Problem 11 - Prealgebra

Question

Replace each with \(<,>,\) or \(=\) to make a true sentence. $$\frac{9}{10} \circ 0.90$$

Step-by-Step Solution

Answer

\( \frac{9}{10} = 0.90 \)

Step 1: Convert the Decimal to a Fraction

First, convert the decimal number 0.90 into a fraction. Since there are two digits after the decimal point, 0.90 can be expressed as \( \frac{90}{100} \).

Step 2: Simplify the Fraction

Simplify the fraction \( \frac{90}{100} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 10. This gives \( \frac{9}{10} \).

Step 3: Compare the Two Fractions

Now compare the original fraction \( \frac{9}{10} \) and the simplified fraction obtained from the decimal, \( \frac{9}{10} \). Since both fractions are identical, they are equal.

Step 4: Fill in the Symbol

Since \( \frac{9}{10} = \frac{9}{10} \), the correct symbol to make the sentence true is \( = \).

Key Concepts

Decimal to Fraction Conversion

Converting a decimal to a fraction involves expressing the decimal number as a ratio of two integers. The process is straightforward and allows us to work with entire numbers, which can be easier to grasp for some calculations.
To convert a decimal like 0.90 to a fraction, first note the number of decimal places. Here, there are two digits after the decimal, so you consider the decimal as being over 100.

• Write 0.90 as a fraction: \(\frac{90}{100}\).
• The number 90 comes from the digits after the decimal, and 100 corresponds to having two decimal places (i.e., 10 to the power of 2).

This step sets the stage for further simplification. Converting decimals into fractions helps when comparing different values or when performing arithmetic operations requiring common denominators.

Fraction Simplification

Simplifying fractions is the process of reducing them to their simplest form. It means finding a fraction equivalent to the original, with the smallest possible numerator and denominator.
In our example, \(\frac{90}{100}\), both numbers have a common factor. To simplify, follow these steps:

• Find the greatest common divisor (GCD) of 90 and 100, which is 10.
• Divide both the numerator (90) and the denominator (100) by 10.
• You get \(\frac{9}{10}\), which is the simplified form.

Simplifying fractions makes them easier to compare and use in further calculations, especially when dealing with other fractions or equations. It also makes interpreting results more intuitive, as smaller numbers are generally easier to manipulate.

Mathematical Symbols

Mathematical symbols are powerful tools used to express equations, relationships, and values. They simplify complex ideas into concise representations, making them crucial in solving math problems.
For comparison tasks, symbols like \(<\), \(>\), and \(=\) are used to denote less than, greater than, and equal to, respectively.

• \(<\): Indicates that the value on the left is smaller than the value on the right.
• \(>\): Shows that the value on the left is greater than the value on the right.
• \(=\): Reveals that both sides are equal in value.

In the exercise, we determined that \(\frac{9}{10} = \frac{9}{10}\). This perfect match led to using the equality symbol, \(=\), reinforcing the equal status of both fractions. Understanding these symbols allows you to convey mathematical relationships effectively and accurately, thus simplifying the process of comparing and analyzing values.