Problem 98 - Algebra A Combined Function

Question

Given the percent \(0.5269 \%,\) round as indicated. a. Round to the nearest tenth percent. b. Round to the nearest hundredth percent.

Step-by-Step Solution

Answer

a. 0.5% b. 0.53%

Step 1: Understanding Percentages

A percentage is a way of expressing a number as a fraction of 100. In the problem, we have the percentage \(0.5269\%\). Our job is to round this number to different levels of precision in the following steps.

Step 2: Identify the Place Values

To round the percentage correctly, identify the digits at each place value: tenths, hundredths, and thousandths.- Tenths place: \(5\)- Hundredths place: \(2\)- Thousandths place: \(6\)- Ten-thousandths place: \(9\)These place values will help us round the number as needed.

Step 3: Rounding to the Nearest Tenth Percent

Look at the digit in the tenths place, \(5\), and the digit in the hundredths place, \(2\). Since \(2\) is less than \(5\), the tenths digit remains \(5\). Therefore, \(0.5269\%\) rounds to \(0.5\%\) when rounded to the nearest tenth percent.

Step 4: Rounding to the Nearest Hundredth Percent

Look at the digit in the hundredths place, \(2\), and the digit in the thousandths place, \(6\). Since \(6\) is 5 or higher, increase the hundredths digit by \(1\). Therefore, \(0.5269\%\) rounds to \(0.53\%\) when rounded to the nearest hundredth percent.

Key Concepts

Place Value

Place value is fundamental in understanding numbers, especially when dealing with rounding. Each digit in a number has a specific place, which determines its value. This concept becomes particularly important when rounding percentages like our given value of \(0.5269\%\). Let's break down the places:

• Tenths Place: In \(0.5269\%\), the tenths place is occupied by \(5\).
• Hundredths Place: The digit in the hundredths place is \(2\).
• Thousandths Place: Here, the \(6\) is situated in the thousandths place.
• Ten-thousandths Place: Finally, \(9\) resides in the ten-thousandths place.

Understanding these places helps us decide how to round a number depending on which precision—tenths or hundredths—we aim for. Knowing the specific location of each digit allows us to apply rounding rules correctly.

Tenths and Hundredths

When we talk about tenths and hundredths, we are referring to parts of a whole in decimals, which correspond to certain decimal places. These two places are crucial when rounding decimals like percentages.

• The tenths place is the first digit to the right of the decimal point and indicates the first division of a whole into ten equally sized parts. For example, in \(0.5\), the \(5\) is in the tenths place, making it five-tenths.
• The hundredths place is the second digit to the right of the decimal point, dividing the whole into 100 parts. In \(0.52\), the \(2\) is in the hundredths place, denoting fifty-two hundredths.

Understanding these places helps in rounding decisions. For \(0.5269\%\), we need to check digits one place further to decide whether to round up or keep the current value based on these critical positions.

Rounding Rules

Rounding is all about simplifying numbers while maintaining their value close to the original. Important rules come into play, especially when deciding whether to round up or down.

• First, identify the specific place you are rounding to, such as tenths or hundredths.
• Look at the digit immediately to the right of your desired place. This is the deciding digit.
• If this digit is 5 or more, you round up: increase the digit in the place you are rounding to by 1. For instance, in rounding \(0.5269\%\) to the nearest hundredth, observe the thousandths digit \(6\) and since it’s 5 or higher, you round \(0.52\) up to \(0.53\).
• If the deciding digit is less than 5, you do not change the digit you are rounding to. For example, rounding \(0.5269\%\) to the tenths is controlled by the hundredths \(2\), therefore it stays \(0.5\).

These rules ensure that rounding is consistent, providing simplicity without losing crucial information. Following them thoughtfully allows you to express numbers accurately at a needed level of precision.