Problem 59 - Prealgebra

Question

Perform the indicated operations. $$12 \div \frac{6}{7} \cdot 7$$

Step-by-Step Solution

Answer

The result of the operation is 98.

Step 1: Division of fractions

Before dividing by a fraction, we can simplify the operation by multiplying by its reciprocal. So, instead of solving \(12 \div \frac{6}{7}\), we'll multiply 12 by \(\frac{7}{6}\). The operation becomes \(12 \times \frac{7}{6}\).

Step 2: Multiplication of whole number and fraction

Multiply the whole number 12 by the fraction \(\frac{7}{6}\). This is done by multiplying the numerator by 12, so we have \(\frac{12 \times 7}{6}\), which simplifies to \(\frac{84}{6}\).

Step 3: Simplification

Simplify the fraction \(\frac{84}{6}\) by dividing both the numerator and the denominator by 6, resulting in \(14\).

Step 4: Multiplication

Now, multiply the result from Step 3 by 7, \(14 \times 7\). This equals 98.

Key Concepts

Multiplying Fractions

When it comes to multiplying fractions, the process might seem complex, but it's quite straightforward. To multiply fractions, you simply need to multiply the numerators (the top numbers) together and then multiply the denominators (the bottom numbers) together. This method makes it relatively easy to handle fractions in math problems.
For instance, if you have two fractions, \(\frac{a}{b}\) and \(\frac{c}{d}\), the product is \(\frac{a \times c}{b \times d}\).
Additionally, mixing fractions with whole numbers isn't as intimidating as it seems. Simply treat the whole number as a fraction by giving it a denominator of 1, transforming it into something easier to handle.

• For example, multiplying 12 by \(\frac{7}{6}\): treat 12 as \(\frac{12}{1}\), then multiply to get \(\frac{12 \times 7}{1 \times 6} = \frac{84}{6}\).

Multiplying fractions is crucial for solving a variety of math problems, especially those involving more than one operation.

Simplifying Expressions

Simplifying expressions often involves reducing fractions to their simplest form. This step is key in algebraic processes to keep your calculations manageable. You simplify a fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD). Doing so reduces the fraction without changing its value.
For instance, given the fraction \(\frac{84}{6}\), finding the GCD of 84 and 6 is essential. In this case, 6 divides both, allowing the fraction to be simplified:

• Divide 84 and 6 by 6.
• The result is \(\frac{84 \div 6}{6 \div 6} = 14\).

It's important to simplify wherever possible to avoid lengthy calculations, and simplification helps to clearly see the final answer in the simplest terms. When handling more complex expressions, simplifying helps to streamline the process by minimizing errors and reducing the number of operations.

Reciprocal of a Fraction

A reciprocal of a fraction is simply the fraction flipped. This means switching the numerator and the denominator. Understanding reciprocals is incredibly useful, particularly when dividing fractions.
If you are faced with a division of fractions, you can simplify the operation by multiplying by the reciprocal of the divisor. This is sometimes referred to as "invert and multiply."

• For example, the reciprocal of \(\frac{6}{7}\) is \(\frac{7}{6}\).
• When dividing 12 by \(\frac{6}{7}\), rather than perform a division, multiply by the reciprocal. So the operation \(12 \div \frac{6}{7}\) becomes \(12 \times \frac{7}{6}\).

Mastering reciprocals simplifies many operational processes in prealgebra and is a fundamental skill for progressing to more advanced mathematical topics.