Parentheses in advanced Order of Operations: Using fractions

Let's simplify this expression while following the order of operations which states that exponents come before multiplication and division, which come before addition and subtraction, and that parentheses come before all of these,

We will start by simplifying the expression in parentheses, since multiplication comes before addition, we will first perform the multiplication in the expression, while remembering that multiplying by a fraction means multiplying by the fraction's numerator, then we will perform the division of the fraction (by reducing it), we will complete simplifying the expression in parentheses by performing the addition within them:

$2\cdot(3+\frac{1}{2}\cdot8)= \\ 2\cdot(3+\frac{1\cdot8}{2})= \\ 2\cdot(3+\frac{\not{8}}{\not{2}})= \\ 2\cdot(3+4)= \\ 2\cdot7 \\$We will finish simplifying the given expression and perform the remaining multiplication:

$2\cdot7= \\ 14$Therefore the correct answer is answer C.

Let's simplify this expression while following the order of operations which states that exponents come before multiplication and division, which come before addition and subtraction, and that parentheses come before all of these,

In the given expression there is a term in parentheses that needs to be multiplied, so we'll start by simplifying the expression within these parentheses, meaning - we'll perform the addition of fractions within this expression, which we'll do by expanding the fractions to their minimal common denominator which is 6 (since it's the minimal common multiple of both fraction denominators in the expression - both 2 and 3), and performing the addition operation in the fraction's numerator, remembering that we know by how much to multiply each of the fraction's numerators when expanding the fractions by answering the question "by how much did we multiply the current denominator to get the common denominator?", then we'll simplify the expression in the fraction's numerator:

$\frac{1}{4}\cdot\big(\frac{1}{3}+\frac{1}{2}\big)= \\ \frac{1}{4}\cdot\frac{1\cdot2+1\cdot3}{6}= \\ \frac{1}{4}\cdot\frac{2+3}{6}= \\ \frac{1}{4}\cdot\frac{5}{6} \\$When simplifying the expression we got in the fraction's numerator, we remembered that according to the order of operations mentioned above, multiplication comes before addition,

We'll continue and perform the multiplication of fractions in the expression we got in the last step, remembering that multiplication of fractions is performed by multiplying numerator by numerator and denominator by denominator while keeping the original fraction line:

$\frac{1}{4}\cdot\frac{5}{6}= \\ \frac{1\cdot5}{4\cdot6}= \\ \frac{5}{24}$

Let's summarize the steps of simplifying the given expression, we got that:

$\frac{1}{4}\cdot\big(\frac{1}{3}+\frac{1}{2}\big)= \\ \frac{1}{4}\cdot\frac{5}{6} = \\ \frac{5}{24}$

Therefore the correct answer is answer C.

Let's simplify this expression while following the order of operations which states that exponents come before multiplication and division, which come before addition and subtraction, and that parentheses come before all of these,

We'll start by simplifying the expression inside the parentheses.

In this expression, there are addition operations between mixed fractions, so in the first step we'll convert all mixed fractions in this expression to improper fractions.

We'll do this by multiplying the whole number by the denominator of the fraction, and adding the result to the numerator.

In the fraction's denominator (which is the divisor) - nothing will change of course.

We'll do this in the following way:

$2\frac{1}{2}=\frac{(2\times2)+1}{2}=\frac{4+1}{2}=\frac{5}{2}$

$1\frac{1}{6}=\frac{(1\times6)+1}{6}=\frac{6+1}{6}=\frac{7}{6}$

Now we'll get the exercise:

$5\cdot\big(\frac{5}{2}+\frac{7}{6}+\frac{3}{4}\big)$

We'll continue and perform the addition of fractions in the expression inside the parentheses.

First, we'll expand each fraction to the common denominator, which is 12 (since it is the least common multiple of all denominators in the expression), we'll do this by multiplying the numerator of the fraction by the number that answers the question: "By how much did we multiply the current denominator to get the common denominator?"

Then we'll perform the addition operations between the expanded numerators:

$5\cdot\big(\frac{5}{2}+\frac{7}{6}+\frac{3}{4}\big) =\\ 5\cdot\frac{5\cdot6+7\cdot2+3\cdot3}{12} =\\ 5\cdot\frac{30+14+9}{12} =\\ 5\cdot\frac{53}{12} =\\$We performed the addition operation between the numerators above, after expanding the fractions mentioned.

Note that since multiplication comes before addition, we first performed the multiplications in the fraction's numerator and only then the addition operations,

We'll continue and simplify the expression we got in the last step, meaning - we'll perform the multiplication we got, while remembering that multiplying a fraction means multiplying the fraction's numerator.

In the next step, we'll write the result as a mixed fraction, we'll do this by finding the whole numbers (the answer to the question "How many complete times does the denominator go into the numerator?") and adding the remainder divided by the divisor:

$5\cdot\frac{53}{12}=\\ \frac{5\cdot53}{12}=\\ \frac{265}{12}=\\ 22\frac{1}{12}$

Let's summarize the steps of simplifying the given expression:

$5\cdot\big(2\frac{1}{2}+1\frac{1}{6}+\frac{3}{4}\big)= \\ 5\cdot\big(\frac{5}{2}+\frac{7}{6}+\frac{3}{4}\big)=\\ 5\cdot\frac{5\cdot6+7\cdot2+3\cdot3}{12} =\\ 5\cdot\frac{53}{12} =\\ 22\frac{1}{12}$

Therefore the correct answer is answer B.