After measuring 120 metres of a rope, it was discovered that the measuring metre rod was 3 cm longer. The true length of the rope measured is

### Concept & Logic This is a measurement error problem based on direct proportionality. If the measuring instrument is longer than its stated value, the actual measured length will be proportionally longer. $$ \text{True Length} = \text{Measured Units} \times \text{True Length per Unit} $$ ### Step-by-Step Solution * **Identify the true unit length:** A standard metre rod is meant to be 1 metre (100 cm). The faulty rod is 3 cm longer, meaning its true length is $100 + 3 = 103$ cm, which equals 1.03 metres. * **Identify the number of units measured:** The rope was measured as "120 metres" using the faulty rod. This means the physical rod was laid down end-to-end exactly 120 times. * **Calculate the actual physical length:** Since each physical lay of the rod actually covered 1.03 metres: $$ \text{True Length} = 120 \times 1.03 \text{ metres} $$ * **Perform the calculation:** $120 \times 1.03 = 120 \times \left(1 + \frac{3}{100}\right) = 120 + 3.6 = 123.6$ metres. * **Format the final unit:** 123.6 metres is mathematically equivalent to 123 metres and 60 centimetres. ### Exam Strategy & Shortcut Use the concept of total error accumulation. The error is +3 cm for every metre measured. For 120 metres, the total accumulated error is $120 \times 3 \text{ cm} = 360 \text{ cm}$. Since 360 cm is exactly 3.6 metres, simply add this total error to the base measurement: $120 + 3.6 = 123.6$ metres. This avoids decimals entirely. ### Common Pitfall A very common mistake is subtracting the error (thinking the rod "overmeasured" so the true length is less). Remember, a longer rod covers more physical ground per "metre" reading, so the actual physical rope must be longer than the stated number. ### Final Answer **Therefore, the correct answer is 123 m 60 cm.**