Approximation and rounding

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:

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) $1.03$

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.

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

A 100-metre 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?

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?

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

On the number line in the figure, the two long marks are two consecutive whole numbers (the ones marked with “?”) and each small mark is one tenth.

1. a) Between which two whole numbers does the price lie?
2. b) Which of the two numbers is it closer to? Round to the nearest unit.

Imagine you are going to pay with a $20$ € note for a purchase that costs $14.23$ €. The cashier has no small cent coins and suggests rounding the price to the nearest tenth of a euro (that is, to a multiple of $10$ cents, with a single decimal digit).

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

In a local stationery shop, Sara wants to buy a notebook that costs $2.85$ €, a pack of markers for $4.15$ €, and a folder for $1.90$ €. She has a $5$ € coin and two $2$ € coins in her purse.

1. a) Before calculating the exact price, estimate the total expenditure by rounding the price of each item to the nearest whole euro. According to your previous estimate, will she have enough money to pay for the purchase?
2. b) Calculate the exact cost of the purchase by adding the three prices.
3. c) Check whether your estimate was accurate by calculating the difference between the exact price and the estimated price. Do you think the estimate was useful for making a quick decision before going to the checkout? Justify your answer.

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 $4.56$ metres and $3.27$ metres.

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 metres and always rounds up (rounding up to the nearest unit) to make sure there is enough material, how many metres of fencing will he buy?