Problem 22 - Contemporary Precalculus

Question

Fill the blank with \(<,=,\) or \(>\) so that the resulting statement is true. \(3 / 4\) ______ .75

Step-by-Step Solution

Answer

} 0.75\) Answer: =

Step 1: Convert the decimal to fraction

Convert 0.75 to a fraction: \(0.75 = 75/100 = 3/4.\)

Step 2: Compare the values

Compare the values of \(3/4\) and \(3/4:\) Since both values are equal, we can use the \(=\) symbol and get: \(3/4 = 0.75.\)

Key Concepts

Decimal to Fraction Conversion

Understanding how to convert decimals into fractions is a foundational skill in math that allows for comparison and calculation across different representations of numbers. This concept is crucial when we encounter numbers in decimal form, like 0.75, and need to relate them to a fraction, such as \(\frac{3}{4}\).

To convert a decimal to a fraction, start by examining the place value of the furthest right digit in the decimal. For 0.75, the 5 is in the hundredths place because it is two places to the right of the decimal point. Then, we can write 0.75 as \(\frac{75}{100}\). This fraction represents 75 hundredths. However, our aim is to simplify the fraction to its lowest terms. Since 75 and 100 share a common factor of 25, we can divide both the numerator and the denominator by 25 to get \(\frac{3}{4}\), which is the simplest form of the fraction.

Keeping the fractions in the lowest terms is necessary for easier comparison and calculation. It's also important when comparing to other fractions or decimals to ensure that we are working with equivalent values.

Comparing Numerical Values

Comparing numerical values is a crucial mathematical skill that involves determining the relationship between two numbers. This can involve comparing sizes, quantities, or values to understand which is larger, smaller, or if they are equal.

When we compare fractions and decimals, we first ensure they are in comparable forms. In our exercise, converting the decimal 0.75 to a fraction gave us \(\frac{3}{4}\). This lets us directly compare it to the other fraction, which is also \(\frac{3}{4}\). With fractions in like terms, it's clear that these two numbers represent the same value, and thus they are equal.

There are other methods to compare numerical values as well. For instance, we could convert both numbers to decimals or both to fractions. Regardless of the method, the goal is to understand how these numbers relate to one another within the context of the problem being solved.

Mathematical Symbols

Mathematical symbols are the shorthand of math and they represent specific mathematical concepts or operations. In the context of comparing numbers, symbols such as \(<\), \(>\), and \(=\) are especially important.

The symbol \(<\) is used when the number on the left is less than the number on the right, while \(>\) indicates the opposite - that the number on the left is greater than the number on the right. The equals sign \(=\) denotes that the numbers on either side of the sign are exactly the same in value.

In our exercise, after comparing the simplified fraction \(\frac{3}{4}\) with the decimal 0.75 which was converted to \(\frac{3}{4}\), we found that they are equivalent. Therefore, the appropriate mathematical symbol to fill in the blank in the statement \(\frac{3}{4}\) ______ 0.75 is the equals sign \(=\), leading us to the solution \(\frac{3}{4} = 0.75\). Being able to accurately use these symbols is essential for clear communication in mathematics.