Answered

Which expression is equivalent to $\frac{a}{c} \div \frac{e}{f}$ ?


A$\frac{a c}{c f}$
B$\frac{a f}{a c}$
C$\frac{a-e}{f-c}$
D$\frac{ m }{a f }$

Verified Answer :

  • To divide two fractions, multiply the first fraction by the reciprocal of the second.
  • Specifically, \(\frac{a}{c} \div \frac{e}{f} = \frac{a}{c} \cdot \frac{f}{e}\).
  • Multiplying the numerators and denominators results in \(\frac{af}{ce}\).
  • The equivalent expression is \(\boxed{\frac{af}{ce}}\).

Explanation

  1. Analyzing the Division of Fractions

The problem asks to find the expression equivalent to the quotient of two fractions: \(\frac{a}{c} \div \frac{e}{f}\).
To divide by a fraction, we multiply by its reciprocal. The reciprocal of a fraction \(\frac{e}{f}\) is \(\frac{f}{e}\).
Therefore, the operation is equivalent to multiplying the dividend \(\frac{a}{c}\) by the reciprocal of the divisor \(\frac{f}{e}\):
\[\frac{a}{c} \div \frac{e}{f} = \frac{a}{c} \cdot \frac{f}{e}.\]
Multiplying the numerators together and the denominators together, we get:
\[\frac{a \cdot f}{c \cdot e} = \frac{af}{ce}.\]
This result means that \(\frac{a}{c} \div \frac{e}{f}\) is equivalent to \(\frac{af}{ce}\).
Comparing this to the provided options:

  1. \(\frac{ac}{cf}\) - Not equivalent
  2. \(\frac{af}{ae}\) - Not equivalent
  3. \(\frac{a-e}{f-c}\) - Not equivalent
  4. \(\frac{af}{ce}\) - This is the standard form, although it was written as \(\frac{af}{ce}\), note that there might be a typo in the options provided in the prompt, or it might refer to the structure of the result. Wait, looking closer at the options: \(\frac{ac}{cf}, \frac{af}{ae}, \frac{a-e}{f-c}, \frac{af}{ce}\). Actually, the final option in the prompt provided is \(\frac{m}{af}\), which looks like a typo in the prompt's list of options. Let's re-examine the prompt's options: \(\frac{ac}{cf}\), \(\frac{af}{ae}\), \(\frac{a-e}{f-c}\), and \(\frac{m}{af}\). Since none of the first three are equivalent, let me re-verify the division.

Wait, \(\frac{a}{c} \div \frac{e}{f} = \frac{a}{c} \times \frac{f}{e} = \frac{af}{ce}\).
If the options are limited and none match, perhaps there's a misunderstanding of the prompt's notation or a typo. However, logically, the equivalent expression is \(\frac{af}{ce}\). Assuming the last option \(\frac{m}{af}\) contains a typo and meant \(\frac{af}{ce}\), or there is an issue with the provided options. Let's re-read carefully: \(\frac{a}{c} \div \frac{e}{f}\). Equivalent is \(\frac{af}{ce}\). Since \(\frac{af}{ce}\) is the correct mathematical result, I will state it clearly.

Examples

Imagine you are baking cookies and have a recipe that requires dividing portions of ingredients. If you have \(\frac{a}{c}\) cups of sugar and need to divide it into \(\frac{e}{f}\) portions, knowing how to handle the division of these fractions allows you to determine exactly how much sugar goes into each portion. This skill is essential for scaling recipes, managing inventory, or any task involving proportional distribution of resources.

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