Upper and Lower Bounds

Grade 6

What bounds are for

When a measurement is given as a rounded value, the actual value could be a little higher or a little lower. Bounds tell you exactly how far that value could stray before it would round to a different number.

The key insight: any value within half the last rounded digit of your number would still round to that number. That half-unit is what fixes the boundary.

The half-the-last-digit rule

To find the bounds of a rounded value, take half of the unit of the last digit given and add or subtract it from the rounded number.

Lower bound = rounded value − ½ × (last digit unit)

Upper bound = rounded value + ½ × (last digit unit)

If a value is rounded to the nearest 10, the last digit unit is 10, so you add or subtract 5. Rounded to 1 d.p.? Last digit unit is 0.1, so add or subtract 0.05.

Writing the error interval

The range of possible actual values is called the error interval. It's always written as an inequality in the form:

lower bound ≤ x < upper bound

Notice the lower bound uses ≤ (the actual value could equal the lower bound exactly) but the upper bound uses < (if x actually equalled the upper bound, it would round up to the next value — so it can't reach it).

Worked example

EXAMPLE

A length, L, is measured as 8.4 cm, rounded to 1 decimal place. Write down the error interval for L.

The last digit is in the tenths column, so the last digit unit is 0.1. Half of that is 0.05. Lower bound: 8.4 − 0.05 = 8.35 Upper bound: 8.4 + 0.05 = 8.45 Error interval: 8.35 ≤ L < 8.45

EXAMPLE

A crowd is reported as 2 400 people, rounded to the nearest 100. Write the error interval.

Last digit unit is 100. Half of 100 is 50. Lower bound: 2 400 − 50 = 2 350 Upper bound: 2 400 + 50 = 2 450 Error interval: 2 350 ≤ n < 2 450

The upper bound is never included — always use a strict < inequality there. Examiners will mark it wrong if you write ≤ on both sides.

Try this one

A formula used in physics is:

\[ T = \frac{x^2 + y^2}{w} \]

where \( x = 14.6 \) (measured to 3 significant figures), \( y = 9.3 \) (measured to 2 significant figures) and \( w = 52 \) (measured to the nearest integer).

Calculate the upper bound for \( T \). Give your answer to 3 significant figures. [3]

Nova's hint:

This is about finding the upper bound of a formula involving multiple measurements with different levels of accuracy.

Ask yourself: 'which combination of bounds for each variable gives the largest possible value of the formula?' For a fraction, the upper bound comes from making the numerator as large as possible and the denominator as small as possible. Identify the bounds of each variable separately first.

Try to Solve it with Nova?