Verified Answer :
- Identify the mixed fraction \(3 \frac{3}{4}\) as the sum of the integer 3 and the fraction \(\frac{3}{4}\).
- Convert the integer 3 to a fraction with a denominator of 4, which gives \(\frac{12}{4}\).
- Add the two fractions together: \(\frac{12}{4} + \frac{3}{4} = \frac{15}{4}\).
- The final improper fraction is \(\boxed{\frac{15}{4}}\).
Explanation
- Understanding the Mixed Fraction Representation
A mixed fraction of the form \(A \frac{B}{C}\) represents the sum of an integer \(A\) and a proper fraction \(\frac{B}{C}\). Specifically, \(A \frac{B}{C} = A + \frac{B}{C}\). In our case, we are given the mixed fraction \(3 \frac{3}{4}\). Our objective is to express this as an improper fraction \(\frac{n}{d}\), where \(n \geq d\).
- Finding a Common Denominator
To convert \(3 \frac{3}{4}\) into an improper fraction, we write it as the sum of its parts:
\[3 \frac{3}{4} = 3 + \frac{3}{4}\]
To add these, we need a common denominator. The integer \(3\) can be rewritten as a fraction with a denominator of \(4\) by multiplying the numerator and denominator by \(4\):
\[3 = \frac{3 \times 4}{4} = \frac{12}{4}\]
- Adding the Fractions and Final Result
Now, substitute the fraction back into the sum:
\[3 + \frac{3}{4} = \frac{12}{4} + \frac{3}{4}\]
Since the denominators are now the same, we can add the numerators:
\[\frac{12 + 3}{4} = \frac{15}{4}\]
Thus, the mixed fraction \(3 \frac{3}{4}\) is equivalent to the improper fraction \(\frac{15}{4}\).
Examples
Improper fractions are frequently used in cooking and baking. For example, if a recipe calls for \(3 \frac{3}{4}\) cups of flour, knowing that this is equal to \(\frac{15}{4}\) cups helps when you are using a measuring scoop that only measures in quarters of a cup. By converting the measurement, you can easily determine that you need to fill your quarter-cup scoop exactly 15 times to get the required amount. This highlights the practical utility of converting between mixed and improper fractions for measuring and portioning ingredients accurately.