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How to Solve $\\left(\\frac{1}{2} \\cdot \\frac{2}{3}\\right) \\div \\frac{5}{6}$: Fraction Operations Practice
Mathematics
College Freshman
Question Content
Perform the indicated operations: $\\left(\\frac{1}{2} \\cdot \\frac{2}{3}\\right) \\div \\frac{5}{6}$. Simplify your answer. Type an integer or a fraction.
Correct Answer
$\\frac{2}{5}$
Detailed Solution Steps
1
Step 1: First solve the multiplication inside the parentheses. Multiply the numerators and denominators of the fractions: $\\frac{1}{2} \\cdot \\frac{2}{3} = \\frac{1 \\times 2}{2 \\times 3} = \\frac{2}{6} = \\frac{1}{3}$ (simplify the fraction by dividing numerator and denominator by their greatest common divisor, 2).
2
Step 2: Convert the division operation to multiplication by the reciprocal of the divisor. The reciprocal of $\\frac{5}{6}$ is $\\frac{6}{5}$, so the expression becomes $\\frac{1}{3} \\times \\frac{6}{5}$.
3
Step 3: Multiply the fractions: $\\frac{1 \\times 6}{3 \\times 5} = \\frac{6}{15}$. Simplify the result by dividing numerator and denominator by their greatest common divisor, 3, to get $\\frac{2}{5}$.
Knowledge Points Involved
1
Multiplication of Fractions
To multiply two fractions, multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator. The result can be simplified by dividing both the numerator and denominator by their greatest common divisor. This applies to both proper and improper fractions, and is used to solve operations within parentheses first per order of operations.
2
Division of Fractions
Dividing by a fraction is equivalent to multiplying by its reciprocal. The reciprocal of a fraction $\\frac{a}{b}$ (where $a$ and $b$ are non-zero) is $\\frac{b}{a}$. This rule converts division problems into multiplication problems, which are easier to solve using fraction multiplication rules.
3
Order of Operations (PEMDAS/BODMAS)
When evaluating mathematical expressions, operations inside parentheses are solved first, followed by exponents, multiplication/division (from left to right), then addition/subtraction (from left to right). For this problem, the multiplication inside the parentheses is calculated before the division operation outside.
4
Simplifying Fractions
A fraction is simplified when the numerator and denominator have no common positive divisors other than 1. To simplify, find the greatest common divisor (GCD) of the numerator and denominator, then divide both by this number. This reduces the fraction to its lowest terms, which is the standard form for presenting fractional answers.
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