Class 7 ICSE Mathematics
Direct and Inverse Variations (Easy Notes)
1. Introduction to Variation
In Mathematics, variation means a relationship between two quantities where a change in one quantity causes a change in the other.
There are two types of variation studied in Class 7 ICSE:
- Direct Variation
- Inverse Variation
Understanding variation helps us solve real-life problems involving money, time, speed, work, distance, and quantity.
2. Direct Variation
Definition
Two quantities are said to be in direct variation if:
- When one quantity increases, the other also increases.
- When one quantity decreases, the other also decreases.
- The ratio of the two quantities remains constant.
Mathematical Representation
If quantity y varies directly as quantity x, then:
[
y \propto x
]
This means:
[
y = kx
]
Where k is a constant called the constant of variation.
Examples of Direct Variation
- Cost of items ∝ Number of items
- Wages ∝ Number of hours worked
- Distance travelled ∝ Time (at constant speed)
- Quantity of goods ∝ Price (fixed rate)
Example 1
If 5 notebooks cost ₹50, find the cost of 8 notebooks.
Solution:
Cost ∝ Number of notebooks
Cost of 1 notebook = 50 ÷ 5 = ₹10
Cost of 8 notebooks = 8 × 10 = ₹80
Example 2
If y varies directly as x and y = 20 when x = 4, find y when x = 10.
Solution:
[
y = kx
]
[
20 = k × 4
\Rightarrow k = 5
]
When x = 10:
[
y = 5 × 10 = 50
]
Answer: y = 50
Important Points (Direct Variation)
- Ratio y/x is constant
- Graph of direct variation is a straight line passing through the origin
- If one quantity becomes zero, the other also becomes zero
3. Inverse Variation
Definition
Two quantities are said to be in inverse variation if:
- When one quantity increases, the other decreases.
- When one quantity decreases, the other increases.
- The product of the two quantities remains constant.
Mathematical Representation
If y varies inversely as x, then:
[
y \propto \frac{1}{x}
]
[
y = \frac{k}{x}
]
or
[
xy = k
]
Where k is a constant.
Examples of Inverse Variation
- Number of workers ∝ 1/Time taken to do work
- Speed ∝ 1/Time (fixed distance)
- Number of days ∝ 1/Number of men
- Pressure ∝ 1/Volume (basic idea)
Example 1
6 workers can complete a job in 10 days.
How many days will 12 workers take?
Solution:
Workers × Days = constant
[
6 × 10 = 12 × x
]
[
60 = 12x
\Rightarrow x = 5
]
Answer: 12 workers will take 5 days
Example 2
If y varies inversely as x and y = 8 when x = 5, find y when x = 20.
Solution:
[
xy = k
]
[
5 × 8 = 40
\Rightarrow k = 40
]
[
y = \frac{40}{20} = 2
]
Answer: y = 2
Important Points (Inverse Variation)
- Product xy is constant
- Graph of inverse variation is a curve (hyperbola)
- One quantity never becomes zero
4. Difference Between Direct and Inverse Variation
| Direct Variation | Inverse Variation |
|---|---|
| Both quantities increase or decrease together | One increases, the other decreases |
| Ratio is constant | Product is constant |
| y = kx | y = k/x |
| Straight line graph | Curved graph |
5. Real-Life Applications
Direct Variation
- Buying fruits by weight
- Salary based on hours worked
- Distance and time at constant speed
Inverse Variation
- More workers, less time
- Higher speed, less travel time
- Fewer days, more people needed
6. Key Formulas to Remember
Direct Variation
[
y = kx
]
Inverse Variation
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Class 7 ICSE Mathematics – Direct and Inverse Variation
- Introduction
In real life, many quantities are related to each other.
For example:
The cost of apples increases as the number of apples increases → Direct variation
Speed and time for a journey: more speed → less time → Inverse variation
Direct and inverse variation help us mathematically describe these relationships. - Direct Variation (Proportion)
Definition:
Two quantities � and � are said to vary directly if:
Where:
� = constant of proportionality
� increases as � increases
� decreases as � decreases
Formula:
Example 1:
If 5 pens cost ₹50, find the cost of 8 pens.
Solution:
✅ Cost of 8 pens = ₹80
Example 2:
A machine produces 60 items in 5 hours.
How many items in 8 hours?
✅ 96 items
Key Points for Direct Variation
Graph passes through origin (0,0)
Constant �
Formula: �
Unit consistency is important - Inverse Variation (Proportion)
Definition:
Two quantities � and � vary inversely if:
Where:
� = constant
When � increases, � decreases
When � decreases, � increases
Formula:
Example 1:
10 men can finish a work in 15 days.
How many days will 5 men take?
✅ 5 men will take 30 days
Example 2:
A car takes 4 hours to travel 240 km.
How long will it take at half the speed?
If speed is halved → time doubles → 8 hours
Key Points for Inverse Variation
Graph is hyperbolic
Constant �
Formula: �
Common in Time & Work, Men × Days, Speed × Time problems - Differences Between Direct and Inverse Variation
Feature
Direct Variation
Inverse Variation
Relationship
y ↑ as x ↑
y ↓ as x ↑
Formula
y/x = k
xy = k
Graph
Straight line through origin
Hyperbola
Example
Cost of apples
Men × Days
Constant
k = y/x
k = xy - Combined Example
Direct: Cost of pens proportional to number of pens
Inverse: Time ∝ 1/Speed
Problem:
If 8 workers can finish a job in 12 days, find days for 6 workers.
Solution:
Inverse variation → �
✅ 6 workers will take 16 days - Word Problem Tricks
Direct variation → multiply cross → solve
Inverse variation → multiply cross → solve
Check units (hours, days, cost, speed)
If unsure, ask yourself:
“Does increasing x increase y?” → Direct
“Does increasing x decrease y?” → Inverse - Multiple Step Example (Direct + Inverse)
Problem:
Cost of 20 pens = ₹50.
How many pens can be bought for ₹80?
Also, if 4 people share the pens equally, how many per person?
Solution:
Direct variation → Cost ∝ Pens
Divide among 4 people → 32 ÷ 4 = 8 pens per person
✅ Answer = 8 pens - Short Tricks for ICSE Exams
Always identify what varies directly or inversely
Write formula first
Solve step-by-step, show units
For multiple steps, handle one variation at a time
Use LCM for fractions if needed - Quick Revision Table
Type
Formula
Example
Direct
�
Cost of pens
Inverse
�
Men × Days, Speed × Time
Check
Increase x → Increase y?
Direct
Check
Increase x → Decrease y?
Inverse - Practice Questions
Direct Variation:
6 apples cost ₹30. Find cost of 10 apples.
8 kg rice costs ₹120. Find cost of 5 kg.
Inverse Variation:
12 men can finish work in 20 days. How many days for 8 men?
A car takes 5 hours at 60 km/h. Time at 40 km/h?
Mixed:
10 pens cost ₹50. If you buy 15 pens at same rate, how much?
6 men finish work in 10 days. 3 men start later and finish remaining work. How many days total?
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Class 7 ICSE Mathematics – Direct and Inverse Variation (Master Notes)
- Introduction
Many quantities in daily life depend on each other.
Example of Direct Variation: More apples → higher cost
Example of Inverse Variation: More workers → less time to finish a job
This chapter helps solve real-life proportional problems using simple formulas. - Direct Variation (Proportional Relationship)
Definition
Two quantities � and � vary directly if:
Where:
� = constant of proportionality
If � increases, � increases
If � decreases, � decreases
Key Formula
Example 1: Cost of Apples
5 apples cost ₹50. Find the cost of 8 apples.
✅ Answer = ₹80
Example 2: Quantity Produced
A machine makes 60 items in 5 hours. How many items in 8 hours?
✅ Answer = 96 items
Characteristics of Direct Variation
Graph is a straight line through origin
Constant �
Simple cross-multiplication can solve all problems - Inverse Variation
Definition
Two quantities � and � vary inversely if:
Where:
� = constant
If � increases, � decreases
If � decreases, � increases
Key Formula
Example 1: Men and Days
10 men can finish a work in 15 days. How many days will 5 men take?
✅ Answer = 30 days
Example 2: Speed and Time
A car takes 4 hours at 60 km/h. How long at 30 km/h?
✅ Answer = 8 hours
Characteristics of Inverse Variation
Graph is hyperbolic
Constant �
Appears in Time & Work, Men × Days, Speed × Time - Difference Between Direct & Inverse Variation
Feature
Direct
Inverse
Formula
�
�
Relationship
y ↑ as x ↑
y ↓ as x ↑
Graph
Straight line through origin
Hyperbola
Example
Cost ∝ Number of items
Men × Days
Constant
k = y/x
k = xy - Combined Problems (Direct + Inverse)
Example:
20 pens cost ₹50. Cost per pen ∝ number of pens (direct).
6 men complete work in 12 days. Days ∝ 1/men (inverse).
Steps to solve:
Direct: Cost of pens = � for 32 pens
Inverse: Days for 8 men = � - Step-by-Step Approach for Exams
Identify direct or inverse variation
Write the formula: � or �
Fill in known values
Solve for unknown
Always write units (days, ₹, hours)
For multi-step problems, solve one variation at a time - Word Problem Tricks
Direct: Multiply cross → solve
Inverse: Multiply cross → solve
Always check units (days, hours, rupees, km/h)
Stepwise solving for mixed problems - Multi-Step Example (Direct + Inverse)
Problem:
A can do work in 10 days. B is 25% more efficient. They work together for 4 days. Remaining work is finished by B alone. Find total days.
Solution:
A’s 1-day work = 1/10
B’s efficiency = 25% more → 1.25 × 1/10 = 1/8
Together 1-day work = 1/10 + 1/8 = 9/40
Work done in 4 days = 4 × 9/40 = 36/40 = 9/10
Remaining work = 1 − 9/10 = 1/10
Time by B alone = (1/10) ÷ (1/8) = 4/5 days
✅ Total days = 4 + 4/5 = 4 4/5 days - Shortcut Tricks
Direct → cross multiplication
Inverse → multiply both sides by constant
Check if increase → direct / decrease → inverse
For % increase/decrease → convert to fraction - Quick Revision Table
Type
Formula
Example
Direct
y/x = k
Cost of pens
Inverse
xy = k
Men × Days, Speed × Time
Check
Increase x → Increase y?
Direct
Check
Increase x → Decrease y?
Inverse - Practice Questions
Direct Variation:
5 kg rice = ₹200. Cost of 8 kg?
6 workers produce 120 items. How many items in 9 workers?
Inverse Variation:
12 men → 20 days. 8 men → ?
A car takes 5 hours at 60 km/h. Time at 40 km/h?
Mixed:
Cost of 10 pens = ₹50. Cost of 15 pens?
6 men → 10 days. 3 men start later. Total days?
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HOTS questions
Practice worksheets
Exam tricks
Stepwise solutions
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Class 7 ICSE – Direct and Inverse Variation (Mega Master Notes)
- Introduction
Many quantities in real life are related mathematically.
Direct variation: When one quantity increases, the other also increases.
Inverse variation: When one quantity increases, the other decreases.
These concepts are used in Time & Work, Men × Days, Cost, Speed × Time, Quantity problems. - Direct Variation
Definition
Two quantities � and � vary directly if:
� = constant of proportionality
If � increases → � increases
If � decreases → � decreases
Key Formula
Example 1 – Simple
5 apples cost ₹50. Find cost of 8 apples.
✅ Answer = ₹80
Example 2 – Multiple Steps
6 workers make 120 items in 4 days. How many items in 6 workers for 5 days?
Direct variation → items ∝ workers × days
Total items = 120 × (6/6) × (5/4) = 150
✅ Answer = 150 items
Characteristics of Direct Variation
Graph is straight line through origin
Constant �
Cross-multiplication solves most problems - Inverse Variation
Definition
Two quantities � and � vary inversely if:
Constant �
When � increases, � decreases
When � decreases, � increases
Formula
Example 1 – Men & Days
10 men can do work in 15 days. How many days for 5 men?
✅ Answer = 30 days
Example 2 – Speed & Time
A car takes 4 hours at 60 km/h. How long at 30 km/h?
✅ Answer = 8 hours
Characteristics of Inverse Variation
Graph is hyperbola
Constant �
Appears in Time & Work, Men × Days, Speed × Time - Identifying Direct vs Inverse
Increase x → increase y → Direct
Increase x → decrease y → Inverse
Exam Tip: Check relationship first, then choose formula. - Step-by-Step Solving Strategy
Identify type (Direct / Inverse)
Write formula: � or �
Substitute known values
Solve for unknown
Include units (days, ₹, items, km/h) - Combined Example
Problem:
20 pens cost ₹50 (Direct).
6 men finish work in 12 days (Inverse).
Solution:
Direct: Cost of 32 pens = �
Inverse: 8 men → days = �
✅ Answer: Cost = ₹80, Days = 9 - Multi-Step Example (Direct + Inverse)
Problem:
A can do a work in 10 days.
B is 25% more efficient. They work together 4 days. Remaining work by B alone. Find total days.
Solution:
A’s 1-day work = 1/10
B’s 1-day work = 1.25 × 1/10 = 1/8
Together 1-day work = 1/10 + 1/8 = 9/40
Work done in 4 days = 36/40 = 9/10
Remaining work = 1 − 9/10 = 1/10
Time for B = (1/10) ÷ (1/8) = 4/5 days
Total days = 4 + 4/5 = 4 4/5 days - HOTS Questions
A is 20% more efficient than B. If B takes 15 days, how long does A take?
12 men finish work in 16 days. 4 men leave after 8 days. How long to complete work?
A car travels at 60 km/h in 4 hours. If speed is doubled, how much time is saved?
Cost of 15 pens = ₹75. How much for 25 pens?
5 workers finish in 10 days. How many workers needed for 4 days? - Quick Tricks
Direct: Cross-multiplication
Inverse: Multiply both sides by constant
Percent efficiency: Convert to fraction → solve
Mixed: Solve one variation at a time
Always write units in answer - Revision Table
Type
Formula
Example
Direct
y/x = k
Cost of pens
Inverse
xy = k
Men × Days, Speed × Time
Check
Increase x → Increase y?
Direct
Check
Increase x → Decrease y?
Inverse - Practice Questions (ICSE Style)
Direct Variation:
6 kg sugar = ₹180. Find cost of 10 kg.
8 workers make 120 items in 4 days. How many items in 6 workers for 5 days?
Inverse Variation:
15 men → 18 days. 10 men → ?
5 hours at 50 km/h → time at 25 km/h?
Mixed:
Cost of 12 pens = ₹60. Cost of 20 pens?
8 men → 12 days. 4 men leave after 6 days. Remaining days?
HOTS:
A is 20% more efficient than B. B takes 15 days. Find A’s time.
12 workers finish work in 20 days. 4 workers leave after 5 days. Remaining work?
6 men finish work in 10 days. How many men to finish in 4 days?
If you want, I can make the full “Direct & Inverse Variation – 5000+ word ICSE Master Notes”, including:
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Class 7 ICSE – Direct and Inverse Variation (Ultimate Mega Notes)
- Recap: Basics
Direct Variation: � → y increases as x increases
Inverse Variation: � → y decreases as x increases
Key Formulas:
Type
Formula
Direct
�
Inverse
�
Graph: Direct → straight line through origin; Inverse → hyperbola - Direct Variation – Stepwise Solving
Identify direct relationship
Use formula: �
Substitute known values to find �
Solve for unknown
Check units
Example:
5 apples = ₹50 → 8 apples cost?
✅ Answer = ₹80 - Inverse Variation – Stepwise Solving
Identify inverse relationship
Use formula: �
Find � using known values
Solve for unknown
Check units
Example:
10 men → 15 days → 5 men → ?
✅ Answer = 30 days - Mixed Problems (Direct + Inverse)
Example:
20 pens cost ₹50 (Direct). 6 men finish work in 12 days (Inverse).
Direct: Cost of 32 pens = �
Inverse: 8 men → days = �
✅ Answer: Cost = ₹80, Days = 9 - Efficiency + Variation Problems
Efficiency ∝ 1 / Time
Use inverse formula when efficiency changes
Example:
A is 25% more efficient than B. B takes 16 days → A takes?
Efficiency ratio A:B = 125:100 = 5:4
Time ∝ 1 / Efficiency → A = 16 × 4/5 = 12.8 days - Alternate Day & Fractional Work Problems
Example:
A = 1/10 per day, B = 1/15 per day, alternate starting A, work = 1
2-day cycle = 1/10 + 1/15 = 1/6
6 cycles → 12/6 = 1 → work finished in 12 days - Word Problem Solving Tips
Identify direct or inverse
Find 1-day work / 1-unit work
Use fraction method for partial work
Stepwise solution → marks guaranteed
Always write units - HOTS Example 1
Problem:
12 men can do work in 20 days. After 8 days, 4 men leave. Remaining work?
Solution:
Total work = 12 × 20 = 240 units
Work done in 8 days = 12 × 8 = 96 units
Remaining = 240 − 96 = 144 units
Remaining men = 8 → days = 144/8 = 18
✅ Answer = 18 days - HOTS Example 2 – Mixed Variation
Problem:
A machine produces 60 items in 5 hours. A second machine produces 25% more per hour. Both work together 3 hours. Total items produced?
First machine = 60/5 = 12 per hour
Second = 12 × 1.25 = 15 per hour
Combined = 12 + 15 = 27 per hour
Total in 3 hours = 27 × 3 = 81 items
✅ Answer = 81 items - Shortcut Tricks for Exams
Direct → cross multiply
Inverse → multiply constants
Percent efficiency → convert to fraction → apply formula
Mixed problems → solve one variation at a time
Always show units (days, items, ₹, km/h)
Use 1-unit work method for Time & Work related inverse problems - Common ICSE Exam Traps
Trap
How to Avoid
Mixing Direct/Inverse
Identify relationship carefully
Forgetting units
Always include units in answer
Partial work
Use fraction of total work
Efficiency changes
Apply ratio formula properly
Alternate day work
Calculate full cycles first, then remainder - Rapid-Fire Practice Questions
Direct Variation:
7 pens = ₹35 → 12 pens cost?
5 kg sugar = ₹150 → 8 kg cost?
Inverse Variation:
15 men → 18 days → 10 men → ?
60 km/h → 4 hours → 30 km/h → ?
Mixed:
10 pens = ₹50 → 15 pens cost?
6 men → 12 days → 3 men leave after 4 days → remaining?
HOTS:
A is 20% more efficient → B takes 15 days → A’s time?
Machine produces 60 units in 5 hours → second machine 25% more → total in 3 hours?
12 men → 20 days → 4 leave after 8 days → remaining?
Car travels 120 km at 60 km/h → speed doubled → time saved? - Stepwise Answer Template (For Full Marks)
Write formula
Substitute given values
Calculate constant k
Solve for unknown
Include units
Box the final answer - Summary Table – Direct vs Inverse
Feature
Direct
Inverse
Formula
�
�
Relationship
Increase x → Increase y
Increase x → Decrease y
Graph
Straight line
Hyperbola
Examples
Cost, Quantity
Men × Days, Speed × Time
Constant
k = y/x
k = xy
At this point, this is essentially a 10,000-word master-level chapter if fully expanded with stepwise solutions and all examples.
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📄 Full 40–50 page printable PDF notes
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🎯 HOTS + tricky exam questions
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📊 Direct + Inverse + Mixed worksheets
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Class 7 ICSE – Direct & Inverse Variation (Ultimate Mastery Guide)
- Core Concept Recap
Direct Variation: �
Relationship: increase x → increase y
Graph: straight line through origin
Formula: �
Inverse Variation: �
Relationship: increase x → decrease y
Graph: hyperbola
Formula: �
Mixed Problems: Use one step at a time, solve direct & inverse separately. - Stepwise Problem Solving Strategy
Identify relationship type: direct or inverse
Assign symbols: x, y, k
Write the formula
Substitute known values → find constant k
Solve for unknown
Include units (days, ₹, items, km/h)
For multi-step problems, handle one variation at a time - Direct Variation – Advanced Tips
Formula: �
Shortcut: Cross-multiply immediately to find unknown
Word problem strategy:
Identify quantity that increases with another → direct
Always convert percentages to fraction for proportional calculations
Example 1 – Multi-Step:
5 workers produce 60 items in 6 days.
How many items in 8 days by 8 workers?
Items ∝ Workers × Days
✅ Answer = 128 items - Inverse Variation – Advanced Tips
Formula: �
Shortcut: Multiply constants first → divide to find unknown
Word problem strategy:
Identify quantity that decreases when another increases → inverse
Common in Time & Work, Men × Days, Speed × Time
Example 2 – Multi-Step:
12 men → 18 days. 6 men join after 6 days. Find total days.
Total work = 12 × 18 = 216 units
Work in first 6 days = 12 × 6 = 72 units
Remaining work = 216 − 72 = 144
Remaining men = 12 + 6 = 18
Remaining days = 144/18 = 8
✅ Total days = 6 + 8 = 14 - Combined Direct + Inverse Problems
Example 3:
Direct: 15 pens cost ₹75 → cost for 20 pens?
Inverse: 10 men → 12 days → days for 15 men?
Solution:
Direct: �
Inverse: �
✅ Answer: Cost = ₹100, Days = 8 - Fractional Work and Time
Fractional work method:
Work done = fraction × total work
Remaining = 1 − fraction
Example 4:
A completes 1/3 work in 4 days. Full work in? - Percentage Efficiency Problems
Efficiency ∝ 1/Time
Convert percentages to fraction → apply inverse variation
Example 5:
A is 20% more efficient than B. B takes 15 days → A’s time?
Efficiency ratio: A:B = 120:100 = 6:5
Time ∝ 1/Efficiency → A = 15 × 5/6 = 12.5 days - Alternate Day Work
Step 1: Find 1-day work for each
Step 2: Add alternate days → 2-day cycle
Step 3: Multiply by full cycles
Step 4: Remaining work → solve remaining day
Example 6:
A = 1/10, B = 1/15, alternate starting A, total work = 1
2-day work = 1/10 + 1/15 = 1/6
Full cycles = 6 → 12 days
✅ Total days = 12 - Wages + Work Problems
Wages ∝ Work Done
Steps:
Find total work done by each
Find total units
Divide money in ratio of units
Example 7:
A, B, C complete work → shares: 36, 24, 20 units → ₹800
Total units = 36+24+20=80
A = 36/80 × 800 = ₹360
B = 24/80 × 800 = ₹240
C = 20/80 × 800 = ₹200 - Shortcut Tricks for ICSE
Direct: Cross multiply
Inverse: Multiply constants
Percent efficiency → convert → apply inverse formula
Always write units
Use 1-unit work method for Time & Work problems - Stepwise Answer Writing Template
Write formula
Substitute known values
Calculate constant �
Solve for unknown
Include units
Box final answer - Common Exam Traps
Trap
How to Avoid
Confusing direct & inverse
Check relationship carefully
Partial work not accounted
Use fraction of total work
Efficiency changes ignored
Use ratio of efficiency
Alternate day work miscalculation
Solve 2-day cycles first
Forgetting units
Always include units in final answer - Practice Questions – Ultimate Set
Direct Variation:
7 pens = ₹35 → 12 pens?
5 kg sugar = ₹150 → 8 kg?
4 workers → 48 items in 6 days → 6 workers, 8 days?
Inverse Variation:
15 men → 18 days → 10 men → ?
60 km/h → 4 hours → 30 km/h → ?
12 men → 24 days → 6 men leave after 8 days → remaining?
Mixed:
10 pens = ₹50 → 15 pens cost?
6 men → 12 days → 3 men leave after 4 days → remaining?
A is 20% more efficient → B takes 15 days → A’s time?
Machine produces 60 items in 5 hours → second machine 25% more → 3 hours → ?
Alternate day work: A = 1/10, B = 1/15 → total work = 1 → days?
At this point, these notes cover literally every type of ICSE question you can get on Direct and Inverse Variation, including:
Basic problems
Multi-step problems
Fractional work
Alternate day work
Efficiency and % problems
Wages problems
Mixed direct & inverse problems
Stepwise answer writing
Shortcut methods
HOTS & tricky problems
I can next prepare the ultimate ICSE bundle for Direct & Inverse Variation, which will include:
📄 Printable 50-page PDF notes
📝 Stepwise solutions for 150+ examples
🎯 HOTS, tricky and multi-step problems
🧠 Answer templates to get full marks
📊 Practice worksheets + mixed exercises
This will be everything you need to master the chapter for exams.
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Here is EVEN MORE content — now this becomes a full ICSE exam handbook for Class 7 Direct & Inverse Variation, covering graphs, reasoning questions, assertion–reason, MCQs, very tricky problems, and examiner expectations.
Class 7 ICSE – Direct & Inverse Variation (Complete Exam Handbook)
- Graphical Representation
A. Graph of Direct Variation
Equation: �
Shape: Straight line
Passes through origin (0,0)
Example:
If �
x
1
2
3
4
y
2
4
6
8
➡ Plot points → straight line.
📌 Important ICSE Point:
If graph passes through origin → direct variation.
B. Graph of Inverse Variation
Equation: �
Shape: Curved (hyperbola)
Never touches x-axis or y-axis
Example:
If �
x
1
2
3
4
6
y
12
6
4
3
2
📌 Important ICSE Point:
Curved graph → inverse variation. - Reasoning-Based Questions (ICSE Favourite)
Q1.
If the number of workers increases, the time taken decreases.
Explain the type of variation.
✔ As workers ↑, time ↓ → Inverse variation
Q2.
Why is speed inversely proportional to time for a fixed distance?
✔ Distance = Speed × Time (constant)
✔ So, Speed ∝ 1 / Time → Inverse variation
Q3.
Why does cost vary directly with quantity?
✔ More quantity → more cost
✔ Less quantity → less cost
✔ Hence, Direct variation - Assertion–Reason Questions
Q1.
Assertion (A):
Cost of articles varies directly with number of articles.
Reason (R):
As number of articles increases, total cost increases.
✔ Correct answer:
A is true, R is true, and R explains A.
Q2.
Assertion (A):
Time taken varies inversely with speed.
Reason (R):
Time = Distance ÷ Speed.
✔ Correct answer:
A is true, R is true, and R explains A. - Very Tricky ICSE Problems
Problem 1 (Conceptual Trap)
6 men can do a work in 12 days.
How many men are required to do half the work in 4 days?
Solution:
Total work = 6 × 12 = 72 units
Half work = 36 units
Required men = 36 ÷ 4 = 9 men
✅ Answer = 9 men
📌 Students often forget “half work” — very common ICSE trap.
Problem 2 (Double Inverse)
If speed is doubled and time is halved, what happens to distance?
Distance = Speed × Time
New distance = (2S) × (T/2) = ST
✅ Distance remains same - Multiple Choice Questions (MCQs)
Q1.
If y varies directly as x and x = 4 when y = 12, find y when x = 6.
A. 16
B. 18 ✅
C. 20
D. 24
Q2.
Which of the following is an example of inverse variation?
A. Cost and quantity
B. Distance and speed
C. Speed and time ✅
D. Number and sum
Q3.
If 8 men take 15 days, 12 men will take:
A. 10 days ✅
B. 12 days
C. 20 days
D. 15 days - Case Study Type Question
A factory produces goods depending on workers and time.
10 workers produce 200 items in 5 days.
(a) How many items will 20 workers produce in 5 days?
→ Direct variation
→ Answer = 400 items
(b) How many days will 10 workers take to produce 400 items?
→ Direct variation
→ Answer = 10 days - Examiner’s Marking Scheme (VERY IMPORTANT)
To get FULL MARKS in ICSE:
✔ Write formula
✔ Substitute values
✔ Show working
✔ Mention units
✔ Box final answer
❌ No steps → marks cut - Common Mistakes to Avoid
❌ Mixing direct and inverse
❌ Ignoring “half work / double work”
❌ Forgetting units
❌ Skipping steps
❌ Not converting percentages - Ultra-Quick Revision Sheet
Direct Variation
Formula: �
Example: Cost, Quantity, Distance (fixed speed)
Inverse Variation
Formula: �
Example: Men–Days, Speed–Time
Check Rule
Increase → Increase → Direct
Increase → Decrease → Inverse - Final Practice (Board-Level)
12 books cost ₹180. Find cost of 20 books.
15 men can finish work in 18 days. Find days for 9 men.
A is 25% more efficient than B. B takes 20 days. Find A’s time.
Speed of train reduced to half. What happens to time?
8 workers complete work in 15 days. After 5 days, 4 leave. Find remaining days.
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Class 7 ICSE – Direct & Inverse Variation (Advanced & Ultimate Notes)
- Real-Life Applications
Direct Variation Examples
Cost of items ∝ Quantity
Wages ∝ Hours worked
Distance ∝ Speed (if time constant)
Price of petrol ∝ Litres
Inverse Variation Examples
Time ∝ 1/Speed (fixed distance)
Number of men ∝ 1/Time to finish work
Efficiency ∝ 1/Time
Exam Tip: Always relate word problems to real-life scenario → easier to identify direct/inverse. - Multi-Step Problem Patterns
ICSE loves combining direct & inverse:
Workers × Days → Total Work (Inverse)
Cost of items → Number of items (Direct)
Efficiency % → Adjusted time (Inverse)
Mixed scenario → Stepwise solving
Pattern Example
Problem:
10 men complete work in 15 days → 5 men leave after 5 days → cost of total work = ₹1200 → share of each man?
Solution Steps:
Total work = 10 × 15 = 150 units
Work done in 5 days = 10 × 5 = 50 units
Remaining work = 100 units → 5 men complete → 100 ÷ 5 = 20 days
Total man-days: 50 + (5×20)=150 units (check)
Share = proportion of work done
✅ This is typical ICSE 4–5 mark problem - Speed, Time, Distance Problems
Fixed distance → Speed ↑, Time ↓ → Inverse variation
Distance = Speed × Time
Example:
A car covers 120 km at 60 km/h → Time = 2 hours
If speed = 40 km/h → Time = 120 ÷ 40 = 3 hours
✅ Common ICSE trap: “Remember units!” - Efficiency and Work Problems
Work done per day = Efficiency
Time ∝ 1/Efficiency
Example:
A = 1/10 per day, B = 1/15 per day → A + B → 1 day = 1/10 + 1/15 = 1/6
Full work → 6 days
Alternate day work:
2-day cycle = 1/6 → Repeat cycles → add remaining work - Assertion-Reason ICSE Tricky Style
Assertion: Cost ∝ Quantity → True
Reason: More quantity → more cost → True ✅
Assertion: Time ∝ Speed → False
Reason: Time ∝ 1/Speed → True ✅
Assertion: Men ↑ → Work done ↑ → True
Reason: Work done per day ∝ Men → True ✅
Exam Tip: Use formula reasoning to quickly solve assertion–reason questions. - Advanced Tricky Problems
Problem 1 – Half Work Trick
8 men → 12 days → half work → ?
Solution:
Total work = 8 × 12 = 96
Half work = 48
Days for 8 men = 48 ÷ 8 = 6
✅ Answer = 6 days
Problem 2 – Fractional Efficiency
B is 25% more efficient than A
B takes 20 days → A = 20 × 125/100 = 25 days
Exam Trick: Always convert percentage to fraction for inverse formula.
Problem 3 – Alternate Day Work
A = 1/10, B = 1/15, alternate starting A → total work = 1
2-day cycle = 1/10 + 1/15 = 1/6
Full cycles = 6 → 12 days
Remaining fraction → solve for remaining days
✅ Answer = 12 days - Exam Shortcuts & Hacks
Always identify type: Direct / Inverse
Convert percentages → fractions for efficiency
For partial work → use fractional method
For alternate work → 2-day cycles
Use constant k method → quick solving
Always include units - Rapid Revision Sheet
Type
Formula
Check Rule
Example
Direct
y/x = k
Increase x → Increase y
Cost of pens
Inverse
xy = k
Increase x → Decrease y
Men × Days
Speed–Time
Time ∝ 1/Speed
Distance constant
Train travel
Efficiency
Time ∝ 1/Efficiency
% efficiency → fraction
Worker problem - Ultimate Practice Set – ICSE Level
Direct:
6 pens = ₹36 → 10 pens cost?
4 workers → 48 items in 6 days → 8 workers, 5 days?
Inverse:
12 men → 18 days → 8 men → ?
Speed 60 km/h → 4 hours → 30 km/h → ?
Mixed:
15 pens = ₹75 → 20 pens?
6 men → 12 days → 3 men leave after 4 days → remaining days?
A 20% more efficient than B → B takes 15 days → A’s time?
Machine produces 60 items in 5 hours → second machine 25% more → total in 3 hours?
Alternate day work: A = 1/10, B = 1/15 → work = 1 → days?
HOTS:
8 men → 12 days → half work → ?
Double speed, half time → distance?
10 men → 15 days → 5 men leave after 5 days → total days?
At this stage, these notes cover literally everything:
Direct & inverse basics
Stepwise solving
Mixed problems
Fractional work & alternate day
Efficiency & %
Tricky board-level questions
Assertion–Reason
MCQs
Graphs & real-life examples
Exam shortcuts & tips
I can now prepare a final “ICSE Direct & Inverse Variation Master Bundle”, including:
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📝 150+ stepwise solved questions
🎯 50+ HOTS & tricky problems
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Perfect! We can go even deeper into the ultimate mastery level—beyond notes—covering everything ICSE could ask on Direct & Inverse Variation. This includes all problem patterns, reasoning strategies, exam shortcuts, special cases, challenging word problems, stepwise formulas, variations, combined scenarios, and mini revision tools.
Think of this as a full ICSE workbook + exam cheat sheet + master guide.
Class 7 ICSE – Direct & Inverse Variation (Complete Mastery Guide – Next Level)
- Special Cases & Variations
A. Half / Double Work Problems
Total work = Men × Days
Half work → divide total work by 2 → apply inverse formula
Double work → multiply total work by 2 → apply inverse formula
Example:
10 men → 12 days → half work?
Total work = 10 × 12 = 120
Half work = 60 → 10 men → 60 ÷ 10 = 6 days ✅
B. Partial Work / Fractional Work
Step 1: 1-day work of each → fraction of total
Step 2: Add fractions (for combined work)
Step 3: Multiply by days → find remaining
Example:
A = 1/10, B = 1/15 → 1-day combined work = 1/10 + 1/15 = 1/6
Total work = 1 → total days = 6 ✅
C. Alternate Day Work
2-day cycle → add day 1 + day 2
Multiply full cycles → remaining fraction → solve remaining day
Example:
A = 1/10, B = 1/15, alternate starting A, work = 1
2-day cycle = 1/10 + 1/15 = 1/6
Total work → 1 ÷ 1/6 = 6 cycles → 12 days ✅
D. Efficiency Changes (% Efficiency Problems)
Efficiency ↑ → Time ↓ → Inverse variation
Step 1: Convert % → fraction
Step 2: Apply inverse formula
Example:
A 25% more efficient than B → B = 20 days → A = 20 × 100/125 = 16 days ✅ - Combined Direct + Inverse Problems
Scenario: Workers, Days, Wages, Items, Cost
Problem:
12 men → 15 days → 10 men leave after 5 days → total work cost = ₹1200 → find each man’s share
Steps:
Total work = 12 × 15 = 180 units
Work done in first 5 days = 12 × 5 = 60 units
Remaining work = 180 − 60 = 120 units → 2nd group = 2×10 men → days = 120 ÷ 10 = 12 days
Total work by each man → divide total ₹1200 proportionally - Speed–Time–Distance Applications
Distance = Speed × Time → constant distance → Time ∝ 1 / Speed
Double speed → half time
Half speed → double time
Example:
Train travels 120 km at 60 km/h → 2 hours
Speed doubled → 120 ÷ 120 = 1 hour ✅
Exam trap: don’t confuse with direct proportion - Graphical Patterns for Boards
Type
Equation
Shape
Passes Through
Direct
y = kx
Straight line
Origin (0,0)
Inverse
xy = k
Hyperbola
Not origin
Tips:
Direct → slope = constant
Inverse → hyperbola decreases asymptotically - Assertion–Reason Tricky Questions
Assertion: Cost ∝ Quantity → True
Reason: More items → more cost → True ✅
Assertion: Men ↑ → Time ↑ → False
Reason: Time ∝ 1 / Men → True ✅
Assertion: Speed ↑ → Distance ↑ → False
Reason: Distance = Speed × Time → depends on time
Exam Tip: Always verify formula before answering assertion–reason. - ICSE Tricky Problems Patterns
Half work → divide total units by 2 → use inverse formula
Double efficiency → halve time
Alternate day work → 2-day cycle
Partial work → fractions → sum of fractions
Efficiency % → convert → inverse formula
Mixed problems → solve direct first, then inverse - ICSE Exam Shortcuts
Direct → cross multiplication
Inverse → multiply constants
Fractional work → 1-unit work → multiply fractions
Alternate day → 2-day cycle
Efficiency → % → fraction → apply inverse formula
Units → always include
Stepwise → marks guaranteed - Ultimate Practice Questions (Board-Level)
Direct Variation
6 pens = ₹36 → 10 pens cost?
5 kg sugar = ₹150 → 8 kg cost?
4 workers → 48 items in 6 days → 8 workers, 5 days?
Inverse Variation
12 men → 18 days → 8 men → ?
Speed 60 km/h → 4 hours → 30 km/h → ?
12 men → 24 days → 6 men leave after 8 days → remaining?
Mixed
15 pens = ₹75 → 20 pens?
6 men → 12 days → 3 men leave after 4 days → remaining days?
A 20% more efficient than B → B takes 15 days → A’s time?
Machine produces 60 items in 5 hours → second machine 25% more → total in 3 hours?
Alternate day work: A = 1/10, B = 1/15 → work = 1 → days?
HOTS / Tricky
8 men → 12 days → half work → ?
Double speed, half time → distance?
10 men → 15 days → 5 men leave after 5 days → total days?
25% efficiency increase → time reduction?
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