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ICSE • Class 8 • Mathematics • Ch 11
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Direct and Inverse Variations

In ICSE Class 8 Mathematics, "Direct and Inverse Variations" provides an authoritative, mathematically rigorous master study guide investigating proportional relationships, functional dependencies, constant of proportionality, and unitary methods. This comprehensive chapter explores Concept of Variation (Two quantities $x$ and $y$ changing in tandem such that an alteration in one induces a corresponding alteration in the other), Direct Variation / Proportion (Two quantities vary directly if their ratio remains strictly constant: $\frac{x}{y} = k$ or $\frac{x_1}{y_1} = \frac{x_2}{y_2}$; Constant of variation $k > 0$; Graphical representation: straight line passing through the origin; Real-world applications: cost of articles vs quantity, distance vs time at uniform speed, petrol consumed vs distance traveled, scale map distances), Inverse Variation / Indirect Proportion (Two quantities vary inversely if their product remains strictly constant: $x \times y = k$ or $x_1 y_1 = x_2 y_2$; As $x$ increases, $y$ decreases proportionally; Constant of variation $k$; Graphical representation: rectangular hyperbola; Real-world applications: speed vs time for fixed distance, number of workers vs days to complete a job, pipes filling a cistern, garrison food provisions and soldiers), Compound / Mixed Variations, Time and Work Problems (Work done in 1 day $= \frac{1}{n}$; Pipes and cisterns: inlet and outlet rates), and Distinguishing Between Direct and Inverse Word Problems aligned with the 2026–27 CISCE ICSE curriculum.

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How Can Increasing the Number of Workers on a Construction Site Make the Total Job Finish Days Earlier Instead of Later?

If you buy 5 mangoes for 100 rupees, 10 mangoes will cost you 200 rupees. Both quantities go UP together—that is DIRECT VARIATION! More items mean more money. But now imagine a construction contractor hiring 10 workers to dig a canal in 30 days. If he doubles the workforce to 20 workers, will it take 60 days? Absolutely not! The job finishes in just 15 days! As the number of workers goes UP, the time required goes DOWN! This is the magic of INVERSE VARIATION! While direct variation keeps the ratio $\frac{x}{y}$ constant, inverse variation keeps the product $x \times y$ constant! If a military garrison of 500 soldiers has food rations for 30 days, and 100 more soldiers arrive, how many days will the food last? How does a pipe filling a water tank work in tandem with a leak draining it? Let's master direct and inverse variations.

Why This Chapter Matters

Variation is the foundation of scientific scaling laws: Ohm's law in physics ($V = IR$), Newton's law of gravitation ($F \propto \frac{1}{r^2}$), Boyle's gas law ($PV = k$), construction project scheduling, and industrial assembly-line logistics. Mastering direct and inverse proportions is essential for ICSE Class 8 mathematics and competitive aptitude examinations.

Before You Begin (Prerequisites)

  • Ratio and proportion fundamentals from Class 7.
  • Solving simple linear equations in one variable.
  • Unitary method.

What You Will Learn (Core Objectives)

  • Identify whether two physical quantities are in direct or inverse variation.
  • Determine the constant of variation $k$ for direct ($\frac{x}{y}=k$) and inverse ($xy=k$) relations.
  • Set up and solve proportion equations: $\frac{x_1}{y_1} = \frac{x_2}{y_2}$ and $x_1 y_1 = x_2 y_2$.
  • Solve multi-person Time and Work problems using unitary day-rates.
  • Solve Pipes and Cisterns problems involving both inlet taps and drainage leaks.
  • Solve food garrison and camp provision variation word problems.

Chapter Roadmap & Progression

1 1. Direct Variation: Ratio Invarian...
2 2. Inverse Variation: Product Invar...
3 3. Time & Work Mathematical Framewo...
4 4. Pipes and Cisterns Mechanics

Complete Concept Guide (100% Curriculum Coverage)

1. Direct Variation: Ratio Invariance

Understand
A. Definition of Direct Variation:

Two quantities $x$ and $y$ are said to be in Direct Variation (or direct proportion) if an increase (or decrease) in $x$ produces a proportional increase (or decrease) in $y$ such that their ratio remains constant:

$$\mathbf{\frac{x}{y} = k \quad (\text{Constant of Variation})} \quad \Longleftrightarrow \quad \mathbf{\frac{x_1}{y_1} = \frac{x_2}{y_2}}$$
  • Examples:
    • Distance traveled and time taken (at constant speed).
    • Total purchase cost and number of articles bought.
    • Weight of a metal rod and its length.
  • Graphical Plot: A straight line passing directly through the coordinate origin $(0, 0)$.

2. Inverse Variation: Product Invariance

Inverse Proportion
A. Definition of Inverse Variation:

Two quantities $x$ and $y$ are said to be in Inverse Variation (or indirect proportion) if an increase in $x$ produces a proportional decrease in $y$ (and vice versa) such that their product remains constant:

$$\mathbf{x \times y = k \quad (\text{Constant})} \quad \Longleftrightarrow \quad \mathbf{x_1 y_1 = x_2 y_2}$$
  • Examples:
    • Speed of a vehicle and time taken to cover a fixed journey distance.
    • Number of workers and time taken to complete a specific task.
    • Number of people sharing food provisions and the days the food lasts.
  • Graphical Plot: A smooth rectangular hyperbola curving asymptotically toward both axes.

3. Time & Work Mathematical Framework

Time & Work
Fundamental Axioms of Work:
  1. If a person can finish a piece of work in $n$ days, then the work done by that person in $1\text{ day}$ is: $$\mathbf{\text{Work done in 1 day} = \frac{1}{n}}$$
  2. Conversely, if a person completes $\frac{1}{n}$ of the work in 1 day, the total time taken to complete the entire job is $n\text{ days}$.
  3. If person $A$ finishes work in $x$ days and person $B$ in $y$ days, their combined 1-day work is: $$\text{Combined 1-day work} = \frac{1}{x} + \frac{1}{y} = \frac{x + y}{xy}$$ $$\mathbf{\text{Total Days Together} = \frac{xy}{x + y}}$$

4. Pipes and Cisterns Mechanics

Pipes & Cisterns
  • Inlet Pipe: Fills the tank. If it fills a tank in $T_1$ hours, its 1-hour filling rate is $+\frac{1}{T_1}$.
  • Outlet / Leak Pipe: Empties the tank. If it drains the tank in $T_2$ hours, its 1-hour emptying rate is $-\frac{1}{T_2}$.
  • If both pipes are open simultaneously: $$\mathbf{\text{Net 1-hour Work} = \frac{1}{T_1} - \frac{1}{T_2} = \frac{T_2 - T_1}{T_1 T_2}}$$ $$\mathbf{\text{Time to Fill Tank} = \frac{T_1 T_2}{T_2 - T_1} \quad (T_2 > T_1)}$$

Key Formulas, Identities & Theorems

Direct Variation Formula
$$\frac{x_1}{y_1} = \frac{x_2}{y_2} \quad \left( \frac{x}{y} = k \right)$$
Ratio remains constant.
Inverse Variation Formula
$$x_1 y_1 = x_2 y_2 \quad (x \cdot y = k)$$
Product remains constant.

Mathematics: Direct vs Inverse Variation Curves

Direct vs Inverse Variations: Algebraic Laws & Graphs DIRECT VARIATION (RATIO CONSTANT) y = kx x / y = k ⇒ x1 / y1 = x2 / y2 • Straight line through origin (0, 0) • More items = More cost • More speed = More distance INVERSE VARIATION (PRODUCT CONSTANT) xy = k x × y = k ⇒ x1 y1 = x2 y2 • Rectangular hyperbola curve • More workers = Fewer days • Faster speed = Less time DIRECT: x1/y1 = x2/y2 (RATIO CONSTANT) • INVERSE: x1*y1 = x2*y2 (PRODUCT CONSTANT)

Chapter Summary & 10 Key Takeaways

Takeaway 1
Direct variation occurs when two quantities increase or decrease together: x/y = k.
Takeaway 2
In direct variation, x1 / y1 = x2 / y2; its graph is a straight line through the origin.
Takeaway 3
Inverse variation occurs when an increase in one causes a decrease in the other: x * y = k.
Takeaway 4
In inverse variation, x1 * y1 = x2 * y2; its graph is a rectangular hyperbola.
Takeaway 5
Worker-time relationships are inverse variations: more workers take fewer days.
Takeaway 6
Speed and travel time for a fixed distance are in inverse variation.
Takeaway 7
In Time and Work, a person completing a job in n days accomplishes 1/n of the work in 1 day.
Takeaway 8
Combined time for two workers is (x * y) / (x + y) days.
Takeaway 9
Pipes filling a cistern add work (+1/T1); drainage leaks subtract work (-1/T2).
Takeaway 10
Always verify whether the problem scenario is direct (ratio) or inverse (product) before calculating.

Check Your Understanding (Diagnostic Practice Questions)

Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.

1
If $15$ men can build a wall in $48\text{ hours}$, how many workers will be required to build the same wall in $30\text{ hours}$?
Reveal Answer & Explanation
Answer:

Step 1: Identify the type of variation:
More workers will require LESS time to build the same wall. Therefore, this is an INVERSE VARIATION problem.
Step 2: Let the number of workers required be $x_2$.
Here $x_1 = 15$ men, $y_1 = 48$ hours; $y_2 = 30$ hours.
Step 3: Apply the Inverse Variation formula ($x_1 y_1 = x_2 y_2$):

$$15 \times 48 = x_2 \times 30$$


$$720 = 30 x_2$$


$$x_2 = \frac{720}{30} = \mathbf{24\text{ men}}$$

.
Exactly $24$ workers will be required.


Inverse variation: $15 \times 48 = x \times 30 \implies x = 720 / 30 = 24\text{ men}$.
2
A loaded truck travels $14\text{ kilometers}$ in $25\text{ minutes}$. If the speed remains constant, how far will it travel in $5\text{ hours}$?
Reveal Answer & Explanation
Answer:

Step 1: Identify the type of variation:
At constant speed, more time means MORE distance covered. Therefore, this is a DIRECT VARIATION problem.
Step 2: Convert time into the same unit (minutes):

$$5\text{ hours} = 5 \times 60 = 300\text{ minutes}$$


Step 3: Let the distance traveled be $x_2\text{ km}$.
Here $x_1 = 14\text{ km}, y_1 = 25\text{ mins}; y_2 = 300\text{ mins}$.
Step 4: Apply the Direct Variation formula ($\frac{x_1}{y_1} = \frac{x_2}{y_2}$):

$$\frac{14}{25} = \frac{x_2}{300}$$


$$x_2 = \frac{14 \times 300}{25} = 14 \times 12 = \mathbf{168\text{ kilometers}}$$

.


Direct variation: $14 / 25 = x / 300 \implies x = 14 \times 12 = 168\text{ km}$.
3
$A$ can complete a piece of work in $12\text{ days}$, while $B$ can complete the same work in $24\text{ days}$. In how many days can they complete the work together?
Reveal Answer & Explanation
Answer: Step 1: Calculate 1-day work for each person:
• Work done by $A$ in $1\text{ day} = \frac{1}{12}$
• Work done by $B$ in $1\text{ day} = \frac{1}{24}$
Step 2: Calculate their combined 1-day work:
$$\text{Work done by } (A + B) \text{ in 1 day} = \frac{1}{12} + \frac{1}{24} = \frac{2 + 1}{24} = \frac{3}{24} = \frac{1}{8}$$
Step 3: Total time required is the reciprocal of their 1-day work:
$$\text{Total Days} = \mathbf{8\text{ days}}$$.
Combined 1-day work is $1/12 + 1/24 = 3/24 = 1/8$. Total time is 8 days.
4
A military garrison of $300$ soldiers had provisions for $45\text{ days}$. After $15\text{ days}$, $50$ more soldiers joined the garrison. How many days will the remaining provisions last?
Reveal Answer & Explanation
Answer: Step 1: Calculate remaining food provisions:
Provisions left after $15\text{ days}$ are sufficient for the original $300$ soldiers for $(45 - 15) = \mathbf{30\text{ days}}$.
Step 2: New total soldiers in garrison:
$$\text{New count} = 300 + 50 = \mathbf{350\text{ soldiers}}$$
Step 3: More soldiers will exhaust food in FEWER days (Inverse Variation):
$$x_1 y_1 = x_2 y_2$$
$$300 \times 30 = 350 \times y_2$$
$$9000 = 350 y_2$$
$$y_2 = \frac{9000}{350} = \frac{900}{35} = \frac{180}{7} = \mathbf{25 \frac{5}{7}\text{ days}}$$.
For 300 men, food lasts 30 days. For 350 men, $300 \times 30 = 350 \times y \implies y = 180 / 7 = 25\frac{5}{7}$ days.
5
A tap can fill a cistern in $8\text{ hours}$, while an outlet pipe can empty it in $12\text{ hours}$. If both pipes are opened simultaneously, in how many hours will the empty cistern be completely filled?
Reveal Answer & Explanation
Answer:

Step 1: Rate of filling in $1\text{ hour} = +\frac{1}{8}$.
Step 2: Rate of emptying in $1\text{ hour} = -\frac{1}{12}$.
Step 3: Net filling work done in $1\text{ hour}$ when both are open:

$$\text{Net Rate} = \frac{1}{8} - \frac{1}{12}$$


LCM of 8 and 12 is 24:

$$= \frac{3 - 2}{24} = \mathbf{\frac{1}{24}}$$


Step 4: Since $\frac{1}{24}$ of the tank fills in 1 hour, the cistern will be completely filled in $24\text{ hours}$.


Net rate is $1/8 - 1/12 = (3 - 2)/24 = 1/24$. Time taken is 24 hours.
6
A train traveling at a uniform speed of $75\text{ km/h}$ covers a certain distance in $4\text{ hours}$. How long will the same journey take if the train speed is increased to $100\text{ km/h}$?
Reveal Answer & Explanation
Answer:

Step 1: Higher speed requires LESS travel time (Inverse Variation):

$$x_1 y_1 = x_2 y_2$$


Here $x_1 = 75\text{ km/h}, y_1 = 4\text{ hours}; x_2 = 100\text{ km/h}, y_2 = ?$

$$75 \times 4 = 100 \times y_2$$


$$300 = 100 y_2 \implies y_2 = \frac{300}{100} = \mathbf{3\text{ hours}}$$

.
The journey will take $3\text{ hours}$.


Inverse variation: $75 \times 4 = 100 \times y \implies y = 300 / 100 = 3\text{ hours}$.
7
If the weight of $12$ sheets of thick paper is $40\text{ grams}$, how many sheets of the same paper would weigh $2 \frac{1}{2}\text{ kilograms}$?
Reveal Answer & Explanation
Answer: Step 1: Direct Variation (more sheets weigh more):
Convert $2 \frac{1}{2}\text{ kg}$ to grams: $2.5 \times 1000 = 2500\text{ grams}$.
Step 2: Let the number of sheets be $x_2$.
$$\frac{x_1}{y_1} = \frac{x_2}{y_2}$$
$$\frac{12}{40} = \frac{x_2}{2500}$$
$$x_2 = \frac{12 \times 2500}{40} = \frac{30,000}{40} = \mathbf{750\text{ sheets}}$$.
Direct variation: $12 / 40 = x / 2500 \implies x = (12 \times 2500) / 40 = 750\text{ sheets}$.
8
Explain how to determine whether a given table of values represents Direct Variation or Inverse Variation.
Reveal Answer & Explanation
Answer:

• Direct Variation Test: Calculate the ratio $\frac{x}{y}$ for every data pair $(x_1, y_1), (x_2, y_2)$. If the ratio $\frac{x}{y} = k$ is constant for all pairs, the relationship is Direct Variation.
• Inverse Variation Test: Calculate the product $x \times y$ for every data pair. If the product $x \times y = k$ is constant for all pairs, the relationship is Inverse Variation.


Check if the ratio $x/y$ is constant (Direct) or if the product $x \times y$ is constant (Inverse).
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