Problem 11 - Calculus: One and Several Variables

Question

Replace the symbol \(*\) by \(<,>,\) or \(=\) to make the statement true. \(\frac{3}{4} * 0.75\).

Step-by-Step Solution

Answer

\(\frac{3}{4} = 0.75\)

Step 1: Convert Fraction to Decimal

First off, we need to convert \(\frac{3}{4}\) to a decimal. This is achieved by simply dividing the numerator by the denominator. That is 3 divided by 4, which equals 0.75.

Step 2: Compare the Values

Next, we compare 0.75 (from step 1) with 0.75. One can clearly see now that 0.75 is equal to 0.75.

Step 3: Replace The Symbol

Having established the relationship between \(\frac{3}{4}\) and 0.75, we can now replace the * symbol with =, since 0.75 equals 0.75. The final statement thus becomes \(\frac{3}{4} = 0.75\)

Key Concepts

Fraction to Decimal Conversion

Fractions and decimals are two ways of representing the same concept: parts of a whole. When you convert a fraction to a decimal, you divide the numerator (the top number) by the denominator (the bottom number). This gives you a decimal representation of the fraction. For example, if you want to convert \( \frac{3}{4} \) to a decimal, divide 3 by 4. The result is 0.75. So, \( \frac{3}{4} \) is equal to 0.75 in decimal form. Knowing how to convert fractions to decimals is useful because it allows you to easily perform operations and comparisons that are more straightforward in decimal form. Converting fractions to decimals can make it easier to distinguish between different values and understand their sizes relative to each other. When learning fraction to decimal conversion, practice with simple fractions first to build confidence. Once you get the hang of it, you'll find this skill very handy in all sorts of mathematical problems.

Decimal Comparison

To compare decimals, start by lining up the numbers by their decimal points. This alignment helps you accurately assess each digit from left to right. Once aligned, compare digit by digit, starting from the leftmost where they differ. For example, compare 0.75 and 0.76:

• First digits: 0 is equal to 0.
• Second digits: 7 is equal to 7.
• Third digits: 5 is less than 6, hence 0.75 is less than 0.76.

Whenever the digits are the same up to the end, the decimals are equal. If you check \( 0.75 \) against itself, they match exactly, meaning the two numbers are equal. The process is similar for any number of decimals. Just remember to carefully check each digit positionally, and you'll be able to establish whether one decimal is greater, lesser, or equal to another.

Mathematical Symbols

Mathematical symbols are shorthand for expressing relations between numbers. They help quickly communicate whether numbers are greater than, less than, or equal to each other. The most common symbols for comparing numbers are:

• \(<\): Less than
• \(>\): Greater than
• \(=\): Equals

To use these symbols, one typically performs a comparison of two numbers or expressions. For example, in the exercise \( \frac{3}{4} * 0.75 \), we determine that 0.75 is equal to 0.75. Therefore, the correct symbol to use is \(=\), forming the equation \( \frac{3}{4} = 0.75 \). Symbols provide clarity and precision in mathematical communication. Understanding and correctly using these symbols is key in solving mathematical expressions and determining relationships between numbers.