Proportional Reasoning — 2
Speed, Time, Distance & Inverse Proportions.
The Motorcycle Puzzle
Your father takes 3 hours to drive from Lucknow to Kanpur at 30 km/h. Now he wants to use a faster route on his motorcycle at 60 km/h. Will it take 2 hours? Or 1.5 hours? Or something else?
The Hidden Pattern:
When you go faster, you reach sooner. But HOW much sooner? Is the relationship the same as direct proportion, where "faster = shorter time" just like "more apples = more cost"? Or is something different happening here?
You've learned about direct proportion: when one quantity doubles, the other doubles too. But now you'll discover something remarkable: sometimes when one quantity doubles, the OTHER quantity is HALVED. This is called inverse proportion.
The Key Insight:
In some relationships, one quantity changing causes the other to change in the OPPOSITE direction, but by the same factor. When speed doubles, time halves. When workers double, days halve. When tap count doubles, filling time halves.
The Mathematical Truth: In inverse proportion, the PRODUCT of the two quantities is always constant.
speed × time = distance (constant!)
Why does this matter? Understanding inverse proportion unlocks solutions to countless real-world problems: how long will food last if more people arrive? How long will the job take with fewer workers?
Logic Ladder 1: Speed-Time-Distance Triangle
The Universal Relationship
Distance = Speed × Time
Therefore: Speed × Time = Distance (constant for a fixed route)
Example: Fixed Distance of 90 km
At 5 km/h (walking): Time = 90 ÷ 5 = 18 hours
At 30 km/h (motorcycle): Time = 90 ÷ 30 = 3 hours
At 60 km/h (car): Time = 90 ÷ 60 = 1.5 hours
Observe the Pattern
Speed 5 → 30 (multiply by 6) and Time 18 → 3 (divide by 6)
Speed 30 → 60 (multiply by 2) and Time 3 → 1.5 (divide by 2)
When speed increases by factor n, time decreases by factor 1/n.
Let's formalize the pattern you just observed.
The Definition:
Two quantities x and y are inversely proportional if:
x × y = k (where k is a constant)
OR equivalently: y = k/x
Why Does This Matter?
In direct proportion, you can find missing values using multiplication. In inverse proportion, you use the PRODUCT instead.
Example: Workers and Work Duration
A group of 20 workers can complete a project in 4 days. If you reduce to 10 workers, how many days will it take?
Key insight: The amount of work is constant. If workers and days are inversely proportional:
Workers × Days = Constant Work
20 × 4 = 10 × d
80 = 10 × d
d = 8 days
With half the workers, it takes twice as long!
"Inverse proportion" does NOT mean "the opposite of proportion." It means the PRODUCT is constant, not the ratio. Don't treat it like reverse multiplication!
Wrong Thinking: If speed doubles, time halves, so I'll divide: 3 ÷ 2 = 1.5 hours. (This happens to work here, but only because of the math!)
Right Thinking: Speed × Time = constant distance, so I use the product rule.
Logic Ladder 2: Testing for Inverse Proportion
Given Data Table
x: 4, 8, 10, 16
y: 20, 10, 8, 5
Are x and y inversely proportional?
Calculate Products
4 × 20 = 80
8 × 10 = 80
10 × 8 = 80
16 × 5 = 80
Check if Constant
All products equal 80. The constant k = 80.
YES, x and y are inversely proportional!
The relationship is: y = 80/x
Sometimes we need to divide a whole into many parts based on a given ratio. For example, to make concrete, we need cement, sand, and gravel in the ratio 1 : 1.5 : 3.
Understanding Multi-Term Ratios:
A ratio like 1 : 1.5 : 3 means:
- For every 1 part cement,
- We need 1.5 parts sand,
- And 3 parts gravel.
How to divide a quantity in a given ratio:
If we need 110 units of concrete in the ratio 1 : 1.5 : 3:
Step 1: Add all ratio parts: 1 + 1.5 + 3 = 5.5
Step 2: Cement = 110 × (1/5.5) = 110 × (1/5.5) = 20 units
Sand = 110 × (1.5/5.5) = 30 units
Gravel = 110 × (3/5.5) = 60 units
Check: 20 + 30 + 60 = 110 ✓
Pie Charts: Visualizing Proportions
A pie chart shows how different parts make up a whole. Each "slice" represents a percentage of the total.
Key principle: A circle has 360°. If students are in the ratio 12 : 10 : 8 (for grades A, B, C), then:
Grade A angle = (12/(12+10+8)) × 360° = (12/30) × 360° = 144°
Grade B angle = (10/30) × 360° = 120°
Grade C angle = (8/30) × 360° = 96°
If the ratio is 12 : 10 : 8, simplify to 6 : 5 : 4 first (divide by 2). Your calculations will be easier!
Logic Ladder 3: Creating a Pie Chart from Data
Given
Survey of favorite subjects: Math (12), Science (10), English (8), Total = 30
Write Ratio
12 : 10 : 8
Convert to Angles
Math: (12/30) × 360° = 144°
Science: (10/30) × 360° = 120°
English: (8/30) × 360° = 96°
Draw with Protractor
Use a protractor to mark angles of 144°, 120°, and 96° from the center.
Verify: 144° + 120° + 96° = 360° ✓
Socratic Sandbox — Test Your Thinking
A car travels 240 km in 4 hours. At the same speed, how long to travel 360 km?
Reveal Answer
Speed = 240 ÷ 4 = 60 km/h
Time for 360 km = 360 ÷ 60 = 6 hours
If you double the number of workers, why does the time to complete a job halve (assuming equal work capacity)?
Reveal Answer
Total work = Workers × Time per worker × (rate per worker)
If the total work and rate stay constant, then Workers × Time = constant.
When workers double (multiply by 2), time must divide by 2 to keep the product constant.
A pump fills a tank in 6 hours. How long will it take 3 identical pumps working together?
Reveal Answer
Pumps × Time = Work (constant)
1 × 6 = 3 × t
t = 6/3 = 2 hours
A school needs to feed 80 students for 15 days. If 20 more students join, for how many days will the food last?
Reveal Answer
Total food = Students × Days (constant)
80 × 15 = 100 × d
1200 = 100 × d
d = 12 days
In the ratio 2 : 3 : 5 for red : blue : white paint, if you need 50 ml of mixed purple paint, how much of each color do you need?
Reveal Answer
Total parts = 2 + 3 + 5 = 10
Red = 50 × (2/10) = 10 ml
Blue = 50 × (3/10) = 15 ml
White = 50 × (5/10) = 25 ml
Check: 10 + 15 + 25 = 50 ✓