Approximation and Rounding

Look at the following decimal numbers and round them to one decimal place (tenths):

1. a) $3.14$
2. b) $9.68$
3. c) $0.45$
4. d) $1.03$

To carry out an experiment in Science class, we need to measure exactly $4.576$ grams of salt. Complete the following table by applying the two requested approximation methods:

Copy and complete the following table in your exercise book, finding the given approximations for the decimal number $n = 14.3728$.

In the supermarket, fruit scales usually round the weight. If we buy a bag of oranges that weighs $2.341$ kg, state what weight would be displayed if the scale:

1. a) Rounds to the units.
2. b) Rounds to the hundredths.

The download speed of an internet connection is $92.578$ Mbps. The technician tells us that, for simplicity, we can truncate the number to two decimal places (hundredths).

1. a) What value will we obtain?
2. b) Draw a small number line and plot the original value and the truncated value.

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

A 100-meter sprinter has recorded a time of $12.45$ seconds on the stopwatch. If we want to express this time with a single decimal digit:

1. a) Round to the nearest tenth.
2. b) Was it an upward or a downward approximation? Explain why you looked at the hundredths digit.

On the number line, the runner's time is marked with a point between $12.4$ and $12.5$, and each small mark represents $0.01$ s.

Approximate the number $\pi \approx 3.141592...$ in the following ways:

1. a) By rounding to three decimal places (thousandths).
2. b) By truncation to three decimal places (thousandths).
3. c) In which case is the result larger?

Look at the following number line where a decimal number $x$ is shown under a magnifying glass.

Knowing that the orange dot represents the number $x = 3.774$:

1. a) Between which two hundredths does the number $x$ lie?
2. b) If you had to approximate $x$ to tenths by rounding, which value would you choose? Justify your answer by looking at the position of the dot on the line relative to the midpoint ($3.75$).

Three athletes took part in a 60-meter sprint at school. The times recorded by the stopwatch were as follows:

• Lucía: $8.429$ seconds.

• Marcos: $8.421$ seconds.

• Adrián: $8.425$ seconds.

1. a) If the results are published on the school website rounded to two decimal places (hundredths), what time will appear for each person?
2. b) Looking at the rounded results, does it seem that they took the same time? Is that true according to the original times?

A mathematics teacher calculates the average mark for an exam and gets $6.4782$.

1. a) Approximate the mark to one decimal place by truncation.
2. b) Approximate the mark to one decimal place by rounding.
3. c) Which of the two approximations is more favourable to the student? Why do you think school report cards usually use rounding instead of truncation?

Think and answer this decimal number riddle:

"I am a number with three decimal places. If you truncate me to two decimal places (hundredths), I am $7.42$. However, if you round me to two decimal places, I am $7.43$."

Write three different numbers that satisfy these conditions. What is the smallest possible number it could be? And the largest?

In a video game store, the price of a poster is $7.82$ pesos. If you only have one-pesos coins and want to know approximately how many whole pesos it costs:

1. a) Between which two integers does the price lie?
2. b) Which of the two numbers is it closer to? Perform rounding to the nearest whole number.

Imagine that you are going to pay with a $20$ pesos bill for a purchase that costs $14.23$ pesos. The cashier has no small change and suggests rounding the price to the nearest 10 cents (that is, to one decimal place).

1. a) What would be the new rounded price?
2. b) How much change would the cashier give you after rounding?

We want to put a protective fence around a small rectangular garden. When measuring with a digital tape measure, we find that the sides measure $4.56$ meters and $3.27$ meters.

1. a) Calculate the perimetre of the garden.
2. b) If the garden owner buys the fence from a DIY store that only sells in whole meters and always rounds up (rounding up to the nearest unit) to make sure there is enough material, how many meters of fencing will he buy?

At a budget petrol station, the price per liter of Unleaded 95 petrol is $1.419$ pesos. A group of friends fills up the tank of their van for a trip with $52.6$ liters.

The petrol station's till is programmed to calculate the total and then round the final result to the nearest pesos cent (two decimal places). However, one of the friends insists that they should first truncate the price per liter to the nearest cent and then multiply by the number of liters.

Calculate the difference in price between both methods. Which of the two benefits the customer and which benefits the petrol station?

A modern chef needs to weigh three ingredients using high-precision scales that measure to thousandths of a gram:

• Salt: $2.341$ g

• Pepper: $1.256$ g

• Saffron: $0.897$ g

The chef wants to find the total weight of the spice mixture. Compare these two procedures:

1. Procedure 1: Add the three exact weights and then round the final result to tenths.

2. Procedure 2: Round each ingredient to tenths and then add them.

Calculate both results. Are they equal? Explain why rounding individual values can lead to a larger error than rounding at the end of the calculation.

In a biology laboratory, the growth of a bacterium is being measured. Its current length is $0.03478$ millimeters.

• Scientist A decides to use an approximation by rounding to three decimal places (thousandths).

• Scientist B prefers to use an approximation by truncation to four decimal places (ten-thousandths).

State which of the two values is larger and calculate the exact difference between both approximations. Which of the two do you think is "closer" to the actual value of the bacterium?

Look at the following number line where two approximations of the number $X$ are shown.

Knowing that the truncation to hundredths is $3.14$ and the rounding to hundredths is $3.15$, state whether the following statements are true or false, justifying each one:

• a) The thousandths digit of $X$ must be greater than or equal to $5$.

• b) The number $X$ could be $3.1449$.

• c) If we add $0.001$ to the number $X$, its truncation to hundredths will still be $3.14$.

A 7th grade class is designing a part for a 3D printer. The software gives a measurement of $12.666...$ (repeating) millimeters for the thickness of a fitting. The printer has three precision modes:

• Eco Mode: Truncates to tenths.

• Standard Mode: Rounds to hundredths.

• Pro Mode: Rounds to thousandths.

Calculate the value that the printer would use in each mode. If the slot where the part must fit measures exactly $12.67$ mm, in which modes will the part fit and in which will it not? (Assume that to fit, the part must be less than or equal to the slot size).

We are investigating a "mystery" decimal number, which we will call $N$, which has three decimal places. We know the following clues about it:

1. Its integer part is the smallest two-digit prime number.

2. If we round it to two decimal places (hundredths), we get $11.38$.

3. If we truncate it to one decimal place (tenths), we get $11.3$.

4. The sum of its three decimal digits is $19$.

Which number is it? Reason your answer by ruling out the options that do not meet the conditions.

Find a decimal number that satisfies these three "boundary" conditions simultaneously:

• It is greater than $7.45$ and less than $7.46$.

• When rounded to thousandths, the result is $7.458$.

• The sum of its four decimal digits is equal to $25$.

Is this number unique or are there several possibilities? Justify your answer.

Create a real-life problem (shopping, sports, science...) where both of these conditions are met simultaneously:

1. It is necessary to use a number with at least four decimal places.

2. The difference between truncating that number to hundredths and rounding it to hundredths is exactly $0.01$.

Write the full question statement and explain how it would be solved using the rounding and truncation rules.