Problem 38 - Prealgebra

Question

Replace each \(\circ\) with \(<,>,\) or \(=\) to make a true sentence. $$1 \frac{1}{20} \circ 1.01$$

Step-by-Step Solution

Answer

1.05 > 1.01, so use \(>\).

Step 1: Convert Fraction to Decimal

Convert the mixed number \(1 \frac{1}{20}\) to a decimal. First, convert \(\frac{1}{20}\) to a decimal by dividing 1 by 20, which is 0.05. Add this to the 1, the whole number, giving the decimal 1.05.

Step 2: Compare the Decimals

Now compare the two decimal values: 1.05 and 1.01. Since 1.05 is greater than 1.01, the symbol \(>\) is the correct choice.

Key Concepts

Mixed Numbers

When dealing with numbers that include both whole parts and fractions, we often use mixed numbers. A mixed number consists of a whole number and a proper fraction. For example, in the mixed number \(1 \frac{1}{20}\), the whole number is 1, and the fraction is \(\frac{1}{20}\). This format is particularly useful when adding or subtracting quantities, as it provides a clear picture of how many whole units we have plus any additional fraction.
To work with mixed numbers in calculations involving decimals, it's essential to convert the fractional part into a decimal. This is because decimals provide a simple and consistent way to handle calculations. Once the fraction is expressed as a decimal, simply add it to the whole number to arrive at a single decimal value.

Comparing Decimals

Decimals are crucial for comparing different numerical values, especially when fractions or mixed numbers are involved. When comparing decimals, the value with the larger number in the tenths place is generally the larger number. If these are equal, we look to the next decimal place (hundredths place), continuing the process until one of the decimals has a larger digit in a subsequent place.
For instance, when comparing 1.05 and 1.01, we first examine the tenths place: both have a 0. Therefore, we move to the hundredths place, where 5 is greater than 1. Thus, 1.05 is larger. Always compare decimals digit by digit from left to right, as this mirrors how we read numbers.

• Ensure all numbers are in decimal form before comparing.
• Compare place by place, left to right.
• Use symbols such as \(>\), \(<\), and \(=\) to represent the relationship between the values.

Understanding decimals is key in many math problems, ensuring accurate and precise comparisons.

Converting Fractions to Decimals

Conversion from fractions to decimals is a vital skill in prealgebra. This process involves dividing the numerator (the top part of the fraction) by the denominator (the bottom part).
For example, to convert \(\frac{1}{20}\) into a decimal, divide 1 by 20. Performing this division gives 0.05. Thus, \(\frac{1}{20}\) is equivalent to 0.05 in decimal form. Once converted, these decimal forms simplify various mathematical operations, like addition, subtraction, and those requiring precise comparisons.

• Ensure you divide the numerator by the denominator accurately.
• If necessary, round the decimal to the desired number of places.
• Decimals provide a straightforward way to perform calculations and comparisons with mixed numbers.

By mastering conversions from fractions to decimals, students can move fluidly between numerical formats, enhancing their ability to solve various mathematical challenges effortlessly.