Problem 53 - Fundamentals of Mathematics

Question

For the following problems, find each value. $$8 \frac{3}{4} \div \frac{7}{8}$$

Step-by-Step Solution

Answer

The value is 10.

Step 1: Convert Mixed Number to Improper Fraction

First, convert the mixed number \(8 \frac{3}{4}\) into an improper fraction. Multiply the whole number (8) by the denominator (4) and add the numerator (3):\[8 \times 4 + 3 = 32 + 3 = 35\]So, \(8 \frac{3}{4} = \frac{35}{4}\).

Step 2: Set Up the Division of Fractions

Now, set up the division of the two fractions: \(\frac{35}{4} \div \frac{7}{8}\).

Step 3: Apply the Division Rule for Fractions

To divide fractions, multiply by the reciprocal of the divisor. The reciprocal of \(\frac{7}{8}\) is \(\frac{8}{7}\). Set up the multiplication:\[\frac{35}{4} \times \frac{8}{7}\]

Step 4: Multiply the Numerators

Multiply the numerators of the fractions:\[35 \times 8 = 280\]

Step 5: Multiply the Denominators

Multiply the denominators of the fractions:\[4 \times 7 = 28\]

Step 6: Simplify the Resulting Fraction

Simplify the fraction \(\frac{280}{28}\) by dividing the numerator and the denominator by their greatest common divisor, which is 28:\[\frac{280 \div 28}{28 \div 28} = \frac{10}{1} = 10\]

Key Concepts

Mixed Numbers

Mixed numbers are numbers that include both a whole number and a fraction. For example, in the mixed number \(8 \frac{3}{4}\), "8" is the whole number, and "\(\frac{3}{4}\)" is the fractional part. These are useful for representing quantities that are more than a whole but not complete to the next whole number.
To perform operations like division, we need to convert mixed numbers to an improper fraction.

• For instance, multiplying the whole number (8) by the denominator (4), we get 32.
• Then, adding the numerator (3) gives 35.
• So \(8 \frac{3}{4}\) becomes \(\frac{35}{4}\).

This conversion simplifies calculations and is necessary for dividing fractions.

Improper Fractions

Improper fractions are fractions where the numerator is greater than or equal to the denominator. For example, \(\frac{35}{4}\) is improper since 35 is greater than 4.

They are common when converting mixed numbers for mathematical operations.

• Improper fractions facilitate easier computation, especially in division and multiplication of fractions.
• They can later be converted back to mixed numbers for interpretation or answers.

Understanding improper fractions is key when working with fractions in mathematical problems.

Reciprocal

When dividing fractions, an important step is to multiply by the reciprocal of the divisor. The reciprocal of a fraction is formed by swapping the numerator and the denominator.

For example, the reciprocal of \(\frac{7}{8}\) is \(\frac{8}{7}\).

• To find the reciprocal, simply invert the fraction.
• This process turns the division problem into a multiplication one, making the math more manageable.

Using reciprocals simplifies the process of dividing fractions and is crucial in ensuring correct calculations.

Simplifying Fractions

Simplifying fractions ensures the result is in its simplest form, making it easier to understand and use. This process involves dividing the numerator and the denominator by their greatest common divisor (GCD).

For example, \(\frac{280}{28}\) simplifies to \(\frac{10}{1}\) by dividing both parts by 28.

• Finding the GCD helps in reducing fractions efficiently.
• A fully simplified fraction represents the same value in a clearer form.

Simplification is a critical step in solving fraction problems, allowing us to present answers more neatly and clearly.