Problem 81 - Prealgebra

Question

The following problems review addition and subtraction with fractions and mixed numbers. $$\frac{7}{15}-\frac{2}{15}$$

Step-by-Step Solution

Answer

The solution is \( \frac{1}{3} \).

Step 1: Identify the Fractions

First, identify the given fractions in the problem. We have two fractions: \( \frac{7}{15} \) and \( \frac{2}{15} \). Since they have a common denominator of 15, we can proceed with subtraction directly.

Step 2: Subtract the Numerators

Since the denominators of the fractions are the same, subtract the numerators. This means we calculate \( 7 - 2 \), which is equal to 5.

Step 3: Write the Resulting Fraction

Place the result from Step 2 over the original common denominator. This gives us the fraction \( \frac{5}{15} \).

Step 4: Simplify the Fraction

To simplify \( \frac{5}{15} \), find the greatest common divisor (GCD) of 5 and 15, which is 5. Divide both the numerator and the denominator by 5. This results in \( \frac{1}{3} \).

Step 5: Verify the Simplification

Check that \( \frac{1}{3} \) cannot be simplified further. Since 1 and 3 have no common factors other than 1, \( \frac{1}{3} \) is indeed fully simplified.

Key Concepts

Simplifying Fractions

When you come across a fraction, simplifying it can be quite helpful to make calculations easier and to ensure your final answer is in its simplest form. Fractions consist of a numerator (top number) and a denominator (bottom number). To simplify a fraction, look for the greatest common divisor (GCD) that both numbers share.

The GCD is the largest number that can evenly divide both the numerator and the denominator. Once found, you can divide both parts of the fraction by this number to simplify it. For instance, with the fraction \( \frac{5}{15} \), notice that both 5 and 15 are divisible by 5. Dividing the numerator and denominator by 5 leads us to \( \frac{1}{3} \).

It's key to simplify fractions whenever possible to maintain consistency in mathematical expressions and to make them much easier to interpret. After simplifying, always verify by checking if the reduced fraction can be divided further by a number other than 1.

Finding Common Denominators

In operations involving fractions, such as addition and subtraction, having a common denominator is crucial. This ensures that all parts of the fractions involved are on an equal footing for accurate calculation. A simple trick is that if two fractions already share a common denominator, you can skip directly to working with the numerators, but in cases where they don't, finding a common denominator is the next step.

To find a common denominator, look for the least common multiple (LCM) of the denominators involved in the calculation. For example, to add or subtract \( \frac{2}{5} \) and \( \frac{1}{3} \), find the LCM of 5 and 3, which is 15. Adjust each fraction by multiplying both its numerator and denominator to transform \( \frac{2}{5} \) to \( \frac{6}{15} \) and \( \frac{1}{3} \) to \( \frac{5}{15} \).

This technique aids in precise calculations and keeps the fractions uniform, allowing stress-free computation.

Fraction Subtraction Steps

Subtracting fractions might seem daunting at first, but with a few clear steps, it can be quite straightforward. The crucial first step is ensuring that the fractions involved share a common denominator. If they do not, you'll need to convert them to have one, as explained before.

Once a common denominator is established, you move on to subtracting the numerators directly while keeping the denominator the same. Take the fractions \( \frac{7}{15} \) and \( \frac{2}{15} \) as an example. Both already have a common denominator of 15. Subtracting the numerators 7 and 2, you get \( \frac{5}{15} \).

After obtaining the result, check if the resulting fraction can be simplified. As noted previously, simplify \( \frac{5}{15} \) by dividing both the numerator and the denominator by their GCD, 5, which gives \( \frac{1}{3} \). Following these steps leads to an accurate and simplified result every time.