Direct & Inverse Proportions Class 8: Concepts & Word Problems

Think about it: if you buy $2$ kg of mangoes instead of $1$ kg, you pay double. If $4$ workers can build a wall in $12$ days, would $8$ workers take more time or less? These are questions about proportions, and they come up in everyday life constantly.

In NCERT Class 8 Math (Chapter 11: Direct and Inverse Proportions), you'll learn to identify whether two quantities are directly or inversely proportional, and use that knowledge to solve problems quickly and accurately. This chapter is a favourite in CBSE exams because the questions are practical and test your logical thinking.

Two quantities $x$ and $y$ are in direct proportion if:
- When $x$ increases, $y$ also increases (by the same factor).
- When $x$ decreases, $y$ also decreases (by the same factor).
- The ratio $\frac{x}{y}$ remains constant.

Mathematically: $\frac{x_1}{y_1} = \frac{x_2}{y_2} = k$ (a constant)

Or equivalently: $x = ky$

Real-life examples of direct proportion:
- More items purchased $\rightarrow$ more money spent (at the same rate).
- More distance to travel $\rightarrow$ more time needed (at constant speed).
- More workers $\rightarrow$ more work done (in the same time period).
- More fuel $\rightarrow$ more distance covered (at same efficiency).

There are two main approaches: the ratio method and the unitary method.

Example 1 (Ratio Method): If $5$ notebooks cost Rs. $120$, how much do $8$ notebooks cost?

Cost and number of notebooks are directly proportional.

So $8$ notebooks cost Rs. $192$.

Example 2 (Unitary Method): If a car travels $240$ km on $15$ litres of petrol, how far can it go on $25$ litres?

Step 1: Find the value for $1$ unit.

Step 2: Multiply for the required quantity.

Example 3: A map has a scale of $1$ cm $= 50$ km. If two cities are $3.5$ cm apart on the map, what is the actual distance?

Maps are a classic example of direct proportion!

Two quantities $x$ and $y$ are in inverse proportion if:
- When $x$ increases, $y$ decreases (and vice versa).
- The product $x \times y$ remains constant.

Mathematically: $x_1 \times y_1 = x_2 \times y_2 = k$ (a constant)

Or equivalently: $xy = k$

Real-life examples of inverse proportion:
- More workers $\rightarrow$ less time to finish the same job.
- Higher speed $\rightarrow$ less time for the same distance.
- More pipes filling a tank $\rightarrow$ less time to fill it.
- Wider road $\rightarrow$ fewer lanes of traffic congestion (for the same flow).

The key question to ask yourself: if I double one quantity, does the other halve? If yes, they're inversely proportional.

Example 1: If $6$ workers can build a wall in $10$ days, how long will $15$ workers take?

More workers means less time, so this is inverse proportion.

Example 2 (Unitary Method): A car travelling at $60$ km/h takes $4$ hours for a journey. How long would it take at $80$ km/h?

Step 1: Total distance $= 60 \times 4 = 240$ km (this is the constant product).

Step 2: At $80$ km/h: $\text{Time} = \frac{240}{80} = 3$ hours.

Example 3: A batch of bottles can be filled by $8$ machines in $6$ hours. Due to breakdown, only $5$ machines are working. How long will they take?

Notice how the answer can be a non-whole number. In real life, machines don't always give you neat answers!

The unitary method works for both direct and inverse proportion. The idea is simple: find the value for one unit, then scale to the required quantity.

For direct proportion:
- $5$ items cost Rs. $200$.
- $1$ item costs Rs. $\frac{200}{5} = 40$.
- $12$ items cost Rs. $40 \times 12 = 480$.

For inverse proportion:
- $6$ workers finish a job in $10$ days.
- $1$ worker would take $6 \times 10 = 60$ days (more workers = less time, so $1$ worker = maximum time).
- $4$ workers would take $\frac{60}{4} = 15$ days.

The unitary method is incredibly versatile and works even when you can't immediately see whether the proportion is direct or inverse. Master it!

Sometimes more than two quantities are involved. This is called compound proportion.

Example: If $8$ workers working $6$ hours a day can build a wall in $10$ days, how many days will $12$ workers working $8$ hours a day take?

Let the required days $= x$.

• More workers $\rightarrow$ fewer days (inverse): $\frac{8}{12}$
  - More hours per day $\rightarrow$ fewer days (inverse): $\frac{6}{8}$

For each pair of quantities, decide whether the relationship with the unknown is direct or inverse, then multiply the ratios accordingly.

Here's your quick-reference summary:

• Direct proportion: $\frac{x_1}{y_1} = \frac{x_2}{y_2}$. Both quantities increase or decrease together.
  - Inverse proportion: $x_1 \times y_1 = x_2 \times y_2$. One increases while the other decreases.
  - The unitary method (find value for $1$ unit) works universally for both types.
  - Compound proportion handles situations with more than two varying quantities.
  - Always identify the type of proportion before solving.
  - Always verify that your answer makes logical sense (e.g., more workers should mean less time).

Head over to SparkEd and practice direct and inverse proportion problems with instant feedback. The more you practice, the more natural these problems become!