Unitary method — solving proportion problems step by step

Question

If 8 notebooks cost Rs 120, find the cost of 15 notebooks using the unitary method.

(CBSE Class 6-7 pattern)

Solution — Step by Step

This is the heart of the unitary method — always find the value of 1 unit first.

\text{Cost of 1 notebook} = \frac{120}{8} = \text{Rs } 15

Now that we know the cost of 1, finding the cost of any number is just multiplication:

\text{Cost of 15 notebooks} = 15 \times 15 = \mathbf{\text{Rs } 225}

We can verify: $\dfrac{8}{120} = \dfrac{15}{x}$

$x = \dfrac{120 \times 15}{8} = \dfrac{1800}{8} = 225$ ✓

flowchart TD
    A["Given: 8 items cost Rs 120"] --> B["Step 1: Find cost of 1 item"]
    B --> C["120 ÷ 8 = Rs 15 per item"]
    C --> D["Step 2: Multiply by required quantity"]
    D --> E["15 × 15 = Rs 225"]
    E --> F["Answer: 15 notebooks cost Rs 225"]

Why This Works

The unitary method works because we are dealing with direct proportion — when the number of items doubles, the cost also doubles. By finding the cost of exactly one unit, we create a “conversion factor” that lets us scale to any quantity.

The word “unitary” comes from “unit” — we reduce everything to one unit first. This simple idea is incredibly powerful and forms the basis for percentages, speed-distance-time, and even advanced ratio problems later.

Alternative Method — Ratio and Proportion

Set up the proportion directly without finding the unit value:

\frac{8}{15} = \frac{120}{x}

Cross multiply: $8x = 120 \times 15 = 1800$

$x = 225$

The unitary method always works in two steps: divide first, then multiply. For direct proportion, divide the given value by the given quantity, then multiply by the required quantity. For inverse proportion (like workers and days), the order reverses — multiply first, then divide.

Common Mistake

Students sometimes divide when they should multiply (or vice versa). Ask yourself: “If I buy MORE notebooks, should the cost be MORE or LESS?” More items = more cost, so the final answer must be bigger than Rs 120. If your calculation gives a number smaller than 120, you have divided when you should have multiplied. This common-sense check catches most errors.