Question

Divide the following fractions and mixed numbers. Reduce to lowest terms. \(\frac{7}{8} \div 14=\) ______

Short answer

\( \frac{1}{16} \)

Step-by-step solution

Change Division to Multiplication

To divide fractions, we change the division problem into a multiplication problem. This is done by multiplying by the reciprocal of the divisor. In this case, the divisor is 14, which can be expressed as a fraction: \( \frac{14}{1} \). The reciprocal of \( \frac{14}{1} \) is \( \frac{1}{14} \). Therefore, our problem changes from \( \frac{7}{8} \div 14 \) to \( \frac{7}{8} \times \frac{1}{14} \).

Multiply the Fractions

Now, multiply the numerators together and the denominators together. Multiply \( 7 \) by \( 1 \) to get the new numerator, and \( 8 \) by \( 14 \) to get the new denominator: \[ \frac{7}{8} \times \frac{1}{14} = \frac{7 \times 1}{8 \times 14} = \frac{7}{112} \].

Simplify the Fraction

To simplify \( \frac{7}{112} \), find the greatest common divisor (GCD) of 7 and 112. The GCD is 7. Divide both the numerator and the denominator by 7: \( \frac{7 \div 7}{112 \div 7} = \frac{1}{16} \). Thus, the simplified form of \( \frac{7}{112} \) is \( \frac{1}{16} \).

Key concepts

Fractions

Fractions are a way to represent parts of a whole. They consist of two numbers: the numerator, which sits on top, and the denominator, which is positioned on the bottom. The numerator indicates how many parts you have, while the denominator tells you how many parts the whole is divided into. For instance, in the fraction \( \frac{7}{8} \), 7 is the numerator and 8 is the denominator. This means you have 7 parts out of a total of 8 parts that make up the whole.

Understanding fractions is crucial in solving problems that involve parts of a whole, such as dividing quantities, measuring, and calculating probabilities. Fractions can also be greater than one when the numerator is larger than the denominator, and they are called improper fractions. To convert improper fractions into mixed numbers or vice versa, a good grasp of fraction operations is needed.

Reciprocal

A reciprocal is a special kind of fraction. It is what you get when you flip a fraction upside down. For example, the reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \). When dealing with whole numbers, like 14, it can be expressed as a fraction: \( \frac{14}{1} \). Its reciprocal is \( \frac{1}{14} \).

Reciprocals are important in fraction division. They allow us to turn a division problem into a multiplication problem. For dividing fractions, you multiply the first fraction by the reciprocal of the second fraction. This simplifies calculations and is fundamental to solving division problems involving fractions.

Simplification of Fractions

Simplification of fractions means reducing them to their lowest terms. A fraction is in its simplest form when the numerator and denominator have no common factors other than 1. To simplify a fraction, you divide both the numerator and the denominator by their greatest common divisor (GCD). For instance, the fraction \( \frac{7}{112} \) simplifies to \( \frac{1}{16} \) by dividing both 7 and 112 by 7, which is their GCD.

Why bother simplifying fractions? Simplification makes fractions easier to understand and can make arithmetic operations less complicated. Plus, fractions in their simplest form are often more intuitive to work with, especially when comparing, adding, subtracting, or multiplying fractions.

Greatest Common Divisor

The greatest common divisor (GCD) of two numbers is the largest number that can divide both without leaving a remainder. Finding the GCD is a key step in simplifying fractions. For example, to simplify \( \frac{7}{112} \), we need the GCD of 7 and 112. Since 7 divides both itself and 112 perfectly, the GCD is 7.

Here’s a quick trick to find the GCD:

• List out the factors of both numbers.
• Identify the largest factor common to both lists.

The concept of GCD is not only useful for simplifying fractions but also in solving problems that involve computations with ratios, or even finding least common multiples. Understanding how to calculate the GCD will help you navigate through various math tasks and exercises with more ease.