7.4 Upper and lower bounds — Learn

Prerequisites: error intervals and substituting values into a formula.

Before you begin

You should be able to write error intervals and substitute values into a formula.

Objectives

• Identify the lower and upper bounds of a rounded measurement.
• Use bounds to find possible limits for a calculated quantity.
• State whether an endpoint can be reached.

Bounds of one measurement

The lower bound is the smallest possible value consistent with the rounding. The upper bound is the greatest boundary; the measurement is less than it.

Example 1. A length is $6.4$ cm to the nearest $0.1$ cm.

So the lower bound is $6.35$ cm and the upper bound is $6.45$ cm.

Bounds in calculations

For positive values, a maximum quotient uses the largest numerator and smallest denominator. A minimum quotient uses the smallest numerator and largest denominator.

Example 2. A journey is $120$ km to the nearest kilometre and takes $2.0$ hours to the nearest tenth. Find the greatest possible speed.

The speed is less than $120.5\div1.95\approx61.7949$ km/h. To 3 significant figures, this is about $61.8$ km/h.

Watch out. A common error is to divide the upper distance by the upper time. For the greatest speed, use the upper distance and lower time: a smaller positive denominator makes the quotient larger.

Progress check. A rectangle has length $5.2$ m and width $3.1$ m, each rounded to the nearest $0.1$ m. Which bounds give its maximum area?

You should now be able to

• find lower and upper bounds from a rounded measurement;
• select bounds to maximise or minimise a positive calculation;
• explain why an endpoint is included or excluded.