Answered

Explain why we can do the following to convert $5 \frac{3}{4}$ to an improper fraction:

$\frac{5 \cdot 4+3}{4}=\frac{23}{4}$



Choose the correct answer below.

AThe mixed number can be converted because $5 \frac{3}{4}=\frac{5}{1}+\frac{3}{4}=\frac{5 \cdot 4}{1}+\frac{3}{4}=\frac{5 \cdot 4+3}{4}$.
BThe mixed number can be converted because $5 \frac{3}{4}=5 \cdot \frac{3}{4}=\frac{5 \cdot 4+3}{4}$.
CThe mixed number can be converted because $5 \frac{3}{4}=5+\frac{3}{4}=\frac{5 \cdot 4}{4}+\frac{3}{4}=\frac{5 \cdot 4+3}{8}$.
DThe mixed number can be converted because $5 \frac{3}{4}=\frac{5}{1}+\frac{3}{4}=\frac{5 \cdot 4+1 \cdot 3}{1 \cdot 4}=\frac{5 \cdot 4+3}{4}$.

Verified Answer :

  • A mixed number is defined as the sum of its integer and fractional parts: \(5 \frac{3}{4} = 5 + \frac{3}{4}\).
  • To add these, express 5 as a fraction with denominator 4: \(5 = \frac{5}{1} = \frac{20}{4}\).
  • Sum the fractions by adding numerators: \(\frac{20}{4} + \frac{3}{4} = \frac{20 + 3}{4} = \frac{23}{4}\).
  • Option A correctly demonstrates this logical process, leading to the result \(\boxed{A}\).

Explanation

  1. Understanding the definition of a mixed number

A mixed number, such as \(5 \frac{3}{4}\), is defined as the sum of its integer part and its fractional part. Specifically, \(5 \frac{3}{4} = 5 + \frac{3}{4}\). To convert this sum into an improper fraction, we need to express the integer as a fraction with the same denominator as the fractional part. In this case, the denominator is 4.

  1. Converting to an improper fraction

To represent the integer 5 as a fraction with a denominator of 4, we multiply the numerator and denominator by 4:
\[5 = \frac{5}{1} = \frac{5 \times 4}{1 \times 4} = \frac{20}{4}\]
Now, we can add this to the fractional part \(\frac{3}{4}\):
\[5 + \frac{3}{4} = \frac{20}{4} + \frac{3}{4}\]
Adding two fractions with the same denominator involves adding their numerators while keeping the common denominator:
\[\frac{20}{4} + \frac{3}{4} = \frac{20 + 3}{4} = \frac{23}{4}\]
This calculation corresponds directly to the expression \(\frac{5 \cdot 4 + 3}{4}\).

  1. Evaluating the options

Reviewing the provided options:

  • Option A: \(5 \frac{3}{4} = \frac{5}{1} + \frac{3}{4} = \frac{5 \cdot 4}{1 \cdot 4} + \frac{3}{4} = \frac{5 \cdot 4 + 3}{4}\). This perfectly matches our logic.
  • Option B: Suggests multiplication between 5 and \(\frac{3}{4}\), which is incorrect.
  • Option C: Calculates the denominator as 8, which is incorrect.
  • Option D: Provides a general fraction addition rule \(\frac{a}{b} + \frac{c}{d} = \frac{ad+bc}{bd}\), which is also mathematically correct for any two fractions, but Option A describes the most direct process for converting a mixed number.

Comparing A and D, both are technically correct representations, but Option A is the standard explanation for this specific conversion. The correct option is A.

Examples

In baking, recipes often use mixed units like 2 1/2 cups of flour. Converting these to improper fractions, like 5/2 cups, makes it easier to multiply the entire recipe when you need to serve a larger group of people. Understanding this conversion ensures your measurements are precise and your baking results are consistent.

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