Variations: How Things Change Together
Hey everyone! Welcome to the chapter on Variations. Ever wondered how things in the world are connected? Like how the more you study, the better your grades get? Or how the faster you travel, the less time your journey takes? That's what variation is all about! It's the mathematics of describing relationships between quantities.
This is a super useful topic, not just for your exams, but for understanding science, economics, and even everyday things like cooking or planning a trip. Don't worry if it sounds complicated; we'll break it down into simple, easy-to-understand parts. Let's get started!
Part 1: Direct Variation - As One Goes Up, So Does the Other!
This is the simplest type of relationship. When two quantities vary directly, it means that as one increases, the other increases at the same rate. And if one decreases, the other decreases too.
Real-world example: Imagine you're buying bubble tea. The more cups you buy, the more money you spend. The number of cups and the total cost are in direct variation.
The Language and the Formula
You'll see phrases like:
- "y varies directly as x"
- "y is directly proportional to x"
Both of these mean the same thing and translate to one simple equation:
$$ y = kx $$
Here, k is the variation constant (or constant of proportionality). It's a fixed number (but not zero!) that connects y and x. Your first job in any variation problem is almost always to find this 'k'.
Step-by-Step Guide to Solving Direct Variation Problems:
- Write the general equation: Start by writing $$y = kx$$. (Sometimes it might be $$y = kx^2$$ or $$y = k\sqrt{x}$$, the question will tell you!).
- Find 'k': Use the first set of information given in the problem (a known value for y and x) to calculate the value of k.
- Write the specific equation: Replace the letter 'k' in your equation with the number you just found. Now you have the specific formula connecting your variables.
- Solve the problem: Use your new specific equation and the final piece of information to find the unknown value.
Let's Try An Example:
Question: y varies directly as x. If y = 24 when x = 8, find the value of y when x = 5.
Step 1: General Equation
We write: $$y = kx$$
Step 2: Find 'k'
We substitute the given values, y = 24 and x = 8:
$$ 24 = k(8) $$
To find k, we divide both sides by 8:
$$ k = \frac{24}{8} = 3 $$
So, our variation constant is 3!
Step 3: Specific Equation
Now we know k, our specific equation is: $$y = 3x$$
Step 4: Solve the Problem
We need to find y when x = 5. We use our new equation:
$$ y = 3(5) $$
$$ y = 15 $$
Answer: The value of y is 15. Easy, right?
Graphs of Direct Variation
The graph of a direct variation (like $$y = kx$$) is always a straight line that passes through the origin (0, 0). The steepness (slope) of the line is your variation constant, k.
Remember: The syllabus notes that the values (domain) can be negative, so the line extends into the negative quadrants too!
Key Takeaway for Direct Variation
Keywords: "varies directly", "is directly proportional to".
Equation: $$y = kx$$
What it means: If x doubles, y doubles. If x is halved, y is halved. They are a team!
Part 2: Inverse Variation - As One Goes Up, the Other Goes Down!
Inverse variation is the opposite. When one quantity increases, the other quantity decreases proportionally.
Real-world example: Think about sharing a pizza. The more friends you share it with, the smaller each person's slice will be. The number of friends and the slice size are in inverse variation.
The Language and the Formula
You'll see phrases like:
- "y varies inversely as x"
- "y is inversely proportional to x"
This translates to the following equation:
$$ y = \frac{k}{x} $$
Again, k is our important variation constant, and we need to find it first.
Let's Try An Example:
Question: The time (T hours) it takes to complete a job varies inversely as the number of workers (N). If 4 workers take 6 hours to finish the job, how long will it take 3 workers?
Step 1: General Equation
The variables are T and N. The equation is: $$T = \frac{k}{N}$$
Step 2: Find 'k'
Substitute T = 6 and N = 4:
$$ 6 = \frac{k}{4} $$
To find k, multiply both sides by 4:
$$ k = 6 \times 4 = 24 $$
Step 3: Specific Equation
The specific formula for this job is: $$T = \frac{24}{N}$$
Step 4: Solve the Problem
We need to find the time (T) for 3 workers (N = 3):
$$ T = \frac{24}{3} $$
$$ T = 8 $$
Answer: It will take 3 workers 8 hours to complete the job.
Graphs of Inverse Variation
The graph of an inverse variation (like $$y = k/x$$) is a curve called a hyperbola.
- It does NOT pass through the origin (0, 0). Think about it: you can't divide by zero!
- The curve gets closer and closer to the x-axis and y-axis, but never touches them.
Did you know?
Many things in physics follow an "inverse square law". For example, the force of gravity between two objects varies inversely with the square of the distance between them ($$F = \frac{k}{d^2}$$). This means if you double your distance from the Earth, the force of gravity pulling on you becomes four times weaker!
Key Takeaway for Inverse Variation
Keywords: "varies inversely", "is inversely proportional to".
Equation: $$y = \frac{k}{x}$$
What it means: If x doubles, y is halved. If x triples, y becomes one-third. They move in opposite directions.
Part 3: Joint Variation - A Team Effort!
Joint variation is just an extension of direct variation, but with more than one variable involved. A quantity varies "jointly" as two or more other quantities.
Real-world example: The interest you earn in a simple interest savings account varies jointly with the amount of money you invest (principal) AND the time you leave it in the bank.
The Language and the Formula
If "z varies jointly as x and y", it means z is directly proportional to the product of x and y.
$$ z = kxy $$
You can even mix and match! For example, "z varies directly as x and inversely as y" would be:
$$ z = \frac{kx}{y} $$
Memory Aid: "Jointly" or "directly" puts the variable on top (numerator). "Inversely" puts the variable on the bottom (denominator).
Let's Try An Example:
Question: C varies jointly as A and the square of B. If C = 72 when A = 4 and B = 3, find C when A = 2 and B = 5.
Step 1: General Equation
"Jointly as A and the square of B" means: $$C = kAB^2$$
Step 2: Find 'k'
Substitute C = 72, A = 4, B = 3:
$$ 72 = k(4)(3^2) $$
$$ 72 = k(4)(9) $$
$$ 72 = 36k $$
$$ k = \frac{72}{36} = 2 $$
Step 3: Specific Equation
Our specific formula is: $$C = 2AB^2$$
Step 4: Solve the Problem
Find C when A = 2 and B = 5:
$$ C = 2(2)(5^2) $$
$$ C = 2(2)(25) $$
$$ C = 4(25) = 100 $$
Answer: The value of C is 100.
Key Takeaway for Joint Variation
Keywords: "varies jointly as...", often involves more than two variables.
Equation: Combines variables, like $$z = kxy$$ or $$z = \frac{kx}{y}$$.
What it means: It's a relationship where one thing depends on several other factors working together.
Part 4: Partial Variation - A Mix and Match!
This one can seem tricky, but the concept is very common in real life. Partial variation means a variable is the sum of a constant part and a variable part.
The BEST real-world example: Your mobile phone bill! You might pay a fixed monthly fee (the constant part) PLUS an amount based on how much data you use (the variable part). Total Bill = Fixed Fee + (Cost per GB × GB used).
The Language and the Formula
The wording is usually quite clear:
- "y is partly constant and partly varies as x"
This means we need TWO constants, one for the fixed part and one for the variable part. Let's call them $$k_1$$ and $$k_2$$.
$$ y = k_1 + k_2x $$
The Big Difference: To solve these, you'll be given two sets of data. You'll use them to create a pair of simultaneous equations to find both $$k_1$$ and $$k_2$$.
Let's Try An Example:
Question: The cost ($C) of a taxi ride is partly constant and partly varies directly as the distance (d km). A 5 km ride costs $60. A 9 km ride costs $92. Find the cost of a 10 km ride.
Step 1: General Equation
"Partly constant and partly varies as d" means: $$C = k_1 + k_2d$$
Step 2: Form Simultaneous Equations
Use the two pieces of information given:
For a 5 km ride (d=5), the cost is $60 (C=60):
$$ 60 = k_1 + k_2(5) \quad \dots (1) $$
For a 9 km ride (d=9), the cost is $92 (C=92):
$$ 92 = k_1 + k_2(9) \quad \dots (2) $$
Step 3: Solve the Simultaneous Equations
Let's use elimination. Subtract equation (1) from equation (2):
$$ (92 - 60) = (k_1 - k_1) + (9k_2 - 5k_2) $$
$$ 32 = 4k_2 $$
$$ k_2 = \frac{32}{4} = 8 $$
Now substitute $$k_2 = 8$$ back into equation (1):
$$ 60 = k_1 + 8(5) $$
$$ 60 = k_1 + 40 $$
$$ k_1 = 60 - 40 = 20 $$
So, the fixed cost is $20 and the cost per km is $8.
Step 4: Write the Specific Equation
Our formula is: $$C = 20 + 8d$$
Step 5: Solve the Problem
Find the cost (C) for a 10 km ride (d=10):
$$ C = 20 + 8(10) $$
$$ C = 20 + 80 = 100 $$
Answer: The cost of a 10 km ride is $100.
Common Mistakes to Avoid
For partial variation, the biggest mistake is forgetting there are TWO constants ($$k_1$$ and $$k_2$$). If you only use one 'k', you won't be able to solve it! Remember, two unknowns ($$k_1, k_2$$) require two equations.
Key Takeaway for Partial Variation
Keywords: "partly constant", "partly varies as".
Equation: $$y = k_1 + k_2x$$ (or similar form).
What it means: A relationship with a fixed starting value plus a variable amount. Requires solving simultaneous equations!
Quick Chapter Summary & Cheat Sheet
Direct Variation
Keywords: varies directly as...
Equation: $$y = kx$$
Idea: They move together.
Inverse Variation
Keywords: varies inversely as...
Equation: $$y = \frac{k}{x}$$
Idea: They move in opposite directions.
Joint Variation
Keywords: varies jointly as...
Equation: $$z = kxy$$ (or $$z = \frac{kx}{y}$$)
Idea: One thing depends on multiple factors.
Partial Variation
Keywords: partly constant and partly varies as...
Equation: $$y = k_1 + k_2x$$
Idea: A fixed part + a variable part. (Think: Simultaneous Equations!)
And that's it! You've learned the four main types of variation. The key to success is to read the question carefully to identify which type of variation it is, write down the correct general equation, and then follow the steps. Keep practicing, and you'll become a pro at spotting these relationships everywhere! Good luck!