This activity is about calculating **upper and lower bounds**.

Suppose we've rounded a number to the nearest whole number and got 14.

Numbers that round **up** to 14 include 13.9, 13.75, 13.694, 13.5, etc.

The smallest of these is 13.5.

This is then called the **lower bound**.

Anything **smaller than 13.5** will round **down **to 13.

Larger numbers that round **down** to 14 include 14.1, 14.34968, 14.49, etc.

The largest of these is 14.4999999...

This is the **upper bound**, **which we write as 14.5.**

**Example 1:**

A room measures 433 cm by 372 cm, measured to the nearest centimetre.

Find the **greatest **and **smallest **possible areas** **of the room.

**Answer:**

Upper bounds of measurements: 433.5 cm and 372.5 cm.

Greatest area is 433.5 × 372.5 = **161,478.75** cm^{2}.

Lower bounds of measurements: 432.5 cm and 371.5 cm.

Smallest area is 432.5 × 371.5 = **160,673.75 **cm^{2}.

**Example 2:**

A sprinter runs 100 m in 12.8 seconds.

The distance is measured to the nearest centimetre and the time to the nearest tenth of a second.

Find the fastest** **and slowest** **possible speeds of the sprinter.

**Answer:**

Upper bounds of measurements: 100.005 m and 12.85 s

Lower bounds of measurements: 99.995 m and 12.75 s.

The formula for finding speed is: speed = distance ÷ time

Be careful here!

For the **fastest** speed, we want the **maximum** distance in the **minimum** time:

The fastest speed = 100.005 ÷ 12.75 = 7.84 m/s (3 s.f.)

For the **slowest*** *speed, we want the* ***minimum distance** in the **maximum** time:

The slowest speed = 99.995 ÷ 12.85 = 7.78 m/s (3 s.f.)

Now it's over to you.