Problem 7 - Prealgebra

Question

Find each quotient. Use an area model if necessary. $$\frac{7}{9} \div(-14)$$

Step-by-Step Solution

Answer

The quotient of \( \frac{7}{9} \div (-14) \) is \( \frac{-1}{18} \).

Step 1: Understand Division in Terms of Multiplication

To divide fractions, it's helpful to recall that dividing by a number is equivalent to multiplying by its reciprocal. Therefore, the problem \( \frac{7}{9} \div (-14) \) can be rewritten as \( \frac{7}{9} \times \left(-\frac{1}{14}\right) \).

Step 2: Multiply the Fractions

Now, multiply the numerators together and the denominators together: \( \frac{7}{9} \times \left(-\frac{1}{14}\right) = \frac{7 \times (-1)}{9 \times 14} = \frac{-7}{126} \).

Step 3: Simplify the Fraction

Simplify the fraction \( \frac{-7}{126} \). Notice that both 7 and 126 are divisible by 7. Divide the numerator and the denominator by 7 to simplify: \( \frac{-7 \div 7}{126 \div 7} = \frac{-1}{18} \).

Key Concepts

Fractions

Fractions represent a way to express parts of a whole. They consist of two main parts: a numerator and a denominator. The numerator is the number above the fraction line, indicating how many parts are considered. The denominator is the number below the fraction line, showing the total number of equal parts that make up the whole. For example, in the fraction \( \frac{3}{4} \), 3 is the numerator and 4 is the denominator, meaning three out of four equal parts are being considered.

• Numerator: Top number, indicates parts chosen.
• Denominator: Bottom number, indicates total parts.

Fractions are versatile and can be used to represent division, ratios, or simply parts of a set. They can be greater than 1, less than 1, or exactly 1. Performing arithmetic with fractions, like addition or division, requires a clear understanding of their structure and meaning.

Division of Fractions

Division of fractions may seem tricky at first but simplifies when understood correctly. Division of a fraction involves multiplying by its reciprocal. Thus, dividing fraction \( a \) by fraction \( b \) is the same as multiplying fraction \( a \) by the reciprocal of fraction \( b \). For example, \( \frac{7}{9} \div \frac{3}{4} \) becomes \( \frac{7}{9} \times \frac{4}{3} \).

• Switch "divide" to "multiply."
• Use the reciprocal of the divisor.

This method keeps the division process consistent and easy by transforming it into a multiplication task. It is crucial to ensure that flipping the divisor to its reciprocal aligns with simplifying further calculations.

Multiplication of Fractions

Multiplying fractions is straightforward and involves a simple two-step process. First, multiply the numerators of the fractions to get a new numerator. Then, multiply the denominators to get a new denominator. The result will be your new fraction. For instance, multiplying \( \frac{3}{8} \times \frac{2}{5} \) involves the numerators \( 3 \times 2 = 6 \) and the denominators \( 8 \times 5 = 40 \), giving you \( \frac{6}{40} \).

• Multiply numerators.
• Multiply denominators.

After multiplying, simplifying the resulting fraction is a good practice, ensuring it is in its simplest form for easy interpretation and use. This might involve finding the greatest common divisor between the new numerator and denominator and dividing both by it.

Reciprocals

Reciprocals are critical in understanding fraction division. Finding the reciprocal of a fraction is simple—swap the numerator and the denominator. So, the reciprocal of \( \frac{5}{7} \) is \( \frac{7}{5} \). For a whole number like 4, imagine it as a fraction \( \frac{4}{1} \), and its reciprocal is \( \frac{1}{4} \).

• Flip numerator and denominator.
• Reciprocal of a whole number \( n \) is \( \frac{1}{n} \).

Reciprocals are fundamental when it comes to dividing fractions. By transforming division into multiplication using reciprocals, the calculations become more direct and manageable. Understanding reciprocals is essential for algebraic manipulations and solving equations involving fractions.