Problem 31 - Fundamentals of Mathematics

Question

For the following 15 problems, convert each of the improper fractions to its corresponding mixed number. $\frac{63}{7}$$

Step-by-Step Solution

Answer

The improper fraction \( \frac{63}{7} \) converts to the mixed number 9.

Step 1: Understand the Problem

We are given an improper fraction \( \frac{63}{7} \) and need to convert it to a mixed number. An improper fraction is when the numerator is greater than the denominator.

Step 2: Divide the Numerator by the Denominator

To convert \( \frac{63}{7} \) to a mixed number, divide the numerator (63) by the denominator (7). Perform the division: 63 divided by 7 equals 9.

Step 3: Interpret the Result

The quotient from the division is 9, with a remainder of 0. When there is no remainder, the mixed number will not have a fractional component.

Step 4: Write the Mixed Number

Since the division left no remainder, the mixed number equivalent of \( \frac{63}{7} \) is simply 9.

Key Concepts

Mixed Numbers

A mixed number combines a whole number with a fraction, presenting a clear way to express quantities greater than one in mixed numerical form. For example, 3 \(\frac{4}{5}\)\ means there are 3 whole parts and another \(\frac{4}{5}\) of a whole. This representation is helpful when dealing with quantities that are not entirely whole numbers, such as measurements or when dividing something into parts. In a mixed number, the whole number represents how many complete units you have, and the fraction shows the remaining portion of the next unit.

Convert Fractions

Converting fractions is a crucial skill in math, allowing us to switch from improper fractions to mixed numbers and vice versa.

• An improper fraction, like \(\frac{9}{4}\), has a numerator larger than the denominator, indicating more than one whole.
• A mixed number, like 2 \(\frac{1}{4}\), splits this into whole parts and a fraction.

To convert \(\frac{9}{4}\) into a mixed number, divide 9 by 4 to get 2 with a remainder of 1. This means 9 divided by 4 is 2 whole parts with 1 left over, resulting in the mixed number 2 \(\frac{1}{4}\). Understanding how to convert between these forms is beneficial for simplifying fractions and for clarity in problem-solving.

Division of Fractions

Dividing fractions might seem tricky at first, but it's a straightforward process once understood. Instead of dividing, you multiply by the reciprocal. For example, to divide \(\frac{1}{2}\) by \(\frac{3}{4}\), multiply \(\frac{1}{2}\) by the reciprocal of \(\frac{3}{4}\), which is \(\frac{4}{3}\).

• The equation becomes \(\frac{1}{2} \times \frac{4}{3}\).
• Multiply the numerators: 1 \(\times\) 4 = 4.
• Multiply the denominators: 2 \(\times\) 3 = 6.
• This results in \(\frac{4}{6}\), which simplifies to \(\frac{2}{3}\).

This method of dividing fractions simplifies calculations and helps solve more complex problems effectively.

Fraction Concepts

Understanding fractions is fundamental in mathematics, representing parts of a whole. Fractions consist of a numerator, the top number indicating how many parts you have, and a denominator, the bottom number indicating how many parts make up a whole.

• A fraction like \(\frac{3}{4}\) denotes 3 out of 4 equal parts.
• Improper fractions and mixed numbers display amounts greater than a whole using these parts.

Fractions are used in various applications, from simple measurements to complex equations, making comprehension vital for mathematical fluency. Understanding equivalent fractions, adding, subtracting, multiplying, and dividing them, allows for more significant mathematical problem-solving and real-world applications.