Associative Property: Using fractions

According to the rules of the order of arithmetic operations, we first place the multiplication exercise inside of parentheses:

$3+(\frac{3}{3}\times\frac{2}{3})-2=$

We then solve the exercise in the parentheses, combining the multiplication into a single exercise:

$(\frac{3}{3}\times\frac{2}{3})=\frac{3\times2}{3\times3}=\frac{6}{9}=\frac{2}{3}$

We obtain the following exercise:

$3+\frac{2}{3}-2=$

Lastly we solve the exercise from left to right:

$3+\frac{2}{3}=3\frac{2}{3}$

$3\frac{2}{3}-2=1\frac{2}{3}$

According to the rules of the order of arithmetic operations, we must first place the two multiplication exercises inside of the parentheses:

$(\frac{1}{4}\times\frac{1}{3})+(4\times\frac{3}{4})=$

We then focus on the left parenthesis and combine the multiplication exercise:

$(\frac{1}{4}\times\frac{1}{3})=\frac{1\times1}{4\times3}=\frac{1}{12}$

Next we focus on the right parenthesis and we again combine the multiplication exercise:

$(4\times\frac{3}{4})=\frac{4\times3}{4}=\frac{12}{4}=3$

Finally we obtain the following exercise:

$\frac{1}{12}+3=3\frac{1}{12}$

To solve the expression $\frac{2}{3}\times\frac{1}{4}+\frac{1}{6}\times\frac{5}{2}$, we need to follow the order of operations (also known as BODMAS/BIDMAS: Brackets, Orders (i.e., powers and roots, etc.), Division and Multiplication, Addition and Subtraction). Multiplication and division should be handled from left to right before addition or subtraction.

First, we perform the multiplication:

• $\frac{2}{3} \times \frac{1}{4}$: To multiply fractions, multiply the numerators and multiply the denominators.
  Numerator: $2 \times 1 = 2$
  Denominator: $3 \times 4 = 12$
  Thus, $\frac{2}{3} \times \frac{1}{4} = \frac{2}{12}$.
• Simplify $\frac{2}{12}$: The greatest common divisor (GCD) of 2 and 12 is 2.
  So, divide both the numerator and the denominator by 2:
  $\frac{2\div2}{12\div2} = \frac{1}{6}$.
• $\frac{1}{6} \times \frac{5}{2}$: Again, multiply the numerators and multiply the denominators.
  Numerator: $1 \times 5 = 5$
  Denominator: $6 \times 2 = 12$
  Thus, $\frac{1}{6} \times \frac{5}{2} = \frac{5}{12}$.

Now, we have the expression $\frac{1}{6} + \frac{5}{12}$.

To add these fractions, find a common denominator. The least common multiple of 6 and 12 is 12.

• Convert $\frac{1}{6}$ to have a denominator of 12.
  Multiply the numerator and denominator by 2:
  $\frac{1\times2}{6\times2} = \frac{2}{12}$.
• Now, add $\frac{2}{12} + \frac{5}{12}$.
  Add the numerators and keep the common denominator:
  $\frac{2+5}{12} = \frac{7}{12}$.

Therefore, the answer is $\frac{7}{12}$, which matches the given correct answer.

To solve the equation $\frac{4}{9}\times\frac{1.5}{2}+\frac{3}{4}\times\frac{3}{3}=$, we will carefully apply the orders of operations, which include handling fractions with attention to multiplication and addition.

Step 1: First, evaluate the multiplication of fractions on the left side of the addition sign. Handle the multiplication $\frac{4}{9}\times\frac{1.5}{2}$. We'll convert the decimal to a fraction: $1.5 = \frac{3}{2}$.

• Thus, $\frac{4}{9}\times\frac{1.5}{2} = \frac{4}{9}\times\frac{3}{2}$
• Multiply the fractions: $\frac{4}{9} \times \frac{3}{2} = \frac{4 \times 3}{9 \times 2} = \frac{12}{18} = \frac{2}{3}$ after simplification.

Step 2: Next, consider the multiplication in the second part: $\frac{3}{4}\times\frac{3}{3}$.

• Since $\frac{3}{3}$ is essentially 1, it does not change the value of the other fraction. Hence, $\frac{3}{4}\times\frac{3}{3} = \frac{3}{4}$.

Step 3: With both products calculated, the equation becomes $\frac{2}{3} + \frac{3}{4}$.

Step 4: Now, you need a common denominator to add the fractions. The least common multiple of 3 and 4 is 12.

• Convert $\frac{2}{3}$ to a fraction with denominator 12: $\frac{2}{3} = \frac{8}{12}$.
• Convert $\frac{3}{4}$ to a fraction with denominator 12: $\frac{3}{4} = \frac{9}{12}$.

Step 5: Add the fractions: $\frac{8}{12} + \frac{9}{12} = \frac{17}{12}$.

Thus, the simplified solution for the equation is $\frac{17}{12}$ or as a mixed number, $1\frac{5}{12}$.

Note that the right-hand side of the addition exercise between the fractions gives a result of a whole number, so we'll start with that:

$6\frac{2}{3}+\frac{1}{3}=7$

Giving us:

$7\frac{5}{6}+7=14\frac{5}{6}$

According to the order of operations, we will solve the exercise from left to right.

Let's note that in the first addition exercise, we have an addition between two halves that will give us a whole number, so:

$\frac{1}{2}+3\frac{1}{2}=4$

Now we will get the exercise:

$4+4\frac{2}{4}=$

Let's note that we can simplify the mixed fraction:

$\frac{2}{4}=\frac{1}{2}$

Now the exercise we get is:

$4+4\frac{1}{2}=8\frac{1}{2}$

According to the rules of the order of operations in arithmetic, we solve the exercise from left to right.

Let's note that:

$\frac{1}{3}+\frac{2}{3}=\frac{3}{3}=1$

We should obtain the following exercise:

$1+2\frac{3}{4}=3\frac{3}{4}$

Let's solve the exercise from left to right.

We will combine the left expression in the following way:

$\frac{6+8}{7}x=\frac{14}{7}x=2x$

Now we get:

$2x+3\frac{2}{3}x=5\frac{2}{3}x$

First, let's convert the mixed fraction to an improper fraction as follows:

$\frac{1}{5}\times\frac{7}{8}\times\frac{3\times2+2}{3}=$

Let's solve the equation in the numerator:

$\frac{1}{5}\times\frac{7}{8}\times\frac{6+2}{3}=$

$\frac{1}{5}\times\frac{7}{8}\times\frac{8}{3}=$

Since the only operation in the equation is multiplication, we'll combine everything into one equation:

$\frac{1\times7\times8}{5\times8\times3}=$

Let's simplify the 8 in the numerator and denominator of the fraction:

$\frac{1\times7}{5\times3}=$

Let's solve the equations in the numerator and denominator and we get:

$\frac{7}{15}$

We move the fraction to the beginning of the exercise and will place the rest of the exercise in parentheses to make solving the equation easier:

$\frac{a}{4}+(2-2)=$

$\frac{a}{4}+0=\frac{a}{4}$

This exercise can be solved in order, but to make it easier, the associative property can be used

$-5+(3+4)=$

$-5+7=$

$7-5=2$

Let's begin by combining the simple fractions into a single multiplication exercise:

$\frac{3\times2}{4\times3}\times2\frac{1}{4}x=$

Let's now proceed to solve the exercise in the numerator and denominator:

$\frac{6}{12}\times2\frac{1}{4}x=$

Finally we'll simplify the simple fraction in order to obtain the following:

$\frac{1}{2}\times2\frac{1}{4}x=1\frac{1}{8}x$

First, let's convert all mixed fractions to simple fractions:

$\frac{3\times6+5}{6}\times\frac{5\times6+5}{6}\times\frac{1}{3}x=$

Let's solve the exercises with the eight fractions:

$\frac{18+5}{6}\times\frac{30+5}{6}\times\frac{1}{3}x=$

$\frac{23}{6}\times\frac{35}{6}\times\frac{1}{3}x=$

Since the exercise only involves multiplication, we'll combine all the numerators and denominators:

$\frac{23\times35}{6\times6\times3}x=\frac{805}{108}x$

First, we'll convert the mixed fractions to simple fractions as follows:

$\frac{2}{3}\times\frac{7\times3+2}{3}\times\frac{3\times2+1}{2}=$

Let's solve the exercises in the fraction multiplier:

$\frac{2}{3}\times\frac{21+2}{3}\times\frac{6+1}{2}=$

$\frac{2}{3}\times\frac{23}{3}\times\frac{7}{2}=$

Since the only operation in the exercise is multiplication, we'll combine everything into one exercise and get:

$\frac{2\times23\times7}{3\times3\times2}=\frac{46\times7}{9\times2}=\frac{322}{18}$

First, let's find a common denominator for 4, 8, and 6: it's 24.

Now, we'll multiply each fraction by the appropriate number to get:

$\frac{5\times4}{6\times4}x+\frac{7\times3}{8\times3}x+\frac{2\times6}{4\times6}x=$

Let's solve the multiplication exercises in the numerator and denominator:

$\frac{20}{24}x+\frac{21}{24}x+\frac{12}{24}x=$

We'll connect all the numerators:

$\frac{20+21+12}{24}x=\frac{41+12}{24}x=\frac{53}{24}x$

Let's break down the numerator into a smaller addition exercise:

$\frac{48+5}{24}=\frac{48}{24}+\frac{5}{24}=2+\frac{5}{24}=2\frac{5}{24}x$

First, let's convert the mixed fraction to a simple fraction as follows:

$\frac{7}{8}\times\frac{8\times2+7}{8}\times\frac{1}{4}=$

Let's solve the exercise in the numerator:

$\frac{7}{8}\times\frac{16+7}{8}\times\frac{1}{4}=$

$\frac{7}{8}\times\frac{23}{8}\times\frac{1}{4}=$

Since the only operation in the exercise is multiplication, we'll combine everything into one exercise:

$\frac{7\times23\times1}{8\times8\times4}=$

Let's solve the exercises in the numerator and denominator:

$\frac{7\times23}{64\times4}=\frac{161}{256}$

We begin by converting the decimal numbers into mixed fractions:

$4\frac{1}{10}\times1\frac{6}{10}\times3\frac{2}{10}+4\frac{7}{10}=$

We then convert the mixed fractions into simple fractions:

$\frac{41}{10}\times\frac{16}{10}\times\frac{32}{10}+\frac{47}{10}=$

We solve the exercise from left to right:

$\frac{41\times16}{10\times10}=\frac{656}{100}$

This results in the following exercise:

$\frac{656}{100}\times\frac{32}{10}+\frac{47}{10}=$

We solve the multiplication exercise:

$\frac{656\times32}{100\times10}=\frac{20,992}{1,000}$

Now we get the exercise:

$\frac{20,992}{1,000}+\frac{47}{10}=$

We then multiply the fraction on the right so that its denominator is also 1000:

$\frac{47\times100}{10\times100}=\frac{4,700}{1,000}$

We obtain the exercise:

$\frac{20,992}{1,000}+\frac{4,700}{1,000}=\frac{20,992+4,700}{1,000}=\frac{25,692}{1,000}$

Lastly we convert the simple fraction into a decimal number:

$\frac{25,692}{1,000}=25.692$