Why rounding errors grow when you multiply, and the bound notation that is examined every session.
A rectangle measured as 12 by 7, each side to the nearest whole unit. The outer box is the largest it could really be, the inner box the smallest. Press play to see the range.
Neither side was out by more than half a unit.
If x = 4.1 to one decimal place, then 4.05 ≤ x < 4.15. Half the rounding unit either side, and note the inequality: the lower bound is included, the upper one is not, because 4.15 would round up.
| Given | Rounding unit | Bounds |
|---|---|---|
| 4.1 to 1 dp | 0.1 | 4.05 ≤ x < 4.15 |
| 12 to the nearest unit | 1 | 11.5 ≤ x < 12.5 |
| 350 to 2 sf | 10 | 345 ≤ x < 355 |
| 0.0046 to 2 sf | 0.0001 | 0.00455 ≤ x < 0.00465 |
Multiply quantities and the percentage errors roughly add. Two sides at about 4% each give an area at about 8%, and the extremes push it to 11.6%. This is why a long calculation from rounded data can end up far less precise than any single measurement in it.
percentage error = |approximate − exact||exact| × 100. Divide by the exact value, not the approximation, and the absolute value keeps it positive.
A plot of land is measured as 38 m by 24 m, each to the nearest metre. A fence costs 450 baht a metre, to the nearest 10 baht.
Find the greatest possible cost of fencing the perimeter.
1. A length is 350 m to 2 significant figures. State the lower bound.
2. 3.14 is used for π. Find the percentage error to 3 significant figures.
3. Sides measured as 12 and 7, each to the nearest unit. The greatest possible area is:
Bounds written with the right inequality signs, ≤ below and < above.
Pairing the bounds correctly: least times least for the smallest area, and for a difference or a quotient the pairing flips, so think rather than copy.
Dividing by the exact value in a percentage error, and rounding only at the very end.
These three are deliberately not all about this page. In an examination the hardest step is deciding which method applies, and questions that arrive straight after the method never make you decide. Answer from memory before you reveal anything.
Come back to these in a week, and again in a month.
Which method?
log 4 + log 25. What is it?
Which method?
A length is 4.1 m to one decimal place. Write down the bounds.
Which method?
A sequence is 3, 6, 12, 24. Find the sum of the first 10 terms.
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