Evaluate, correct to the nearest whole number \(7\frac{1}{2}-\left(2\frac{1}{2}+3\right)\div\frac{33}{2}\)
Answer Details
To evaluate the expression, we must follow the order of operations, which is also known as PEMDAS (parentheses, exponents, multiplication and division, addition and subtraction).
Starting with the parentheses, we have:
$$7\frac{1}{2}-\left(2\frac{1}{2}+3\right)\div\frac{33}{2}$$
$$=7\frac{1}{2}-\left(5+\frac{3}{1}\right)\div\frac{33}{2}$$
$$=7\frac{1}{2}-\left(5+\frac{3}{1}\right)\times\frac{2}{33}$$
$$=7\frac{1}{2}-\left(5\times\frac{2}{33}+\frac{3}{1}\times\frac{2}{33}\right)$$
$$=7\frac{1}{2}-\left(\frac{10}{33}+\frac{6}{33}\right)$$
$$=7\frac{1}{2}-\frac{16}{33}$$
Now, we need to find a common denominator to subtract the fractions. We can convert the mixed number to an improper fraction and multiply by $\frac{33}{33}$ to get:
$$=7\frac{1}{2}\times\frac{33}{33}-\frac{16}{33}$$
$$=\frac{15}{2}\times\frac{33}{33}-\frac{16}{33}$$
$$=\frac{495}{66}-\frac{16}{33}$$
$$=\frac{495-32}{66}$$
$$=\frac{463}{66}$$
To round to the nearest whole number, we divide 463 by 66 and round to the nearest whole number:
$$\frac{463}{66}\approx7$$
Therefore, the correct answer is 7.