CLASS 8 MATHS · PART 2 · CHAPTER 3 · STUDY NOTES
Proportional Reasoning – 2
Understand proportional reasoning – 2 through short explanations, labelled models and original practice. Read one topic, explain it in your own words, then check.
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A shared scale factor preserves the recipe
Compare matching terms or simplify both ratios.Equivalent ingredient shares
- Proportional ratios
- Ratios whose matching quantities differ by the same scale factor.
6:9 and 10:15 both simplify to 2:3. The multiplier from 6 to 10 is the same as from 9 to 15.
For a:b and c:d, with b and d nonzero, test ad = bc. Equal ratios describe proportions; real taste also depends on ingredients and preparation.
Take a moment. What do you think?
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Question
Are 8:12 and 14:21 equivalent?
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Map scale compares lengths in the same units
Convert the stated map distance using the representative fraction.Read a scale bar
- Representative fraction (RF)
- Map distance : corresponding ground distance, using the same length unit.
At RF 1:100,000, 1 cm on the original map represents 100,000 cm = 1 km on the ground. So 3.5 cm represents 3.5 km.
This is a map-distance model, not necessarily road distance. Resizing an image changes a printed RF; a graphic scale bar resizes with the image.
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Question
At 1:50,000, what does 4 cm represent?
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Scale every ingredient, including the third
Keep the order of all quantities in a multi-term ratio.Three ingredients; one scale factor
- Multi-term ratio
- An ordered comparison such as a:b:c, involving more than two quantities.
A model mixture has A:B:C = 2:3:5. If C is 15 units, one part is 15/5 = 3, so A = 6 and B = 9.
The total is 30 units. A mixture-volume model assumes the ingredient amounts add; real mixtures do not always have perfectly additive volumes.
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Question
For 3:2:1 with first amount 18, find the other amounts.
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Use the sum of all ratio parts
Dividing a total differs from being given one ingredient.The total consists of ten equal parts
- Ratio share
- For total T and positive ratio a:b:c, the first share is Ta/(a + b + c).
Share 120 minutes as 2:3:5. Ten parts make the whole, so one part is 12 minutes: 24, 36 and 60 minutes.
Triangle angles in ratio 1:2:3 share 180°, giving 30°, 60°, 90°. Side ratios need an extra check: any two side lengths must add to more than the third.
- 1
Add the ratio terms.
- 2
Divide the total by that sum.
- 3
Multiply the value of one part by each term.
- 4
Check all shares add to the original total.
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Question
Can 1:2:4 be the side ratio of a triangle?
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Turn each share into an angle out of 360°
A full circle represents the whole group.24 students share one circle
- Sector angle
- The central angle of a pie-chart slice, equal to share/total × 360°.
| Choice | Count | Share | Angle |
|---|---|---|---|
| Reading | 12 | 50% | 180° |
| Sport | 6 | 25% | 90° |
| Art | 6 | 25% | 90° |
In a fictional survey of 24 students, 12 choose reading, 6 sport and 6 art. Their angles are 180°, 90° and 90°.
Categories must account for the intended whole. If each student selects exactly one option, slices do not overlap and their angles add to 360°.
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Question
What angle represents 15% of the whole?
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Inverse proportion keeps the product constant
One quantity grows by a factor while the other shrinks by that factor.Fixed distance, changing speed
Travel time (hours)
Speed (km/h)
- 120 km journey · Time = 120/speed.
- 130 × 4 = 120 · (30, 4)
- 260 × 2 = 120 · (60, 2)
Only positive speeds are meaningful. The sampled curve represents an ideal constant-speed journey.
Read the plotted values
| Speed | Travel time |
|---|---|
| 20 | 6 |
| 24 | 5 |
| 28 | 4.285714285714286 |
| 32 | 3.75 |
| 36 | 3.3333333333333335 |
| 40 | 3 |
| 44 | 2.727272727272727 |
| 48 | 2.5 |
| 52 | 2.3076923076923075 |
| 56 | 2.142857142857143 |
| 60 | 2 |
| 64 | 1.875 |
| 68 | 1.7647058823529411 |
| 72 | 1.6666666666666667 |
| 76 | 1.5789473684210527 |
| 80 | 1.5 |
| 84 | 1.4285714285714286 |
| 88 | 1.3636363636363635 |
| 92 | 1.3043478260869565 |
| 96 | 1.25 |
| 100 | 1.2 |
| 104 | 1.1538461538461537 |
| 108 | 1.1111111111111112 |
| 112 | 1.0714285714285714 |
| 116 | 1.0344827586206897 |
| 120 | 1 |
- Inverse proportion
- For positive quantities, xy = k for a fixed positive constant k, so y = k/x.
For a fixed 120 km journey at constant speed, 30 km/h needs 4 h, 40 needs 3 h, and 60 needs 2 h. Speed × time stays 120 km.
Direct proportion keeps a ratio constant. Inverse proportion keeps a product constant. Merely moving in opposite directions does not prove inverse proportion.
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Question
At 80 km/h, how long is that same 120 km journey?
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Equal workers share a fixed amount of work
Use worker-days only when productivity and working hours are equal.Same work; different team size
- Worker-day
- One worker’s work in one day under the stated equal-rate model.
Six equal workers finish a job in 8 days: the job needs 48 worker-days. Twelve such workers need 48/12 = 4 days.
This assumes independent additive work, no delays, the same job, and equal daily hours. Twice as many people may not halve time in a real crowded workplace.
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Question
Four identical pumps fill a tank in 6 hours. How long for three?
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Add work rates, not completion times
Convert each completion time into work completed per hour.Work completed after equal time
Filled amount (tanks)
Time (hours)
- Pump A · Rate 1/3 tank/h.
- Pump B · Rate 1/6 tank/h.
- Together · Rate 1/2 tank/h.
- 1Full after 2 h · (2, 1)
Same tank, constant independent inflows and no leakage. The model ends at the joint filling time.
Read the plotted values
| Time | Filled amount |
|---|---|
| 0 | 0 |
| 2 | 0.6666666666666666 |
| Time | Filled amount |
|---|---|
| 0 | 0 |
| 2 | 0.3333333333333333 |
| Time | Filled amount |
|---|---|
| 0 | 0 |
| 2 | 1 |
- Combined rate
- The sum of independent work rates when people or machines work together.
One pump takes 3 h for a tank, so its rate is 1/3 tank per hour. Another takes 6 h, rate 1/6. Together the rate is 1/2 tank per hour.
Time for one tank is 1 ÷ (1/2) = 2 h. Adding 3 + 6 does not give the joint completion time.
- 1
Write each rate as 1 ÷ its completion time.
- 2
Add rates when the tasks can proceed independently.
- 3
Divide the total work by the joint rate.
- 4
Check the joint time is shorter than either individual time.
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Question
Two workers can each do a job alone in 4 h and 12 h. Find the joint time under additive-rate assumptions.
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