Solving Word Problems with Direct and Inverse Variation

1. Understanding Direct Variation

Direct variation (or direct proportion) describes a relationship where two quantities increase or decrease together at the same rate. If you buy more pens, the total cost goes up. If you work for more hours, you earn more money (assuming a fixed hourly wage).

The key relationship is that the ratio of the two quantities is always constant.

If a quantity $y$ varies directly with a quantity $x$, we write this as:
$y\propto x$
This is translated into the equation:
$y=kx$
where $k$ is the "constant of proportionality". By rearranging this, we get the core principle for direct variation:
$\frac{y}{x}=k(\text{The ratio is constant})$

This constant ratio allows us to solve problems without necessarily finding k. If ($x_1,y_1$) and ($x_2,y_2$) are two pairs of corresponding values, then:
$\frac{y_1}{x_1}=\frac{y_2}{x_2}$

This is often the fastest way to solve a problem.

Let's look at the basic definitions and graphs.

To solidify the core idea, please read the short introductory section in the following resource.

Now, let's see how to apply this to a word problem. The process generally involves two steps:

1. Use a given set of complete information (e.g., "when x is 2, y is 10") to find the constant k.
2. Use that constant k to find the missing value in an incomplete set of information (e.g., "find y when x is 5").

This video will walk you through a typical word problem.

2. Understanding Inverse Variation

Inverse variation (or inverse proportion) describes a relationship where one quantity increases as the other decreases. For example, if you increase the speed of your car, the time it takes to cover a fixed distance decreases. If you hire more workers for a job, the time to complete the job decreases.

Here, the product of the two quantities is always constant.

If a quantity $y$ varies inversely with a quantity $x$, we write this as:
$y\propto \frac{1}{x}$
This is translated into the equation:
$y=\frac{k}{x}$
Rearranging gives the core principle for inverse variation:
$xy=k(\text{The product is constant})$

For two pairs of values, ($x_1,y_1$) and ($x_2,y_2$), we have:
$x_1{y}_1=x_2{y}_2$

This is the fastest way to solve inverse variation problems. Let's watch the application.

3. Joint and Combined Variation

CAT problems often involve more than two quantities. This is where joint and combined variation come in.

• Joint Variation: A quantity varies directly with the product of two or more other quantities.

  • Example: "y varies jointly with x and z" means $y=kxz$.
  • Our partnership formula, Profit = k × Capital × Time, is a perfect example of joint variation.

• Combined Variation: A quantity varies directly with some quantities and inversely with others.

  • Example: "y varies directly with x and inversely with z" means $y=\frac{kx}{z}$.

The strategy is the same: translate the sentence into a single equation and solve for k.

Let's look at a typical CAT-level problem involving combined variation.

Worked Example:
The duration of a railway journey varies as the distance and inversely as the velocity. The velocity varies directly as the square root of the quantity of coal used, and inversely as the number of carriages in the train.

How do we combine this into one formula?
Let $T$ = duration, $D$ = distance, $V$ = velocity, $C$ = coal used, and $N$ = number of carriages.

1. From the first sentence: $T\propto \frac{D}{V}$ which means $T=k_1\frac{D}{V}$.
2. From the second sentence: $V\propto \frac{\sqrt{C}}{N}$ which means $V=k_2\frac{\sqrt{C}}{N}$.

Now, substitute the expression for $V$ from the second equation into the first one:
$T=k_1\frac{D}{k_2\frac{\sqrt{C}}{N}}=\left(\frac{k_1}{k_2}\right)\frac{D\cdot N}{V}$
Wait, I substituted V with an expression that contains V. Let's re-read. The second sentence defines V. So I should substitute the expression for V into the first equation.
$T=k_1\frac{D}{k_2\frac{\sqrt{C}}{N}}=\left(\frac{k_1}{k_2}\right)\frac{D\cdot N}{\sqrt{C}}$
Since $k_1$ and $k_2$ are both constants, their ratio ($k_1/k_2$) is just another constant. Let's call it $K$.
$T=K\frac{DN}{\sqrt{C}}$
This single equation captures all the relationships described. From here, you would use a set of given values to find $K$, and then use $K$ to solve for an unknown. This problem is similar to Problem 2 in the "Level of Difficulty (II)" section of the "Ratio, Proportion and Variation" PDF resource (LINK), which you can review later.

4. Practice Problems

Let's try a couple of problems from a CAT preparation resource. Use the proportionality/ratio method to solve them quickly.

Problem 1 (Question 3):
If x varies inversely as $y^3–1$ and is equal to 3 when y = 2, find x when y = 4.

Problem 2 (Question 4):
If x varies as y, and y = 4 when x = 12, find x when y = 15.

Conclusion

Today we've demystified variation. By recognizing whether quantities are related directly or inversely, you can quickly set up an equation and solve for unknowns.

Key Takeaways:

• Direct Variation: As one goes up, the other goes up. The ratio is constant. $\frac{y_1}{x_1}=\frac{y_2}{x_2}$
• Inverse Variation: As one goes up, the other goes down. The product is constant. $x_1{y}_1=x_2{y}_2$
• Joint/Combined Variation: Used for relationships involving more than two quantities. Combine all relationships into a single equation with one constant, $K$.
• Problem-Solving: The ratio/product shortcut is almost always faster than the two-step method of finding k first.

This lesson concludes our module on Commercial Mathematics. You've built a strong foundation in percentages, interest, partnerships, and variation.

In our next lesson, we will begin the Averages and Mixtures module. We'll start by learning to solve problems with simple, weighted, and combined averages, which are essential skills for the CAT.