Problem 28 - Fundamentals of Mathematics

Question

Simplify \(\frac{\frac{5}{6}-\frac{1}{4}}{\frac{1}{12}}\).

Step-by-Step Solution

Answer

The simplified expression is 7.

Step 1: Subtract Fractions in the Numerator

Start by finding a common denominator for the two fractions in the numerator, \(\frac{5}{6}\) and \(\frac{1}{4}\). The least common denominator is 12. Rewrite the fractions: \(\frac{5}{6} = \frac{10}{12}\) and \(\frac{1}{4} = \frac{3}{12}\). Now subtract: \(\frac{10}{12} - \frac{3}{12} = \frac{7}{12}\).

Step 2: Division of Fractions

Now, divide the simplified numerator \(\frac{7}{12}\) by the denominator \(\frac{1}{12}\). Remember that dividing by a fraction is the same as multiplying by its reciprocal. Therefore, \(\frac{7}{12} \div \frac{1}{12} = \frac{7}{12} \times \frac{12}{1}\).

Step 3: Simplify the Multiplication

Multiply the fractions: \(\frac{7}{12} \times \frac{12}{1}\). You can cancel out the 12 in both the numerator and denominator: \(7 \times 1 = 7\). Therefore, the expression simplifies to 7.

Key Concepts

Least Common Denominator

When working with fractions, especially when adding or subtracting them, finding the least common denominator (LCD) is an essential step. The LCD is the smallest number that each of the denominators can divide into without leaving a remainder.

To find it, consider each fraction's denominator and find their smallest common multiple. In the problem \( \frac{5}{6} \) and \( \frac{1}{4} \), the denominators are 6 and 4.

• List the multiples of 6: 6, 12, 18, 24...
• List the multiples of 4: 4, 8, 12, 16...
• The smallest common multiple is 12, which becomes the least common denominator.

By converting each fraction to have this common denominator, calculations become possible without loss of information. Using the LCD helps in aligning the fractions for straightforward addition or subtraction.

Subtracting Fractions

Subtracting fractions may be simpler than you think, once you have the same denominator for all fractions involved. After finding the least common denominator, you can rewrite each fraction accordingly.

For example, turning \(\frac{5}{6}\) into \(\frac{10}{12}\) and \(\frac{1}{4}\) into \(\frac{3}{12}\) enables you to directly subtract them: \[ \frac{10}{12} - \frac{3}{12} = \frac{7}{12} \].

The process is straightforward:

• Ensure all fractions have the same denominator.
• Subtract the numerators, while keeping the common denominator.
• The result is a new fraction with the same denominator but with the difference of the numerators.

In our example, the result is \(\frac{7}{12}\), a single fraction simplified and ready for further calculations.

Dividing Fractions

Dividing fractions can be approached by understanding that division by a fraction involves multiplying by its reciprocal. The reciprocal of a fraction is obtained by swapping its numerator and denominator.

Consider the expression \(\frac{7}{12} \div \frac{1}{12}\). To divide these fractions:

• Convert the division into multiplication by the reciprocal: \(\frac{7}{12} \times \frac{12}{1}\).
• Notice that the reciprocal of \(\frac{1}{12}\) is \(\frac{12}{1}\).
• Multiply the fractions using simple multiplication rules.Cancel out common terms like '12' in both fraction's numerator and denominator.

This easy trick of flipping and multiplying keeps the math manageable and offers a clear pathway from division to multiplication.

Multiplying Fractions

Multiplying fractions is often one of the simplest operations because you are just working with two numerators and two denominators. The key to multiplication is straightforward: multiply the numerators together, and the denominators together.

For example, when you had \(\frac{7}{12} \times \frac{12}{1}\), here's how you multiply:

• Multiply the numerators together: 7 and 12, giving 84.Multiply the denominators together: 12 and 1, giving 12.
• Next, you can simplify the fraction \(\frac{84}{12}\) by canceling common terms, which here would effectively boil it down to simply 7.

Canceling common terms such as '12' in the above calculation makes the process efficient, turning many steps into just a few clicks of reasoning.