Simplify 3 1/2 x 3 1/2
The Correct Answer and Explanation is :
To simplify ( 3 \frac{1}{2} \times 3 \frac{1}{2} ), follow these steps:
- Convert the mixed number to an improper fraction:
A mixed number consists of a whole number and a fraction. To convert ( 3 \frac{1}{2} ) to an improper fraction:
[
3 \frac{1}{2} = 3 \times 2 + 1 = 6 + 1 = 7
]
Thus, ( 3 \frac{1}{2} = \frac{7}{2} ). - Multiply the fractions:
Now that we have both numbers as improper fractions, we can multiply them:
[
\frac{7}{2} \times \frac{7}{2} = \frac{7 \times 7}{2 \times 2} = \frac{49}{4}
] - Convert back to a mixed number:
To express ( \frac{49}{4} ) as a mixed number, divide the numerator by the denominator:
[
49 \div 4 = 12 \quad \text{with a remainder of } 1.
]
This can be expressed as:
[
12 \frac{1}{4}.
]
Thus, the result of ( 3 \frac{1}{2} \times 3 \frac{1}{2} ) is ( 12 \frac{1}{4} ).
Explanation
When multiplying mixed numbers, converting them to improper fractions simplifies the process. Mixed numbers can be cumbersome to work with directly. By transforming ( 3 \frac{1}{2} ) into an improper fraction ( \frac{7}{2} ), you can focus solely on the multiplication of fractions, which follows the straightforward rule of multiplying the numerators and denominators separately.
In this case, after the conversion, we have:
[
\frac{7}{2} \times \frac{7}{2} = \frac{49}{4}.
]
The next step is crucial: converting the result back into a mixed number for clarity and ease of interpretation. This final transformation allows us to communicate the answer in a more digestible form, particularly useful in everyday contexts, such as cooking or construction, where measurements are often expressed as mixed numbers. The final result ( 12 \frac{1}{4} ) clearly indicates the quantity without the complications of improper fractions.