Problem 31 - Prealgebra

Question

Replace each \(\circ\) with \(<,>,\) or \(=\) to make a true sentence. $$\frac{2}{5} \circ 0.4$$

Step-by-Step Solution

Answer

Replace \(\circ\) with \(=\).

Step 1: Convert Fraction to Decimal

Convert the fraction \(\frac{2}{5}\) to its decimal form. Divide 2 by 5, giving \(0.4\).

Step 2: Compare Decimal Values

Now compare the decimal values: \(0.4\) and \(0.4\). Since both values are equal, use the \(=\) symbol.

Step 3: Write the Final Equation

Replace the \(\circ\) with \(=\) to complete the equation: \(\frac{2}{5} = 0.4\).

Key Concepts

Decimal Conversion

A decimal is a way of representing numbers that are not whole. These are numbers with a dot called a decimal point, followed by digits that show values smaller than one. Understanding decimals is essential in everyday math, as they are used in various real-life applications such as money and measurements.
To identify a decimal, look for the decimal point. The numbers to the left of the point are whole numbers, and the numbers to the right are the decimal parts that represent fractional values. Each digit after the decimal point denotes a fraction of a power of ten. Here's a quick refresher on reading decimal numbers:

• The first place to the right of the decimal point is the "tenths" place.
• The second place is the "hundredths" place.
• The third place is the "thousandths" place, and so on.

For example, in the number 0.456, 4 is in the tenths place, 5 is in the hundredths place, and 6 is in the thousandths place.

Fraction to Decimal Conversion

Fractions represent numbers as parts of a whole. They are constructed with a numerator (top number) and a denominator (bottom number). Converting fractions to decimals can simplify arithmetic operations and make comparisons easier.
To convert a fraction to a decimal, you need to divide the numerator by the denominator. Let's illustrate this through our exercise example:

• Take the fraction \( \frac{2}{5} \).
• Divide the numerator (2) by the denominator (5).
• This division gives you 0.4, which is the decimal equivalent of the fraction \( \frac{2}{5} \).

This method is straightforward and can be applied to any fraction. By converting fractions into decimals, you can easily compare them, perform calculations, or use them in equations.

Equality in Mathematics

In mathematics, equality is a fundamental concept that shows two expressions have the same value. We use the symbol \( = \) to denote equality. In our exercise, \(\frac{2}{5} \) becomes \(0.4\) through conversion, and we find that \(0.4 = 0.4\).
Understanding equality involves recognizing that the two sides of an equation are identical in value, even if they appear different at first glance. Equality lets us manipulate equations and expressions in math while maintaining their integrity.

• To establish equality, ensure both sides of an equation have been simplified to their simplest form.
• This often involves operations like addition, subtraction, multiplication, division, or conversion, as seen with fractions and decimals.

Appreciating equality will enhance your mathematical thinking and problem-solving skills, helping you to see beyond the surface of expressions and calculations.