Proportional Reasoning-2 · Lesson 8 of 8
Chapter Summary and Practice
“Revisit the complete chapter, connect direct and inverse proportion, maps, multi-term ratios, pie charts, and work-rate reasoning, then solve mixed problems with full explanations.”
• Reconnect the chapter’s main ideas about ratios, proportions, scale, and parts of a whole. • Distinguish direct proportion, inverse proportion, and situations that are not proportional. • Choose suitable methods for maps, multi-term ratios, ratio division, pie charts, and work-rate problems. • Check answers using units, totals, constant ratios, and constant products. • Explain common mistakes and solve mixed chapter problems with clear reasoning.
This chapter has used one central idea in several different forms: quantities can be linked by a relationship that stays consistent as the numbers change. Sometimes the ratio stays constant, sometimes the product stays constant, and sometimes a whole must be divided in a fixed ratio. The skill is not only calculating an answer; it is deciding which relationship the situation actually describes.
A good revision strategy is to ask four questions before calculating: What quantities are changing? What stays fixed? Should the quantities move in the same direction or opposite directions? What check should the final answer satisfy? Those questions help separate direct proportion from inverse proportion and prevent formula-based guessing.
The Big Picture of Proportional Reasoning
| Situation | What stays consistent? | Useful relationship |
|---|---|---|
| Equivalent ratios | Ratio between corresponding quantities | a : b = c : d |
| Map scale | Map-to-actual scale factor | Actual distance = map distance × scale factor |
| Dividing a whole | Share of total represented by each ratio term | Share = total × term / sum of terms |
| Pie chart | Fraction of the whole | Slice angle = category / total × 360° |
| Direct proportion | Quotient y/x | y = kx |
| Inverse proportion | Product xy | xy = k |
| Combined work | Total work rate | Rates add, not completion times |
Equivalent Ratios and Proportions
Two ratios are equivalent when they describe the same relative comparison. Multiplying or dividing every term by the same non-zero factor preserves the ratio. Cross-multiplication is a compact way to test a two-term proportion because a/b = c/d exactly when ad = bc.
Problem
Solve 7 : 12 = x : 36.
- 1.Write the equality as 7/12 = x/36.
- 2.Cross-multiply: 12x = 7 × 36 = 252.
- 3.Divide by 12: x = 21.
- 4.Check by simplifying 21 : 36 to 7 : 12.
Equivalent ratios are formed by multiplying or dividing every term by the same factor. Adding the same number to both terms generally changes the ratio.
Map Scales and Unit Discipline
A representative fraction such as 1 : 50,00,000 compares a map distance with the corresponding actual distance in the same unit. If 1 cm on the map represents 50,00,000 cm on the ground, then it represents 500 km because 100,000 cm = 1 km.
Problem
A map has scale 1 : 20,00,000. Two towns are 6.5 cm apart on the map. Find the straight-line geographical distance.
- 1.1 cm represents 20,00,000 cm.
- 2.6.5 cm represents 6.5 × 20,00,000 = 1,30,00,000 cm.
- 3.Convert centimetres to kilometres: 1,30,00,000 ÷ 100,000 = 130 km.
- 4.This is the scale-based straight-line distance; a road route may be longer.
Problem
On a map with scale 1 : 25,00,000, what map distance represents 75 km?
- 1.Convert 75 km to centimetres: 75 × 100,000 = 75,00,000 cm.
- 2.Divide by the scale factor: 75,00,000 ÷ 25,00,000 = 3.
- 3.The map distance is 3 cm.
Multi-Term Ratios and Dividing a Whole
Ratios can compare more than two quantities. In a : b : c, the order matters because each term refers to a particular quantity. When a whole is divided in that ratio, first add the terms to find the total number of equal ratio-parts. Then give each share its stated number of parts.
Problem
₹5400 is divided in the ratio 2 : 3 : 4. Find the three shares.
- 1.Total ratio-parts = 2 + 3 + 4 = 9.
- 2.One part = 5400 ÷ 9 = 600.
- 3.The shares are 2 × 600 = ₹1200, 3 × 600 = ₹1800, and 4 × 600 = ₹2400.
- 4.Check: 1200 + 1800 + 2400 = 5400.
The same method works for angles of a triangle because the total is fixed at 180°. For side lengths, however, satisfying the stated ratio is not enough. The resulting lengths must also satisfy the triangle inequality: the sum of any two sides must be greater than the third.
Pie Charts as Proportional Pictures
A pie chart turns category shares into central angles. The whole data set corresponds to 360°, so a category with one quarter of the data gets one quarter of 360°, or 90°. The arithmetic and the picture must agree: all category angles together must total 360°.
Problem
A survey has category counts 18, 12, 6, and 4. Find the pie-chart angles.
- 1.The total is 18 + 12 + 6 + 4 = 40.
- 2.Angles are 18/40 × 360° = 162°, 12/40 × 360° = 108°, 6/40 × 360° = 54°, and 4/40 × 360° = 36°.
- 3.Check: 162 + 108 + 54 + 36 = 360°.
Direct and Inverse Proportion
In direct proportion, multiplying one quantity by a factor multiplies the other by the same factor. The quotient y/x stays constant. In inverse proportion, multiplying one quantity by a factor divides the other by that factor. The product xy stays constant.
| Feature | Direct proportion | Inverse proportion |
|---|---|---|
| Direction of change | Usually same direction | Opposite directions |
| Constant | y/x | xy |
| Equation | y = kx | xy = k |
| Doubling x | Doubles y | Halves y |
| Typical example | Cost at fixed price per item | Time for a fixed job with more equally productive workers |
Problem
24 equally productive workers complete a fixed job in 15 days. How many days would 30 such workers need?
- 1.The total job is fixed, so workers and days are inversely proportional under the stated equal-productivity assumption.
- 2.Use 24 × 15 = 30 × d.
- 3.360 = 30d, so d = 12 days.
- 4.The answer is sensible: more workers reduce the time.
Working Together Means Adding Rates
When two people, pipes, or machines complete the same job together, their completion times are not added. Convert each time into a rate: the fraction of one whole job completed per unit time. Then add those rates.
Problem
One pump fills a tank in 6 hours and another fills it in 4 hours. How long do they take together?
- 1.The first rate is 1/6 tank per hour and the second is 1/4 tank per hour.
- 2.Combined rate = 1/6 + 1/4 = 2/12 + 3/12 = 5/12 tank per hour.
- 3.Time = 1 ÷ (5/12) = 12/5 hours = 2.4 hours.
- 4.The combined time is less than 4 hours, which is reasonable because both pumps work together.
Choosing the Right Relationship
| Clue in the situation | Likely model | Question to check |
|---|---|---|
| Fixed price per item | Direct proportion | Does doubling items double cost? |
| Fixed journey at different constant speeds | Inverse proportion | Does speed × time stay fixed? |
| Fixed job with equally productive workers | Inverse proportion | Does workers × days stay fixed? |
| Mixture recipe scaled up | Equivalent multi-term ratio | Did every ingredient use the same scale factor? |
| Part of a fixed total | Ratio division | Do all shares add back to the total? |
| Several machines working at once | Rate addition | Have the individual rates been added? |
Real situations only follow proportional models when the assumptions support them. Workers may have different productivity, traffic may prevent constant speed, and bulk pricing may change the cost per item. A mathematically correct formula can still be a poor model if the assumptions are false.
Explaining and Correcting Mistakes
| Claim | Correction and reason |
|---|---|
| Adding the same number to both terms makes an equivalent ratio. | Equivalent ratios come from multiplying or dividing every term by the same non-zero factor. |
| RF 1 : 20,00,000 means 1 cm equals 20,00,000 km. | The two sides use the same unit first; 20,00,000 cm is 20 km. |
| To divide ₹600 in 2 : 3, calculate 600/2 and 600/3. | The ratio has 5 total parts. Shares are 2/5 and 3/5 of the whole. |
| A pie-chart category value is automatically its angle. | Convert the category fraction into the same fraction of 360°. |
| If one quantity rises while another falls, the relationship must be inverse. | A true inverse relationship requires a constant product, together with appropriate assumptions. |
| Two pumps taking 3 h and 6 h separately take 4.5 h together. | Add their rates 1/3 and 1/6, giving 1/2 tank per hour, so together they take 2 h. |
| A formula is enough even if the answer moves in the wrong direction. | A final answer must also fit the situation: more workers should not increase the time for a fixed job under equal-productivity assumptions. |
Quiz
Which condition confirms that x and y are inversely proportional?
A map scale is 1 : 10,00,000. What actual distance does 4 cm represent?
A total of 840 is divided in the ratio 3 : 4. What is the larger share?
A category has 25% of the data. What angle should its pie-chart slice have?
If 12 workers take 20 days for a fixed job, how long would 15 equally productive workers take?
Which situation is most naturally modelled by direct proportion?
One machine completes a job in 8 hours and another in 12 hours. What is their combined rate?
Which check is strongest after dividing a total in a ratio?
Practice Problems
- Decide whether 18 : 30 and 27 : 45 are proportional, and justify your answer. Solution: 1. Cross-products are 18 × 45 = 810 and 30 × 27 = 810. 2. The cross-products match, so the ratios are proportional.
- A map has scale 1 : 15,00,000. Two points are 8 cm apart on the map. Find the actual straight-line distance. Solution: 1. 8 × 15,00,000 = 1,20,00,000 cm. 2. Divide by 100,000 to convert to kilometres. 3. The distance is 120 km.
- A drink uses syrup, water, and lime in the ratio 2 : 7 : 1. If 3 litres of lime are used, find the syrup and water amounts. Solution: 1. One ratio-part equals 3 litres because lime represents 1 part. 2. Syrup = 2 × 3 = 6 L. 3. Water = 7 × 3 = 21 L.
- Divide ₹9600 in the ratio 5 : 3 : 2. Solution: 1. Total parts = 10. 2. One part = ₹960. 3. Shares are ₹4800, ₹2880, and ₹1920. 4. Their sum is ₹9600.
- A pie chart represents 240 students. A slice measures 72°. How many students are in that category? Solution: 1. Fraction = 72/360 = 1/5. 2. Number of students = 1/5 × 240 = 48.
- Test whether the pairs (4,30), (5,24), (8,15), and (10,12) show inverse proportion. Solution: 1. Products are 120, 120, 120, and 120. 2. The product is constant, so the table shows inverse proportion.
- A car covers a fixed distance in 6 hours at 50 km/h. How long would it take at 75 km/h, assuming constant speed throughout? Solution: 1. Distance is fixed, so speed × time is constant. 2. 50 × 6 = 75 × t. 3. t = 4 hours.
- 18 workers can finish a job in 14 days. After planning, only 12 equally productive workers are available. Find the required number of days. Solution: 1. Workers × days stays constant. 2. 18 × 14 = 12 × d. 3. d = 21 days.
- One pipe fills a tank in 10 hours and another in 15 hours. Find their combined filling time. Solution: 1. Rates are 1/10 and 1/15 tank per hour. 2. Combined rate = 3/30 + 2/30 = 5/30 = 1/6. 3. Combined time = 6 hours.
- A student says, 'Whenever one quantity increases and another decreases, they are inversely proportional.' Give a counterexample and explain the correct test. Solution: 1. Temperature may rise while remaining battery charge falls, but their product need not stay constant. 2. Inverse proportion requires a constant product xy = k and assumptions that make the model meaningful.
Key Takeaways
• Equivalent ratios preserve the same relative comparison and can be checked with equal cross-products. • Map scales require careful unit conversion before interpreting real distance. • To divide a whole in a ratio, add the ratio terms, find one part, and check that the shares return to the whole. • Pie charts translate category fractions into angles out of 360°. • Direct proportion has a constant quotient; inverse proportion has a constant product. • Real proportional models depend on assumptions, not just on a convenient formula. • When agents work together, add their work rates rather than their separate completion times.
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Solving Proportion and Work-Rate Problems
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