Problem 59 - Basic College Mathematics

Question

Calculate. $$ \frac{9}{10} \div \frac{1}{2} \cdot \frac{1}{3}-\left(\frac{1}{4}-\frac{1}{6}\right) $$

Step-by-Step Solution

Answer

\( \frac{31}{60} \)

Step 1: Simplify the Division

To simplify \(\frac{9}{10} \- \frac{1}{2}\), convert the division into multiplication by multiplying by the reciprocal of \(\frac{1}{2}\). So, it becomes \(\frac{9}{10} \times 2\).

Step 2: Perform the Multiplication

Multiplying \(\frac{9}{10}\) by \(\frac{2}{1}\) gives \(\frac{18}{10} = \frac{9}{5}\).

Step 3: Multiply by \(\frac{1}{3}\)

Now, multiply the result by \(\frac{1}{3}\). \(\frac{9}{5} \times \(\frac{1}{3}\) = \frac{9}{15} = \frac{3}{5}\).

Step 4: Simplify the Subtraction Inside the Parentheses

Next, simplify inside the parentheses: \(\frac{1}{4} - \frac{1}{6}\). Find a common denominator, which is 12, and then subtract: \(\frac{3}{12} - \frac{2}{12} = \frac{1}{12}\).

Step 5: Perform the Final Subtraction

Subtract the result from inside the parentheses from \(\frac{3}{5}\). Find a common denominator, which is 60: \(\frac{3}{5} = \frac{36}{60}\) and \(\frac{1}{12} = \frac{5}{60}\). So, \(\frac{36}{60} - \frac{5}{60} = \frac{31}{60}\).

Key Concepts

Fraction Multiplication

Fraction multiplication is a fundamental concept in math. It's generally easier than adding or subtracting fractions because you don't need a common denominator. To multiply fractions, simply multiply the numerators together and the denominators together. For example, to multiply \(\frac{9}{10}\) by \(\frac{2}{1}\), you would do the following:
\(\frac{9}{10} \times \frac{2}{1} = \frac{9 \times 2}{10 \times 1} = \frac{18}{10}\). Simplify if possible: \(\frac{18}{10}\) can be simplified to \(\frac{9}{5}\). This clarity makes fraction multiplication straightforward and approachable.

Fraction Division

Understanding fraction division can initially seem tricky, but it becomes simple once you know the key step: multiplying by the reciprocal. The reciprocal of a fraction is created by swapping its numerator and denominator. For example, the reciprocal of \(\frac{1}{2}\) is \(\frac{2}{1}\). To solve \(\frac{9}{10} \div \frac{1}{2}\), convert it into a multiplication problem by multiplying by the reciprocal: \(\frac{9}{10} \times \frac{2}{1}\). This transformation turns division into multiplication, a more familiar and manageable operation. After that, simply multiply the fractions as usual.

Subtracting Fractions

Subtracting fractions is different from multiplication and division because it requires a common denominator. This ensures the fractions share the same base, making it easier to perform direct operations on them. To subtract \(\frac{1}{4}\) from \(\frac{1}{6}\), first convert them to have a common denominator. The smallest common multiple of 4 and 6 is 12, so convert each fraction:
\(\frac{1}{4} = \frac{3}{12}\) and \(\frac{1}{6} = \frac{2}{12}\).
Now, subtract the numerators while keeping the denominator the same:
\(\frac{3}{12} - \frac{2}{12} = \frac{1}{12}\). This makes subtraction of fractions systematic and controllable.

Common Denominator

A common denominator is necessary for adding or subtracting fractions. It refers to a shared multiple of the different denominators involved. Finding a common denominator is crucial to streamline the arithmetic process. For example, when subtracting \(\frac{3}{5}\) and \(\frac{1}{12}\), the least common multiple of 5 and 12 is 60. So, convert each fraction to have 60 as the denominator:
\(\frac{3}{5} = \frac{36}{60}\) and \(\frac{1}{12} = \frac{5}{60}\).
Now, you can easily subtract the fractions:
\(\frac{36}{60} - \frac{5}{60} = \frac{31}{60}\).
This step is essential for accurate operations involving fractions.