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Chapter 4: Problem 70

Fill in the blanks with \(<,>,\) or \(=\). $$\frac{15}{4} \square \frac{21}{6}$$

Short Answer

Expert verified
\(\frac{15}{4} > \frac{21}{6}\).

Step by step solution

01

Convert the fractions to decimals

First, convert each fraction to its decimal form. For \(\frac{15}{4}\): Perform the division: \(15 \div 4 = 3.75\) For \(\frac{21}{6}\): Perform the division: \(21 \div 6 = 3.5\)
02

Compare the decimal values

Now, compare the decimal values obtained: \(\frac{15}{4} = 3.75\) and \(\frac{21}{6} = 3.5\). Since \3.75 > 3.5\, it follows that \(\frac{15}{4} \) is greater than \(\frac{21}{6}\).
03

Fill in the blank

Based on the comparison: \(\frac{15}{4} > \frac{21}{6}\) Place the greater-than sign (\(>\)) in the blank.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fractions
Fractions represent parts of a whole. They consist of two numbers: the numerator (top number) and the denominator (bottom number). For example, in the fraction \(\frac{15}{4}\), 15 is the numerator and 4 is the denominator. This means 15 parts of a whole that is divided into 4 parts.
Fractions can be proper or improper based on the relationship between the numerator and the denominator. A proper fraction has a numerator smaller than the denominator, such as \(\frac{3}{4}\). An improper fraction has a numerator larger than or equal to the denominator, such as \(\frac{15}{4}\).
Fractions can also be converted to decimals by performing division: numerator divided by denominator.
Decimals
Decimals are another way to represent parts of a whole. They use a decimal point to separate the whole number part from the fractional part, like 3.75 or 3.5. Decimals and fractions are interconnected; you can convert between them depending on your need.
To convert a fraction such as \(\frac{15}{4}\) to a decimal, you divide the numerator by the denominator: \(\frac{15}{4} = 15 \div 4 = 3.75\).
Decimals make it easier to compare values, as the size of the numbers can be assessed by looking at their digits from left to right, just like with whole numbers.
Inequalities
Inequalities compare values and determine their relative size. In mathematics, we use symbols like \(<\) for less than, \(>\) for greater than, and \(=\) for equal to. These symbols help in comparing two numbers or expressions.
In the given exercise, we compared the decimals obtained from fractions \(\frac{15}{4}\) and \(\frac{21}{6}\). We found that 3.75 is greater than 3.5, leading to the inequality symbol \(>\).
Using inequalities helps us understand relationships between numbers, an essential skill for solving many mathematical problems.
Division
Division is one of the fundamental arithmetic operations. It involves splitting a number into equal parts. When dividing, the number you start with is called the dividend, and the number you divide by is called the divisor.
For instance, in the division \15 \div 4 = 3.75\, 15 is the dividend, and 4 is the divisor. The result is called the quotient.
Division is crucial for converting fractions to decimals. It helps us understand how many times a divisor fits into a dividend, which is why we used it to convert \(\frac{15}{4}\) and \(\frac{21}{6}\) to their decimal forms.

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