Approximation and Rounding
A middle school runner finishes the 100 meters in $12.45$ seconds. The scale at the corner store reads $2.341$ kg of oranges. You almost never need all those digits: knowing how to round and truncate gives you just the right number, without losing what matters. What is the runner's time with a single decimal place?
Rounding and truncating
Move the slider and compare how many decimal places are kept.
Look at the following decimal numbers and round them to one decimal place (tenths): a) $3.14$ b) $9.68$ c) $0.45$ d) $1.03$
Look at the following decimal numbers and round them to one decimal place (tenths):
- a) $3.14$
- b) $9.68$
- c) $0.45$
- 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…
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:
| Approximation to hundredths | Result |
|---|---|
| Truncation | |
| Rounding |
Copy and complete the following table in your exercise book, finding the given approximations for the decimal number $n = 14.3728$. | Approximation to... | Truncation |…
Copy and complete the following table in your exercise book, finding the given approximations for the decimal number $n = 14.3728$.
| Approximation to... | Truncation | Rounding |
|---|---|---|
| Tenths | ||
| Hundredths | ||
| Thousandths |
When do they give the same result and when not?
Truncating and rounding sometimes give the same number and sometimes they don't. Look at the case of $7.428$ approximated to the hundredths: truncating gives $7.42$ and rounding gives $7.43$. On the number line, $7.428$ is closer to $7.43$.
A 100-meter sprinter recorded a time of $12.45$ seconds on the stopwatch. If we want to express this time to one decimal place: a) Round to one decimal place (tenths).…
A 100-meter sprinter recorded a time of $12.45$ seconds on the stopwatch. If we want to express this time to one decimal place:
- a) Round to one decimal place (tenths).
- b) Was it rounded up or rounded down? Explain why you looked at the hundredths digit.
Approximate the number $\pi \approx 3.141592...$ in the following ways: a) By rounding to three decimal places (thousandths). b) By truncation to three decimal places…
Approximate the number $\pi \approx 3.141592...$ in the following ways:
- a) By rounding to three decimal places (thousandths).
- b) By truncation to three decimal places (thousandths).
- c) In which case is the result larger?
A mathematics teacher calculates the average mark for an exam and gets $6.4782$. a) Approximate the mark to one decimal place by truncation. b) Approximate the mark to…
A mathematics teacher calculates the average mark for an exam and gets $6.4782$.
- a) Approximate the mark to one decimal place by truncation.
- b) Approximate the mark to one decimal place by rounding.
- 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?
At a budget gas station, the price per liter of Unleaded 95 gas is $1.419$ pesos. A group of friends fills up the tank of their van for a trip with $52.6$ liters. The…
At a budget gas station, the price per liter of Unleaded 95 gas is $1.419$ pesos. A group of friends fills up the tank of their van for a trip with $52.6$ liters.
The gas station's register 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 gas station?
Rounding in real life
Sometimes it is better to approximate in only one direction (always up or always down), not to the nearest number.
Money is rounded too: prices are shown to the cents, that is, to the hundredths of a peso. For example, $\$12.957$ is charged as $\$12.96$.
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: a) Between…
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:
- a) Between which two integers does the price lie?
- b) Which of the two numbers is it closer to? Perform rounding to the nearest whole number.
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 $3.27$…
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.
- a) Calculate the perimeter of the garden.
- b) If the garden owner buys the fence from a hardware 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?
In an app store for video games, the price of a new texture pack is $\$12.957…
In an app store for video games, the price of a new texture pack is $\$12.957$.
- a) If the store decides to truncate the price to two decimal places to display it in the catalog, what price will appear?
- b) If the payment system rounds to two decimal places (pesos cents) when taking the final payment, how much will the user actually pay?
- c) What is the difference in price between the truncated price and the rounded price?
Back to the start: to the tenths, the runner's time ($12.45$ s) rounds to $12.5$ s, because the hundredths digit is $5$. The weight of the oranges ($2.341$ kg) becomes $2.3$ kg, because the hundredths digit is less than $5$.