Approximation and rounding

You are checking the prices on an online shop and find an accessory for your console that costs $24,9632$ €. Approximate this price by truncation to hundredths (that is, to euro cents).

In a Design and Technology class, you have measured the thickness of a part using digital callipers, obtaining the value $3,749$ mm. Approximate this value to tenths in two different ways:

1. a) By truncation.
2. b) By rounding.

Complete the following table in your notebook for the number $7,4628$. For each case, indicate whether the approximation obtained by rounding is rounded down or rounded up.

A precision sensor at a weather station records the relative humidity of the air. For data storage reasons, the computer system automatically rounds the value to the hundredths. If the screen shows a value of $64,38\,\%$, determine the exact interval (minimum value and maximum value) in which the actual humidity captured by the sensor lies before being rounded.

The figure shows the number line with the value on the screen and the two values with hundredths next to it.

The official time for a 100-metre sprint is measured to thousandths of a second. A runner took $10,437$ seconds.

1. a) If the news only reports the time rounded to the nearest hundredth of a second, what time will they state?
2. b) If another runner took $10,432$ seconds, what time would they be assigned using the same rounding criterion?
3. c) Would it be fair to use this rounding if both runners were competing in the same heat? Reason your answer based on precision.

The space probe Voyager 1 is approximately $2,348 \cdot 10^{10}$ kilometres from Earth. An astronomer decides to approximate this distance by truncating the number to the nearest thousand million (that is, keeping only two significant figures: $2,3 \cdot 10^{10}$ km).

In the figure, both values, the exact one and the approximate one, are marked on a number line.

Calculate the absolute error and the relative error made with this approximation. Express the relative error as a percentage rounded to the nearest thousandth.

In an electronics component factory, a rectangular part must measure $12,456$ mm long and $8,321$ mm wide. The machine operator rounds both measurements to one decimal place before making the cut.

1. Calculate the area of the part with the original measurements (unapproximated).

2. Calculate the area of the part with the rounded measurements.

3. Determine the absolute error made in the area calculation due to these prior roundings.