Bounds from Rounding

Statement

When a value is rounded to the nearest unit \(u\), the original number must lie within a range (or bound) around the rounded value. This range is given by:

\[ x \in \left[x_{\text{stated}} - \tfrac{u}{2}, \; x_{\text{stated}} + \tfrac{u}{2}\right) \]

The lower bound is the stated value minus half the unit, and the upper bound is the stated value plus half the unit (but not including it).

Why it’s true (short reason)

• Rounding chooses the nearest multiple of a unit \(u\).
• Any number within half a unit above or below would round to the same stated value.
• Thus the interval of possible original values is \([x_{\text{stated}} - u/2, \; x_{\text{stated}} + u/2)\).

Recipe (how to use it)

1. Identify the stated rounded value.
2. Determine the rounding unit \(u\) (e.g. nearest 10 → \(u=10\), nearest 0.1 → \(u=0.1\)).
3. Lower bound = stated value \(- u/2\).
4. Upper bound = stated value \(+ u/2\).
5. Write the result as an interval.

Spotting it

Use bounds when:

• A value is given rounded (to nearest integer, 10, 0.1, etc.).
• You are asked for the “range of possible values.”
• The question mentions error bounds, limits, or inequalities.

Common pairings

• Measurement problems in geometry and physics.
• Percentage error calculations.
• Upper and lower bounds in multiplication or division.

Mini examples

1. \(x = 37\) rounded to the nearest 1 → bounds: \([36.5, 37.5)\).
2. \(y = 120\) rounded to nearest 10 → bounds: \([115, 125)\).
3. \(z = 5.4\) rounded to nearest 0.1 → bounds: \([5.35, 5.45)\).

Pitfalls

• Forgetting to halve the unit \(u\).
• Including the upper bound when it should be excluded.
• Using wrong unit (e.g. thinking nearest 10 means \(u=1\)).

Exam strategy

• Underline the unit of rounding before working.
• Always write both the lower and upper bound.
• Check whether the interval is open at the top \(( )\) or closed \([ ]\).
• In calculations, use upper × upper and lower × lower for extreme cases.

Summary

Bounds describe the range of possible original values after rounding. The lower bound is half a unit below the rounded value, and the upper bound is half a unit above. This ensures precise answers in exam problems involving measurements and error analysis.