When an error of 1% is made in the length and breadth of a rectangle, the…

Concept: When a quantity is the product of two measured quantities, the relative (fractional) errors combine through the product. If A = l × b, and l carries a relative error a while b carries a relative error c, then the measured area is A′ = l(1 + a) × b(1 + c) = A(1 + a + c + ac), so the fractional error in the product is a + c + ac. A measurement error is quoted as a magnitude, not a signed value, so the percentage error in a product is reported by convention as the MAXIMUM possible error: add the magnitudes of the individual percentage errors and drop the second-order cross term ac, which is negligible for small errors. In short, for a product the percentage errors add.

Application: Each dimension here is quoted with an error of magnitude 1%, so |a| = |c| = 0.01. Apply the rule and then check it against the exact product.

1. Write each measured dimension as its true value times a scaling factor, the sign of the error being unknown: l′ = l(1 ± 0.01) and b′ = b(1 ± 0.01).

2. Multiply the two to get the measured area: A′ = l′b′ = A(1 ± 0.01)(1 ± 0.01).

3. Apply the first-order rule — add the magnitudes of the two percentage errors: maximum percentage error in the area = 1% + 1% = 2%.

4. Confirm against the exact worst case, both dimensions erring the same way: (1.01)2 = 1 + 0.01 + 0.01 + 0.0001 = 1.0201, so A′ − A = 0.0201 A, an error of 2.01%. Both erring low gives (0.99)2 = 0.9801, an error of 1.99% in magnitude.

5. The 0.0001 term is the second-order cross term the convention discards. Dropping it leaves 0.02, i.e. 2%, which is the offered value.

Cross-check: Verify the arithmetic on a concrete rectangle, taking the worst case in which both dimensions are measured 1% high.

The measured area is out by 10201 − 10000 = 201 units, and 201 / 10000 = 0.0201 = 2.01%, which is 2% to first order. Note why the maximum is the quantity reported: had one dimension been measured 1% high and the other 1% low, the area would read 101 × 99 = 9999, an error of only 0.01%. Because the signs are not given, the percentage error of a product is quoted over the sign combination that makes it largest.

Contrast: It is worth seeing what each of the other percentages would actually require.

• 0% would need the two scaling factors to multiply to exactly 1. Even equal-magnitude errors in opposite directions do not achieve that: 1.01 × 0.99 = 0.9999, leaving a residual 0.01%. Beyond that, this is the sign combination that makes the error smallest, not the maximum the convention reports.

• 1% is the maximum shift a single scaled dimension produces on its own — the figure if only length, or only breadth, were mismeasured and the other were exact.

• 4% comes from double-counting: either applying the doubling rule twice (1% → 2% → 4%), or charging the error to all four sides of the rectangle instead of to its two independent dimensions.

Result: The maximum percentage error in the area is 1% + 1% = 2% (2.01% if the second-order term is kept), so the offered value is 2. The general rule is worth memorising: for a product or a quotient of two quantities the percentage errors add, and for a quantity raised to the power n the percentage error is multiplied by n.