Problem 80 - Elementary Algebra

Question

\(\frac{1}{10} \div \frac{1}{5}-\frac{3}{5}\)

Step-by-Step Solution

Answer

\( \frac{-1}{10} \)

Step 1 - Solve the Division

Convert the division into multiplication by multiplying by the reciprocal. \ \( \frac{1}{10} \times \frac{5}{1} \)

Step 2 - Simplify the Multiplication

Multiply the fractions: \ \( \frac{1 \times 5}{10 \times 1} = \frac{5}{10} \)

Step 3 - Simplify the Fraction

Simplify \( \frac{5}{10} \) by dividing both the numerator and the denominator by 5: \ \( \frac{5 \/\ 5}{10 \/\ 5} = \frac{1}{2} \)

Step 4 - Perform the Subtraction

Subtract \( \frac{3}{5} \) from \( \frac{1}{2} \) by finding a common denominator. In this case, the common denominator is 10. Rewrite the fractions as follows: \ \( \frac{1}{2} = \frac{5}{10} \) and \( \frac{3}{5} = \frac{6}{10} \) \ Then perform the subtraction: \ \( \frac{5}{10} - \frac{6}{10} = \frac{5 - 6}{10} = \frac{-1}{10} \)

Key Concepts

Dividing Fractions

Dividing fractions might seem tricky at first, but it becomes straightforward when you know the right method. To divide fractions, you simply multiply by the reciprocal of the fraction you're dividing by. For example, in the exercise, we are given \ \( \frac{1}{10} \ \div \ \frac{1}{5} \ \ \), which we convert into a multiplication problem: \ \( \frac{1}{10} \ \times \ \frac{5}{1} \ \). The reciprocal of a fraction is just swapping the numerator and the denominator. This step is essential because it turns a division problem into a multiplication problem, making it easier to solve.

Multiplying Fractions

Multiplying fractions is one of the simplest operations with fractions. When multiplying two fractions, you just multiply their numerators and their denominators. For example, in the previous step, we changed the division into multiplication: \ \( \frac{1}{10} \ \times \ \frac{5}{1} \ \). Multiplying the numerators together (1 and 5) and the denominators together (10 and 1), we get \ \( \frac{1 \ \times 5}{10 \ \times 1} = \frac{5}{10} \ \). This simplifies to \ \( \frac{1}{2} \ \). Remember, always multiply straight across: top with top and bottom with bottom.

Subtracting Fractions

Subtracting fractions involves finding a common denominator so that the fractions have the same base. Once you have that, you can subtract the numerators directly. For instance, in our example, we need to subtract \ \( \frac{3}{5} \ \) from \ \( \frac{1}{2} \ \). First, we find a common denominator for these fractions, which in this case is 10. We rewrite \ \( \frac{1}{2} \ \) as \ \( \frac{5}{10} \ \) and \ \( \frac{3}{5} \ \) as \ \( \frac{6}{10} \ \). Then, we subtract the numerators: \ \( \frac{5 - 6}{10} = \frac{-1}{10} \ \). The fraction \ \( \frac{-1}{10} \ \) is the result of the subtraction.

Common Denominator

A common denominator is necessary when adding or subtracting fractions because it aligns the fractions' denominators, making the numerators directly comparable. To find a common denominator, you can simply multiply the denominators of the fractions together, but often a smaller common multiple is more convenient. For example, in our problem, we need to subtract \ \( \frac{1}{2} \ \) and \ \( \frac{3}{5}\ \). The smallest common multiple of 2 and 5 is 10. Thus, we convert \ \( \frac{1}{2} \ \) to \ \( \frac{5}{10} \ \) and \ \( \frac{3}{5} \ \) to \ \( \frac{6}{10} \ \). Now, both fractions have the same denominator, allowing us to easily subtract them.