Problem 26 - Prealgebra

Question

Find the following quotients. $$\frac{7}{8} \div\left(1 \frac{1}{4} \div 4\right)$$

Step-by-Step Solution

Answer

The quotient is \( \frac{14}{5} \) or as a mixed number, \( 2 \frac{4}{5} \).

Step 1: Convert Mixed Number to Improper Fraction

The mixed number to be converted is \( 1 \frac{1}{4} \). To convert, multiply the whole number (1) by the denominator (4) and add the numerator (1). This gives us \( 1 \times 4 + 1 = 5 \). Thus, \( 1 \frac{1}{4} \equiv \frac{5}{4} \).

Step 2: Simplify the Inner Division

The expression now looks like \( \frac{5}{4} \div 4 \). Remember that dividing by a whole number is the same as multiplying by its reciprocal. So, \( \frac{5}{4} \div 4 = \frac{5}{4} \times \frac{1}{4} = \frac{5}{16} \).

Step 3: Set Up the Outer Division

With the inner division simplified, we substitute back into the main expression to get \( \frac{7}{8} \div \frac{5}{16} \).

Step 4: Perform the Division of Fractions

To divide fractions, multiply by the reciprocal. Thus, \( \frac{7}{8} \div \frac{5}{16} = \frac{7}{8} \times \frac{16}{5} \).

Step 5: Multiply the Fractions

When multiplying fractions, multiply the numerators and denominators separately: \( \frac{7 \times 16}{8 \times 5} = \frac{112}{40} \).

Step 6: Simplify the Resulting Fraction

To simplify \( \frac{112}{40} \), divide both the numerator and the denominator by their greatest common divisor (GCD), which is 8. Thus, \( \frac{112 \div 8}{40 \div 8} = \frac{14}{5} \).

Step 7: Convert to Mixed Number (optional)

For completeness, convert \( \frac{14}{5} \) to a mixed number. Divide 14 by 5 to get 2 whole, with a remainder of 4, so \( \frac{14}{5} = 2 \frac{4}{5} \).

Key Concepts

Mixed Numbers

Mixed numbers show the sum of a whole number and a fraction. They are quite common in everyday math and help in understanding quantities better. For example, if you have 1 and a quarter pizzas, you would write it as \(1\frac{1}{4}\). This means you have 1 whole pizza and a quarter of another pizza. To perform mathematical operations with mixed numbers, you often need to convert them into improper fractions first.

Converting a mixed number to an improper fraction is simple:

• Multiply the whole number by the denominator of the fraction part.
• Add the numerator of the fraction to this product.
• For example, for \(1\frac{1}{4}\), multiply 1 by 4 to get 4, then add the numerator 1, resulting in 5. So, \(1\frac{1}{4}\) becomes \(\frac{5}{4}\).

This conversion is essential when dividing or multiplying fractions, ensuring calculations are manageable.

Improper Fractions

An improper fraction occurs when the numerator is greater than or equal to the denominator. Although they might seem intimidating at first, improper fractions are incredibly useful in mathematics. They are key when dividing or multiplying fractions and are especially useful when needing to express mixed numbers in a different format.

For instance, in the exercise, \(\frac{5}{4}\) is an improper fraction. This means you have more than one whole. Improper fractions provide flexibility when calculating, as they don't require continuous conversion if you are already within a fraction operation.

Moreover, when dividing fractions, using improper fractions simplifies the process immensely, as it aligns with the inversion (turning to reciprocal) rule necessary for such division. It's vital to feel comfortable manipulating improper fractions as they are foundational to more advanced math operations.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form, where no more common factors exist between the numerator and denominator. Simplifying helps make fractions more understandable and often easier to work with.

To simplify a fraction:

• Find the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the numerator and the denominator by the GCD.
• For instance, simplify \(\frac{112}{40}\): the GCD is 8. Dividing both parts by 8, you get \(\frac{14}{5}\), which is already in its simplest form.

This simplification is crucial in expressing results clearly; it transforms complex fractions into easy-to-read formats and is often the final step in fraction operations.

Mastering simplification not only helps in exams but is practical in daily life when needing to adjust portions or measurements.