately, keeping each grade matched to its angle.

Successive circles show radii added around centre A. The completed angle diagram displays 108°, 90°, 72°, 54° and 36°. A colouring example labels the grade A and grade B slices.

Reference: NCERT Class 8, pages 61 to 62, unnumbered figures

Distinguish the centre label A from grade A: one names a point and the other names a category. Measure each new angle from the preceding radius, so the sectors occupy consecutive portions of the circle.

How does an angle reveal a proportion?

A sector’s angle divided by 360° gives its fraction of the whole. The school-transport chart has bus 120°, walk 90°, cycle 60° and two-wheeler 60°. The remaining car angle is 30°. Bus has the largest sector; cycle and two-wheeler have equal sectors.

Cars therefore represent 30/360 = 1/12 of the children. If 18 children travel by car, the total is 18 × 12 = 216 children. This reverses the angle calculation: a known part and its fraction determine the whole.

When should you use direct proportion?

Two quantities are in direct proportion when both change by the same factor and their quotient remains the same. A quotient is the result of division. In these examples, multiplying one quantity by a factor multiplies the other by that factor too.

Result: Direct proportion keeps the quotient constant

Let x and y stand for two directly proportional quantities. A constant is a value that stays unchanged. If k denotes this constant quotient, then x/y = k for their corresponding values. The quantities and their order must stay consistent.

The rule of three finds a missing fourth quantity from three given quantities in a proportion. For a : b :: c : d, where a, b and c are known and d is unknown, cross-multiplication gives a × d = b × c.

Dividing by a gives d = (b × c)/a. Before using this rule, confirm that the situation supports direct proportion. Merely finding four numbers in a word problem does not establish the relationship between them.

A rate of work measures the amount completed per unit time, such as bricks moved in one day.

Worked example 9. Five workers move 4500 bricks in one day. At the same rate of work per worker, how many workers are needed to move 18000 bricks in one day?

Answer: The brick quantity increases by the factor 18000 ÷ 4500 = 4. The worker count increases by the same factor, giving 5 × 4 = 20 workers. Equivalently, the required number is (18000 × 5)/4500 = 20.

The time available stays at one day. This fixed condition matters: the comparison is between workers and bricks moved in that time. By contrast, asking how long different worker groups take for the same fixed job leads to an inverse relationship.

What is inverse proportion and why is its product constant?

Two quantities are in inverse proportion when one changes by a factor and the other changes by the reciprocal factor. A reciprocal is one divided by a non-zero number. If n denotes the changing factor, its reciprocal is 1/n.

For a fixed journey, increasing speed reduces the time taken. Speed means distance travelled per unit time. The unit km/h means kilometres per hour. In the given Lucknow to Kanpur journey, every speed and time pair represents the same distance of 90 km.

Result: Inverse proportion keeps the product constant

Let x denote speed and y denote travel time. Their product is the result of multiplying them. For this fixed journey, x × y = k, where k denotes the constant distance. Each row gives 90 km when speed is multiplied by time.

Write x₁ and y₁ for one corresponding pair, and x₂ and y₂ for another. The small numerals distinguish the pairs. Then x₁ × y₁ = x₂ × y₂, or x₁/x₂ = y₂/y₁. Notice the reversed order of the time values in the second equation.

Worked example 10. Puneeth’s father takes 3 hours at 30 km/h for a journey. How long will the same journey take at 60 km/h?

Answer: The speed doubles, so the time halves: 3 ÷ 2 = 1.5 hours. The product check is 30 × 3 = 60 × 1.5 = 90 km. Both speed-time pairs represent the same journey distance.

Walking to cycling gives another check. Speed changes from 5 to 15 km/h, multiplying by 3. Time changes from 18 to 6 hours, dividing by 3. It is this reciprocal change, together with a fixed distance, that establishes inverse proportion.

How can you test whether a table shows inverse proportion?

Multiply the two quantities in each corresponding pair. If every product is the same, the table satisfies the constant-product test for inverse proportion. Do not decide from a single pair or merely because some values rise while others fall.

In the tables below, x and y name the two quantities being compared. Each row contains one corresponding pair. The first table has the following values, with products calculated separately after the table.

The products are 40 × 20 = 800, 80 × 10 = 800, 25 × 32 = 800 and 16 × 50 = 800. All four agree, so these values of x and y are inversely proportional.

Why is a partial match insufficient?

Here the products are 800, 800, 312.5 and 128. The first two pairs fit the same product, but the remaining pairs do not. Therefore, the complete table does not show inverse proportion. Check every pair before making the conclusion.

A further table pairs 30 with 15, 90 with 5, 150 with 3, and 10 with 45. Each product is 450, so this table also represents inverse proportion. The constant can differ between different situations; it must stay unchanged within one relationship.

For a missing entry, first calculate the product from a complete pair. Divide that constant product by the known member of the incomplete pair. This uses the same condition as the test, rather than a separate rule to memorise.

How does inverse proportion solve workers, pumps and provisions problems?

Look first for the quantity that stays fixed. It may be a length of road, a tank to fill, or a stock of food. Then identify the two quantities that change. Equal working rates or equal consumption rates are needed for these proportional models.

Worked example 11. Twenty workers lay a road in 4 days. How many days do 10 workers need for the same road, assuming the same rate per worker and the same working time each day?

Answer: The worker count halves, so the time doubles. Required days = (20 × 4)/10 = 8 days. The products 20 × 4 and 10 × 8 are equal, which checks the inverse relationship.

How do additional pumps change filling time?

Worked example 12. Two pumps fill a tank in 18 hours. Two more pumps of the same kind are added. Find the time for all four pumps to fill that tank, working together at unchanged rates.

Answer: The number of pumps becomes 2 + 2 = 4. Required time = (2 × 18)/4 = 9 hours. Doubling the number of identical pumps halves the filling time for the same tank.

The phrase “two more” gives an increase, not the final count. Form the new total before using the product equation. Keep the tank size unchanged; changing both the pump count and the amount of water would require further reasoning.

How does a larger group affect provisions?

Provisions means the available stock of food.

Worked example 13. Food provisions feed 80 students for 15 days. Twenty more students join. With the same daily food allowance per student and no additional provisions, how long will the food last?

Answer: There are now 100 students. Required days = (80 × 15)/100 = 12 days. The larger group consumes the fixed stock faster, so the provisions last for only 12 days.

The same reasoning applies to water stored for families when each family’s daily use is taken to be equal and unchanged. Stating such assumptions explains why a constant-product calculation is appropriate.

These problems differ in context but share the same structure. A fixed total is shared across more or fewer equal contributors or consumers. The resulting time changes reciprocally, so the product of the count and the time remains constant.

How do you calculate the time taken when people work together?

A rate of work is the amount of work completed per unit time. When people have different individual rates, calculate each person’s work in one hour before adding their contributions. Adding their completion times does not give the time they need together.

Let the complete quantity of vegetables count as one unit of work. Ram cuts that quantity in 1 hour. Shyam cuts the same quantity in 1.5 hours. Their hourly contributions therefore differ, even though their individual tasks have the same total size.

Worked example 14. Ram takes 1 hour and Shyam takes 1.5 hours to cut the same quantity of vegetables individually. Find their time when working together, assuming their individual rates remain unchanged and their contributions add.

Answer: Ram completes 1 unit per hour. Shyam completes 1 ÷ 1.5 = 2/3 unit per hour. Together they complete 1 + 2/3 = 5/3 units per hour. One unit takes 1 ÷ (5/3) = 3/5 hour.

Why does direct proportion appear in this calculation?

Once the combined rate is fixed, the amount of work and the time taken are directly proportional. Completing 5/3 units takes one hour. Completing one unit takes the same fraction of that hour, giving 3/5 hour.

This is different from comparing the number of equal workers with time for a fixed job. There the job stays unchanged while the total working rate changes. Here the combined rate stays unchanged while the amount of work is scaled down to one unit.

Note: Identify what remains fixed before choosing direct or inverse proportion. Equal pumps filling one tank and one pump filling different numbers of equal tanks compare different pairs of quantities.

A useful solution sequence is to name the complete task, calculate each hourly contribution, add those rates, and divide the complete task by the combined rate. State the time with its unit and relate it back to the original task.

Glossary

• Ratio — A comparison of quantities in a specified order, expressed using terms separated by colons.
• Proportional relationship — A relationship maintained when the related quantities change by the same factor.
• Representative Fraction — The ratio of a map distance to the corresponding actual ground distance, measured in the same unit.
• Multi-term ratio — A ratio comparing several quantities whose corresponding amounts scale by the same factor.
• One part — The amount represented by one unit of a ratio when calculating actual shares.
• Highest common factor — The greatest whole number that divides every given whole-number term exactly.
• Pie chart — A circular representation in which the slices show the different proportions of a whole.
• Sector — A slice of a circle between two radii and the curved boundary joining their ends.
• Direct proportion — A relationship in which both quantities change by the same factor and their quotient stays constant.
• Inverse proportion — A relationship in which quantities change by reciprocal factors and their product stays constant.
• Constant — A value that stays unchanged across the corresponding pairs in a particular proportional relationship.
• Reciprocal — The number obtained by dividing one by a given non-zero number.
• Rate of work — The amount of a specified task completed in one unit of time.

Common errors and misconceptions

• Misconception: A map scale directly gives the road distance between cities. Correct: RF compares map distance with geographical ground distance. Keep units consistent, and distinguish this geographical distance from the distance travelled along roads.
• Misconception: Changing only one ingredient preserves a mixture’s ratio. Correct: Every corresponding ingredient must change by the same factor. In the spice mixture, halving the chillies requires halving the other ingredient quantities too.
• Misconception: Ten litres of white paint in the ratio 2 : 3 : 5 represents the whole mixture. Correct: It represents five parts. The complete mixture contains ten parts and totals 20 litres.
• Misconception: Pie-chart angles use the triangle total of 180°. Correct: A pie chart divides the complete angle of 360° around its centre. The total of 180° applies to the angles of a triangle.
• Misconception: Any decrease in one quantity as another increases proves inverse proportion. Correct: The changes must occur by reciprocal factors. Test whether every corresponding pair has the same product.
• Misconception: Two extra pumps means the new total is two pumps. Correct: Add the extra pumps to the original count. Starting with two and adding two gives four pumps for the calculation.
• Misconception: Add Ram’s and Shyam’s individual times to find their time together. Correct: Add their hourly work rates, then divide one complete task by that combined rate to find the required time.

Exam-style questions with model answers

Q1. Viswanath uses 6 cups of rice and 3 cups of urad dal; Puneet uses 4 cups of rice and 2 cups of urad dal. Use cross-multiplication to test whether their ratios are proportional. [2 marks]

1. The ratios, in rice-to-urad-dal order, are 6 : 3 and 4 : 2. Their cross-products are 6 × 2 = 12 and 3 × 4 = 12.
2. The cross-products are equal, so the two ratios are proportional. Both simplify to 2 : 1.

Q2. Yasmin mixes red, blue and white paint in the ratio 2 : 3 : 5. If she uses 10 litres of white paint, find the red paint, blue paint and total volume. [3 marks]

1. White paint represents five parts. One part is therefore 10 ÷ 5 = 2 litres. Red paint represents two parts, so she needs 2 × 2 = 4 litres of red.
2. Blue paint represents three parts, so she needs 3 × 2 = 6 litres of blue.
3. The total volume is 4 + 6 + 10 = 20 litres, including the given white paint.

Q3. A concrete mixture requires cement, sand and gravel in the ratio 1 : 1.5 : 3. Divide a total of 110 units into these components and check the total. [4 marks]

1. The ratio terms add to 1 + 1.5 + 3 = 5.5 parts. Dividing the required 110 units by 5.5 gives 20 units for each part.
2. Cement represents one part, so the cement quantity is 1 × 20 = 20 units.
3. Sand represents 1.5 parts, so the sand quantity is 1.5 × 20 = 30 units.
4. Gravel represents three parts, so it requires 3 × 20 = 60 units. The check is 20 + 30 + 60 = 110 units.

Q4. A class has 40 students: grade A has 12, B has 10, C has 8, D has 6 and E has 4. Calculate all five pie-chart sector angles, using the full-circle angle of 360°, and check their sum. [5 marks]

1. For each grade, divide its student count by the total of 40 and multiply by 360°. Grade A has 12 students, giving (12/40) × 360° = 108°.
2. Grade B has 10 students. Its sector angle is (10/40) × 360° = 90°, representing its share of the whole class.
3. Grade C has 8 students. Its sector angle is (8/40) × 360° = 72°.
4. Grade D has 6 students. Its sector angle is (6/40) × 360° = 54°.
5. Grade E has 4 students. Its angle is (4/40) × 360° = 36°. Checking the complete chart gives 108° + 90° + 72° + 54° + 36° = 360°.

Q5. Two quantities x and y have the corresponding pairs (40, 20), (80, 10), (25, 12.5) and (16, 8), with x listed first in each pair. Are they inversely proportional? Justify using every pair. [3 marks]

1. For inverse proportion, the product of corresponding values of x and y must stay constant. The first two products are 40 × 20 = 800 and 80 × 10 = 800.
2. The remaining products are 25 × 12.5 = 312.5 and 16 × 8 = 128.
3. These products are not all equal, so the table does not show inverse proportion. Agreement between just the first two pairs is insufficient.

Q6. Two pumps fill a tank in 18 hours. Two more pumps of the same kind are added, and all work together at their unchanged rates to fill the same tank. Find the new time and explain the relationship. [4 marks]

1. The new number of pumps is 2 + 2 = 4. The additional pumps must be included in the total before forming the equation.
2. For the same tank and unchanged equal pumping rates, the number of pumps and filling time are inversely proportional.
3. Let t denote the new time in hours. The constant-product equation is 2 × 18 = 4 × t.
4. Thus t = 36 ÷ 4 = 9 hours. Doubling the pump count halves the original filling time.

Q7. A fixed stock of food feeds 80 students for 15 days. Twenty more students join immediately. With equal, unchanged daily consumption per student and no new food supplied, how long will the stock last? [3 marks]

1. The new group contains 80 + 20 = 100 students. Since the stock and daily consumption per student stay fixed, student count and duration are inversely proportional.
2. Let t be the new duration in days. The constant-product equation is 80 × 15 = 100 × t.
3. Therefore, t = 1200 ÷ 100 = 12 days. The larger group uses the same provisions in fewer days.

Q8. Ram takes 1 hour and Shyam takes 1.5 hours to cut the same quantity of vegetables individually. If they work together at unchanged rates and their contributions add, find their completion time. Explain why work and time are directly proportional at their combined rate. [5 marks]

1. Take cutting the given quantity of vegetables as one complete unit of work. Ram completes that unit in one hour, so his rate is 1 unit per hour.
2. Shyam takes 1.5 hours for one unit. His hourly work is therefore 1 ÷ 1.5 = 2/3 unit.
3. Add their contributions to find the combined rate: 1 + 2/3 = 5/3 units per hour.
4. At this fixed combined rate, the amount of work and the time taken change by the same factor. Therefore, work and time are directly proportional.
5. One unit takes 1 ÷ (5/3) = 3/5 hour. This is their completion time for the given quantity when both work together.

Key takeaways

• A proportional ratio preserves the order of quantities and multiplies every corresponding term by the same factor.
• A Representative Fraction compares map and ground distances in the same unit; it represents geographical distance, not road distance.
• To divide a whole, add the ratio terms, find one part, and multiply it by each term.
• A triangle’s angles total 180°, while pie-chart sectors divide the complete angle of 360° around a circle’s centre.
• Direct proportion preserves a quotient; inverse proportion preserves a product and changes the quantities by reciprocal factors.
• Test every corresponding pair before deciding that a table represents inverse proportion, even when some pairs fit.
• Worker, pump and provision calculations depend on fixed totals and the stated assumptions about working or consumption rates.
• For people working together at different rates, add their work per hour before calculating the time for one complete task.

Test yourself

What does RF 1 : 60,00,000 mean for a map distance of 1 cm?

It represents 60,00,000 cm on the ground, equivalent to 60 km of geographical distance.

How does halving every term of 8 : 4 : 2 : 1 change the ratio?

It becomes 4 : 2 : 1 : 0.5, preserving the same proportional relationship among all four quantities.

What quantities make 50 ml of paint in the red : blue : white ratio 2 : 3 : 5?

Use 10 ml of red, 15 ml of blue and 25 ml of white. Together they total 50 ml.

What are the angles of a triangle whose angle ratio is 1 : 3 : 5?

They are 20°, 60° and 100°. The ratio has nine parts, with 20° in each part.

What pie-chart angle represents 10 students out of a total of 40?

The angle is (10/40) × 360° = 90°, representing one quarter of the whole class.

If a fixed journey takes 3 hours at 30 km/h, how long does it take at 60 km/h?

It takes 1.5 hours. Doubling the speed halves the time while keeping the journey distance unchanged.

Why do the pairs (40, 20), (80, 10), (25, 32) and (16, 50) show inverse proportion?

Every corresponding pair has product 800, so the product remains constant across all four pairs.

Ram completes a task in 1 hour and Shyam in 1.5 hours. What combined hourly rate do their unchanged individual rates give?

Their combined rate is 1 + 2/3 = 5/3 units of the complete task per hour.