Problem 29 - Fundamentals of Mathematics

Question

For the following 15 problems, convert each of the improper fractions to its corresponding mixed number. $$\frac{35}{4}$$

Step-by-Step Solution

Answer

\( 8 \frac{3}{4} \)

Step 1: Identify the Improper Fraction

The given improper fraction is \( \frac{35}{4} \). An improper fraction is one where the numerator (top number) is larger than the denominator (bottom number). In this case, 35 is larger than 4.

Step 2: Divide the Numerator by the Denominator

Divide 35 by 4. This division will help us determine the whole number part of the mixed number. Calculate \( 35 \div 4 = 8 \) with a remainder.

Step 3: Determine the Quotient and Remainder

From the division, the quotient is 8 and the remainder is 3 because \( 35 - 4 \times 8 = 3 \).

Step 4: Express as a Mixed Number

Combine the quotient and the remainder to form a mixed number. The quotient 8 becomes the whole number part, and the remainder 3 over the original denominator 4 becomes the fractional part. Thus, \( \frac{35}{4} \) as a mixed number is \( 8 \frac{3}{4} \).

Key Concepts

Mixed Numbers

Mixed numbers are composed of two parts: a whole number and a proper fraction. If you think of a pie, the whole number represents the complete pies, while the fraction signifies the remaining slices. For example, in the mixed number \( 8 \frac{3}{4} \), 8 is the whole number, and \( \frac{3}{4} \) is the fraction part. Mixed numbers are especially useful for representing numbers greater than one, providing a clear understanding of both the whole and fractional portions in a single expression. Converting from improper fractions to mixed numbers involves division, where you determine how many times the denominator can fit into the numerator.

Numerator and Denominator

The numerator and denominator are essential components of a fraction. The numerator is the top number, indicating how many parts you have. The denominator is the bottom number, which shows how many equal parts there are in a whole.

• Numerator: Tells you the part of the whole or the dividend in division.
• Denominator: Represents the total parts into which the whole is divided or the divisor.

In improper fractions, like \( \frac{35}{4} \), the numerator (35) is larger than the denominator (4). This signals that the fraction is greater than one whole, hence the need to convert it into a mixed number for better clarity.

Division in Mathematics

Division is a fundamental operation in mathematics involving two numbers. It is the process of determining how many times one number, the divisor, is contained within another number, the dividend. In the case of improper fractions, division helps split the numerator by the denominator. For \( \frac{35}{4} \), division tells us how many whole groups of 4 fit into 35.Division is crucial because it helps separate the whole number from the fractional part, enabling an easier conversion to a mixed number. The process involves:

• Performing the division to find the quotient.
• Calculating any remainder left after division.

Quotient and Remainder

The result of a division operation gives you a quotient and sometimes a remainder. These terms are vital in converting improper fractions to mixed numbers. Here's how they work:

• Quotient: The number of complete groups formed when dividing. In our example, \( 35 \div 4 \) gives a quotient of 8.
• Remainder: What's left after subtracting the total value of the quotient times the divisor from the dividend. From \( 35 - 4 \times 8 = 3 \), we get a remainder of 3.

In mixed number conversion, the quotient becomes the whole number part, while the remainder over the original denominator becomes the fractional part. Thus, the improper fraction \( \frac{35}{4} \) converts neatly into the mixed number \( 8 \frac{3}{4} \).