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Inverse Proportion
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Foundational Skills
Recognise inverse proportion
\[ \text{More workers} \rightarrow \text{less time} \]
Identify inversely proportional relationships.
Identify the constant product
\[ k = 4 \times 6 = 24 \]
Find k in inverse proportion.
Use constant product
\[ xy = 36, \; x = 9 \Rightarrow y = 4 \]
Find a missing value using k.
Multiplier relationship
\[ \times 3 \rightarrow \div 3 \]
Understand inverse multipliers.
Workers and Time
Time when workers increase
\[ 6 \text{ workers} \rightarrow 12 \text{ workers} \]
More workers means less time.
Time when workers decrease
\[ 8 \text{ workers} \rightarrow 4 \text{ workers} \]
Fewer workers means more time.
Workers for target time
\[ \text{Complete in 4 days?} \]
Find workers needed for a deadline.
Workers given time ratio
\[ \text{Half the time} = \; ? \]
Use ratio reasoning.
Two-step workers problem
\[ \text{More workers join halfway} \]
Changes partway through.
Speed and Time
Time when speed increases
\[ 60 \text{ km/h} \rightarrow 80 \text{ km/h} \]
Faster means less time.
Time when speed decreases
\[ 50 \text{ mph} \rightarrow 30 \text{ mph} \]
Slower means more time.
Speed for target time
\[ \text{Complete in 2 hours?} \]
Find required speed.
Compare journey distances
\[ \text{Which journey is longer?} \]
Compare scenarios.
Time saved by speed increase
\[ \text{Time saved} = \; ? \]
Calculate time difference.
Pipes and Tanks
Time with more pipes
\[ 2 \text{ pipes} \rightarrow 3 \text{ pipes} \]
More pipes fill faster.
Pipes needed for target time
\[ \text{Fill in 1 hour?} \]
Find taps needed.
Combined filling rate
\[ \frac{1}{4} + \frac{1}{6} = \; ? \]
Add rates to find time.
Other Contexts
Sharing equally
\[ £60 \div 6 = \; ? \]
More people = less each.
Gears
\[ \text{teeth} \times \text{speed} = k \]
Connected gear speeds.
Levers
\[ \text{mass} \times \text{distance} = k \]
Balance problems.
Food supplies
\[ \text{More people} \rightarrow \text{fewer days} \]
Consumption problems.
Cutting into pieces
\[ \text{More pieces} \rightarrow \text{shorter} \]
Division problems.
Comparing Direct and Inverse
Classify as direct or inverse
\[ \text{Direct or inverse?} \]
Distinguish proportion types.
Classify real-world relationships
\[ \text{A) Direct B) Inverse C) Neither} \]
Identify proportion in context.
Proportion type from values
\[ x=2, y=18; \; x=6, y=6 \]
Check xy or y/x constant.
Choose correct method
\[ \text{Direct or inverse method?} \]
Pick the right approach.
Verify inverse proportion
\[ 3 \times 8 = 6 \times 4 \; ? \]
Check products are equal.
Problem Solving
Find original value
\[ \text{Speed} \times 1.5 \rightarrow \text{time} \div 1.5 \]
Work backwards.
Percentage change
\[ \text{Workers double} \rightarrow \text{?}\% \]
Connect to percentages.
Multi-step problems
\[ \text{workers} \times \text{days} \times \text{hours} \]
Multiple variables.
Compare scenarios
\[ \text{Team A vs Team B} \]
Compare total work.
Special Cases
Valid inverse proportion
\[ \text{2× speed} \rightarrow \text{half time?} \]
Recognise where it applies.
Invalid inverse proportion
\[ \text{2 people walking} = \; ? \]
Recognise where it doesn’t apply.
Complete in 1 day
\[ 5 \times 5 = 25 \; \text{workers for 1 day} \]
Handle value of 1.
Set up calculation
\[ (3 \times 8) \div 5 \]
Write but don’t evaluate.
Verify a claim
\[ \text{Is Sam correct?} \]
Check reasoning.
Timer (Optional)
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