Approximation and Rounding

In the decimal place value system, each digit has a value according to its position. Given the number $58.347$, state which digit occupies each of the following positions:

1. a) The tenths.
2. b) The hundredths.
3. c) The thousandths.
4. d) The units.

Look at the following number line where point $P$ is marked. Write down the exact decimal number represented by $P$ and then round this value to the nearest whole number.

You are checking the prices on an online store and find an accessory for your console that costs $24.9632$ pesos. Approximate this price by truncation to hundredths (that is, to pesos 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.

State whether the following statements about approximating $4.572$ are true or false:

1. a) Truncating to tenths gives $4.5$.
2. b) Rounding to tenths gives $4.5$.
3. c) Rounding to hundredths gives $4.57$.
4. d) Truncating to units gives $4$.

An athlete recorded a time of $10.438$ seconds in a 100-meter sprint. Round their time to hundredths. Was the rounding an overestimate or an underestimate?

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.

Consider the irrational number $\pi \approx 3.14159265...$. In the figure you can see the value of each decimal place.

1. a) Truncate to the ten-thousandths.

1. b) Round to the ten-thousandths.

1. c) Compare both results and explain why, in this particular case, rounding and truncation do not coincide.

Approximate the number $0.666...$ (pure recurring) in the following ways:

1. a) Truncating to hundredths.
2. b) Rounding to hundredths.
3. c) Rounding to tenths.

If the side of a square measures $6.8$ cm, estimate its area by first rounding the side length to the nearest whole number. Draw a sketch of the square showing the approximate measurement.

You can do it on paper and send a photo with your answer.

In a computer store, the prices of three components are $15.99$ pesos, $32.45$ pesos and $12.80$ pesos.

If you decide to estimate the total cost by rounding to the nearest whole pesos before adding, what is the difference between the actual total price and your estimate? Have you overestimated or underestimated?

A tailor needs to buy fabric to make 7 cushion covers. Each cover requires exactly $1.428$ meters of fabric.

1. a) Calculate the exact total amount of fabric needed.
2. b) If the tailor decides to buy the fabric by rounding the total obtained to one decimal place, will he have fabric left over or be short? Calculate the exact value of that difference.

The official time for a 100-meter 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? Explain your answer based on precision.

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.

A farmer has measured the area of his rectangular plot of land and obtained a value of $124.3756$ $m^2$.

1. a) Calculate the value of the area approximated by rounding to two decimal places.
2. b) Calculate the absolute error made when making this approximation. Remember that the absolute error is $E_a = |V_{real} - V_{aproximado}|$.

The space probe Voyager 1 is approximately $2.348 \times 10^{10}$ kilometers 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: $2.3 \times 10^{10}$ 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.

A transport company calculates the cost of fuel for a long trip. The price of diesel is $1.387$ pesos/liter and the truck consumes $32.4$ liters per 100 km. The trip is 450 km long.

1. a) Calculate the exact total cost of fuel for the trip.
2. b) If the company estimates the cost by rounding the price per liter to hundredths and the consumption to units (liters), what would the estimated budget be?
3. c) Calculate the difference between the actual cost and the estimated cost. Is it a safe estimation for the company (an overestimate) or a risky one (an underestimate)?

Two methods of approximation for the irrational number $\pi$ are being compared.

• Method A uses rounding to four decimal places ($3.1416$).

• Method B uses Archimedes' fraction $\frac{22}{7}$.

Calculate the absolute error of each method given that the value of $\pi$ to seven decimal places is $3.1415926$. Which of the two methods is more precise? Justify your answer by comparing the errors.

Consider the mixed recurring decimal $n = 4.12\widehat{6}$, that is, $n = 4.12666\ldots$ (the 6 repeats indefinitely).

1. a) Obtain its generating fraction simplified as much as possible.

1. b) Approximate the number $n$ by rounding to the thousandths.

1. c) Calculate the absolute value of the difference between the actual value (using the generating fraction) and the approximation from part b). Which error concept does this result correspond to?

Represent the pure recurring decimal $2,\widehat{6}$ on a number line. To do this:

1. a) State which two integers it lies between.
2. b) State which number you would get if you rounded to one decimal place.
3. c) What is the maximum error bound when rounding to one decimal place?

An influencer is analysing their follower growth. On Monday they had $1,240,500$ followers. On Tuesday, their audience grew by $0.125\,\%$.

1. Calculate the exact number of new followers.

2. If the analytics system only displays whole numbers and uses truncation, what total number of followers will the influencer see on Tuesday?

3. What is the relative error (as a percentage) made by the platform when displaying the truncated figure compared to the actual growth?

We want to design a square plate whose exact area is $30$ cm². Since we cannot measure with infinite precision, we approximate the side of the square ($\sqrt{30}$) by rounding to the hundredths.

1. a) What is the value of the approximate side?

1. b) If we make the plate with that approximate side, what will its actual area be?

1. c) Calculate the absolute error made in the area with respect to the original $30$ cm².

A courier company uses drones to deliver parcels. The navigation software estimates the arrival time by dividing the distance by the average speed.

For a journey of $15$ km at a speed of $35$ km/h:

1. Calculate the exact time in hours (in fraction form and as a recurring decimal).

2. The software provides an estimate by rounding the time to whole minutes.

3. Calculate the relative error made by the software when informing the customer in minutes, comparing it with the actual time (express both times in the same unit).