Exercises: Approximation and rounding

In a video game store, the price of a poster is $\$7,82$. If you only have one-dollar bills and want to know approximately how many whole dollars it costs:

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

To carry out an experiment in Science class, we need to measure exactly $4,576$ grams of salt. Complete the following table applying the two requested approximation methods:

Approximation to the nearest hundredth	Result
Truncation
Rounding

A 100-meter sprint runner has recorded a time of $12,45$ seconds on the stopwatch. If we want to express this time with a single decimal place:

1. a) Round to the nearest tenth.
2. b) Has it been an approximation up or down? Explain why you looked at the hundredths digit.

Look at the following decimal numbers and approximate them to the nearest tenth using the rounding technique:

1. a) $3,14$
2. b) $9,68$
3. c) $0,45$
4. d) $1,03$

The download speed of an internet connection is $92,578$ Mbps. The technician tells us that, to simplify, we can truncate the number to the nearest hundredth.

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

In the supermarket, fruit scales usually round the weight. If we buy a bag of oranges weighing $2,341$ lb, indicate what the displayed weight would be if the scale:

1. a) Rounds to the nearest unit.
2. b) Rounds to the nearest hundredth.

Approximate the number $\pi \approx 3,141592...$ in the following ways:

1. a) By rounding to the nearest thousandth.
2. b) By truncation to the nearest thousandth.
3. c) In which case is the result greater?

Imagine you are going to pay with a $\$20$ bill for a purchase that costs $\$14,23$. The cashier does not have small cent coins and suggests rounding the price to the nearest ten cents (that is, to the tenths).

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

Copy and complete the following table in your notebook, carrying out the indicated approximations for the decimal number $n = 14,3728$.

Approximation to the...	Truncation	Rounding
Tenths
Hundredths
Thousandths

Look at the following number line where a decimal number $x$ has been represented with a magnifying glass.

Knowing that the orange point 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 the tenths by rounding, which value would you choose? Justify your answer by observing the position of the point on the line with respect to the midpoint ($3,75$).

In a video game app store, the price of a new texture pack is $\$12,957$.

1. a) If the store decides to truncate the price to the nearest hundredth to show it in the catalog, what price will appear?
2. b) If the payment system rounds to the nearest hundredth (cents) when making the final charge, how much will the user actually pay?
3. c) What is the difference in money between the truncated price and the rounded price?

Three athletes have taken part in a 60-meter sprint at the school. The times recorded by the stopwatch were the following:

• Chloe: $8,429$ seconds.

• Mason: $8,421$ seconds.

• Andrew: $8,425$ seconds.

1. a) If the results are published on the school website rounded to the nearest hundredth, what time will appear for each one?
2. b) Looking at the rounded results, does it seem like they took the same amount of time? Is this true according to the original times?

A math teacher calculates the mean grade of an exam and gets $6,4782$.

1. a) Approximate the grade to the nearest tenth by truncation.
2. b) Approximate the grade to the nearest tenth by rounding.
3. c) Which of the two approximations benefits the student more? Why do you think rounding is usually used instead of truncation on report cards?

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 $4,56$ feet and $3,27$ feet.

1. a) Calculate the perimeter of the garden.
2. b) If the owner of the garden buys the fence at a hardware store that only sells by whole feet and always rounds up (approximation by excess to the nearest unit) to make sure there is no shortage of material, how many feet of fence will they buy?

Think and answer this riddle about decimal numbers:

"I am a number with three decimal places. If you truncate me to the nearest hundredth, I am $7,42$. However, if you round me to the nearest hundredth, I am $7,43$".

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

A precision scale shows that a bag of apples weighs $2,345$ kg.

1. a) Express the weight in grams (remember that $1$ kg = $1.000$ g).
2. b) If the shopkeeper tells us that the weight is approximately $2,3$ kg, what type of approximation have they made?
3. c) Calculate the error made in the shopkeeper's approximation (the difference between the real weight and the approximate weight). Express the result in kilograms and in grams.

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

1. Its whole-number part is the first two-digit prime number.

2. If we round it to the hundredths, we get $11,38$.

3. If we truncate it to the tenths, we get $11,3$.

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

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

At a discount gas station, the price of a gallon of regular unleaded is $\$1,419$. A group of friends fills the tank of their van for a trip with $52,6$ gallons.

The gas station's register is programmed to calculate the total and then round the final result to the nearest cent (two decimal places). However, one of the friends insists that the price per gallon should first be truncated to the cents and then multiplied by the gallons.

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

In a biology laboratory, the growth of a bacterium is being measured. The current length is $0,03478$ millimeters.

• Scientist A decides to use an approximation by rounding to the thousandths.

• Scientist B prefers to use an approximation by truncation to the ten-thousandths.

State which of the two values is greater and calculate the exact difference between the two approximations. Which of the two do you think is "more faithful" to the real value of the bacterium?

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

• It is greater than $7,45$ and less than $7,46$.

• When rounded to the 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.

A 6th grade class is designing a part for a 3D printer. The software gives them a measurement of $12,666...$ (repeating) millimeters for the thickness of a fitting. The printer has three precision modes:

• Eco Mode: Truncates to the tenths.

• Standard Mode: Rounds to the hundredths.

• Pro Mode: Rounds to the thousandths.

Calculate the value that the printer would use in each mode. If the hole where the part must fit measures exactly $12,67$ mm, in which modes will the part fit and in which will it not? (Consider that to fit, the part must be less than or equal to the hole).

Look at the following number line where two approximations of the number $X$ have been represented.

Knowing that the truncation to the hundredths is $3,14$ and the rounding to the 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 the hundredths will still be $3,14$.

A modern cuisine chef needs to weigh three ingredients with a high-precision scale that gives thousandths of a gram:

• Salt: $2,341$ g

• Pepper: $1,256$ g

• Saffron: $0,897$ g

The chef wants to know the total weight of the spice mix. Compare these two procedures:

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

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

Calculate both results. Are they the same? Explain why individual rounding can lead to a greater error than rounding at the end of the operation.

Make up an everyday life problem (shopping, sports, science...) where these two conditions occur simultaneously:

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

2. The difference between truncating that number to the hundredths and rounding it to the hundredths is exactly $0,01$.

Write the complete problem statement and explain how it would be solved applying the rules of approximation.