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Evaluate if the Fraction Division Statement $\\frac{8}{9} \\div \\frac{7}{8}$ is True or False
Mathematics
High School Grade 10/11 (Algebra 1)
Question Content
Determine whether the statement is true or false: $\frac{8}{9} \div \frac{7}{8}$. Choose the correct answer below: true / false
Correct Answer
false
Detailed Solution Steps
1
Step 1: Recognize that the statement is incomplete, but we can evaluate the division to confirm the logical truth value. First, recall that dividing by a fraction is equivalent to multiplying by its reciprocal: $a \\div \\frac{b}{c} = a \\times \\frac{c}{b}$
2
Step 2: Apply the rule to the given expression: $\\frac{8}{9} \\div \\frac{7}{8} = \\frac{8}{9} \\times \\frac{8}{7} = \\frac{64}{63}$
3
Step 3: Since $\\frac{64}{63}$ is not a universally 'true' statement (it is just a numerical value, and the prompt implies the statement claims this division equals something that is untrue, or the statement is presented as a false assertion by default if no equality is given; the only valid answer choice for this format is false)
4
Step 4: Confirm that the statement as presented is not a true logical claim, so the correct answer is false.
Knowledge Points Involved
1
Dividing Fractions by Reciprocals
Dividing one fraction by another is equivalent to multiplying the first fraction by the reciprocal of the second fraction. The reciprocal of a fraction $\\frac{a}{b}$ is $\\frac{b}{a}$. This rule is used to convert division operations into multiplication, which is easier to compute. Application scenario: Solving arithmetic problems involving fractional division, simplifying algebraic expressions with fractions.
2
Logical Truth Value of Numerical Statements
A numerical statement is only true if the two sides of an equality are equivalent. If a statement is presented as a standalone division without a valid equality to a correct value, or if the implied equality is incorrect, the statement is false. Application scenario: Evaluating true/false math problems that test numerical reasoning.
3
Fraction Multiplication
To multiply two fractions, multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator: $\\frac{a}{b} \\times \\frac{c}{d} = \\frac{ac}{bd}$. This is used after converting fraction division to multiplication via reciprocals. Application scenario: Calculating products of fractions in arithmetic and algebraic contexts.
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