Question

Subtract the following fractions and mixed numbers. Reduce to lowest terms. \(\frac{5}{6}-\frac{7}{12}=\) ______

Short answer

The result is \( \frac{1}{4} \).

Step-by-step solution

Find a Common Denominator

Look at the fractions \( \frac{5}{6} \) and \( \frac{7}{12} \). The denominators are 6 and 12. The least common denominator is 12 because it is the smallest number that both 6 and 12 divide without a remainder.

Adjust the Fractions

Convert \( \frac{5}{6} \) to a fraction with a denominator of 12. To do this, multiply both the numerator and the denominator by 2: \( \frac{5 \times 2}{6 \times 2} = \frac{10}{12} \). Now the fractions are \( \frac{10}{12} \) and \( \frac{7}{12} \).

Subtract the Fractions

Now that both fractions have the same denominator, subtract the numerators: \( 10 - 7 = 3 \). The result is \( \frac{3}{12} \).

Reduce to Lowest Terms

To reduce \( \frac{3}{12} \) to its lowest terms, find the greatest common divisor of 3 and 12, which is 3. Divide both the numerator and the denominator by 3: \( \frac{3 \div 3}{12 \div 3} = \frac{1}{4} \).

Conclusion

Thus, \( \frac{5}{6} - \frac{7}{12} = \frac{1}{4} \) when reduced to the lowest terms.

Key concepts

Common Denominator

To subtract fractions effectively, we need a common denominator. This is crucial because it allows us to bring different fractions to the same footing. Simply put, the common denominator is a shared multiple of the denominators that lets you align the fractions. Think of it like making sure each ingredient in a recipe is measured in the same units before combining.

Finding the least common denominator (LCD) ensures minimal changes to the fractions. Begin by listing multiples of each denominator and identify the smallest shared multiple. For example, with the denominators 6 and 12, we have:

• Multiples of 6: 6, 12, 18, 24, ...
• Multiples of 12: 12, 24, 36, ...

The smallest common multiple is 12. Adjust the fractions by converting each to an equivalent fraction with this LCD, maintaining their value. In our exercise, this meant converting \( \frac{5}{6} \) to \( \frac{10}{12} \) by multiplying both the top (numerator) and bottom (denominator) by 2. Thus, you obtain a setup where subtraction can smoothly proceed.

Reduce Fractions

Reducing fractions means to make them as simple as possible, while still maintaining their value. This might sound a bit like simplifying a math problem, which it is!

A fraction is reduced to its lowest terms when you can't divide the numerator and the denominator by any other number except 1 and itself or if both numbers are the most basic form. Take \( \frac{3}{12} \). Here, you find the greatest common divisor (GCD), which is the highest number that divides both the numerator and the denominator evenly. For 3 and 12, it's 3.

By dividing both by their GCD, you can present the fraction in a simpler form:

• \( \frac{3 \div 3}{12 \div 3} = \frac{1}{4} \)

This fraction \( \frac{1}{4} \) is equivalent to the original but much simpler and more comprehensible. Reducing fractions is not just a personal preference, but a standardized math approach for clarity.

Mixed Numbers

Mixed numbers are like a blend between a whole number and a fraction. They resemble how we often think about quantities in real life – part whole, part fraction. For instance, 1 and 3/4 of a pizza means one whole pizza and three-quarters of another.

When dealing with subtraction and mixed numbers, the process requires converting them into improper fractions. **Why?** Because it creates a uniform setup for calculation, eliminating the complexity that mixed numbers can introduce.

To transform a mixed number into an improper fraction, multiply the whole number by the denominator of the fraction, then add the numerator. Place this new number over the original denominator. Take the mixed number 3 2/5:

• Multiply: 3 x 5 = 15
• Add the numerator (2): 15 + 2 = 17
• The improper fraction is \( \frac{17}{5} \).

This standardizes the form, so whether you are adding, subtracting, or performing other operations, it remains consistent and manageable.