All Operations in the Order of Operations: Using fractions

In the first stage of the exercise, you need to calculate the multiplication.

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

From here you can continue with the rest of the addition and subtraction operations, from right to left.

$5-1+1=5$

To solve the equation $\frac{25+25}{10}=$, we need to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). In this problem, we will tackle the following steps:

• Parentheses: Solve any operations inside the parentheses first. Here, we have a simple addition $25 + 25$.
• So, $25 + 25 = 50$.
• Next, we address the division, which comes after addition:
• Compute the division of the result by 10: $\frac{50}{10}$.
• The result is $5$.

Thus, the value of $\frac{25+25}{10}$ is $5$.

Let's begin by solving the numerator of the fraction from left to right, according to the order of operations:

$90-15=75$

$75-3=72$

We should obtain the following exercise:

$\frac{72}{8}=72:8=9$

To solve the expression $\frac{0.5 + 2}{5}$, we must follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). Here, we need to focus on the addition within the fraction, and then the division that forms the fraction.

Let's break down the steps:

• Start with the expression inside the numerator: $0.5 + 2$.
• Perform the addition: $0.5 + 2 = 2.5$.
• The expression now becomes: $\frac{2.5}{5}$.
• Next, perform the division: divide 2.5 by 5. To do this, consider the division operation:
• $2.5 \div 5$
• Convert 2.5 to a fraction: $\frac{5}{2}$
• Divide by 5: $\frac{5}{2} \times \frac{1}{5}$ (since dividing by a number is the same as multiplying by its reciprocal).
• This becomes: $\frac{5 \times 1}{2 \times 5} = \frac{5}{10}$
• Simplify the fraction $\frac{5}{10}$ to its lowest terms by dividing both the numerator and the denominator by their greatest common divisor (5), resulting in: $\frac{1}{2}$.

Therefore, the value of the expression $\frac{0.5+2}{5}$ is $\frac{1}{2}$, as given.

To solve the expression $\frac{18}{18+36}$, we need to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). Here we have only addition and division.

First, we perform the operation inside the parentheses, which is addition in this case:

• Add the numbers in the denominator: $18 + 36 = 54$.

Now, we substitute back into the fraction:$\frac{18}{54}$.

Next, simplify the fraction:

• We look for the greatest common divisor (GCD) of 18 and 54. The GCD is 18.

• Divide both the numerator and the denominator by the GCD:

  • $\frac{18}{18} = 1$

  • $\frac{54}{18} = 3$

Thus, the simplified fraction is $\frac{1}{3}$.

The final answer is: $\frac{1}{3}$.

To solve the expression $\frac{9}{42+7}$, we need to follow the order of operations, commonly known by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). In this question, we focus on Parentheses and Addition.

Step-by-Step Solution:

• First, we identify the operation inside the parentheses: $42 + 7$.
• According to the order of operations, we must solve what is inside the parentheses before dealing with any division. So, we perform the addition first.
• Calculate $42 + 7$ to get $49$.
• We then substitute $49$ back into the original expression in the place of $42 + 7$.
• This gives us a simplified expression: $\frac{9}{49}$.
• Since $9$ and $49$ do not have any common factors aside from $1$, this fraction cannot be simplified further.

Therefore, the final answer is $\frac{9}{49}$.

We are given the expression $\frac{100+1}{25}$ and we need to evaluate it step by step according to the order of operations.

Step 1: Evaluate the expression inside the fraction.
We first perform the addition within the numerator:
$100 + 1 = 101$

Step 2: Divide the result by the denominator.
Now we can simplify the fraction:
$\frac{101}{25}$

Step 3: Convert the improper fraction to a mixed number.
To convert $\frac{101}{25}$ to a mixed number, we divide 101 by 25.
25 goes into 101 four times with a remainder:

• 25 times 4 equals 100
• 101 minus 100 equals 1

Therefore, $\frac{101}{25}$ is equivalent to $4 \frac{1}{25}$.

According to the order of operations rules, we solve the exercise from left to right:

$7-1=6$

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

According to the order of operations rules, we'll solve the exercise from left to right:

$7+1=8$

$8+0.2=8.2$

According to the rules of the order of arithmetic operations, we must first enclose both the multiplication and division exercises inside of parentheses:

$1+(2\times3)-(7:4)=$

We then solve the exercises within the parentheses:

$2\times3=6$

$7:4=\frac{7}{4}$

We obtain the following:

$1+6-\frac{7}{4}=$

We continue by solving the exercise from left to right:

$1+6=7$

$7-\frac{7}{4}=$

Lastly we break down the numerator of the fraction with a sum exercise as seen below:

$7-(\frac{4+3}{4})$

$7-(\frac{4}{4}+\frac{3}{4})$

$7-(1+\frac{3}{4})$

$7-1\frac{3}{4}=5\frac{1}{4}$

Let's begin by solving the numerator of the fraction according to the order of operations, from left to right:

$5+3=8$

$8-2=6$

We should obtain the following exercise:

$\frac{6}{3}=6:3=2$

Let's begin by solving the numerator of the fraction, from left to right, according to the order of operations:

$12+8=20$

We should obtain the following exercise:

$\frac{20}{5}=20:5=4$

According to the order of operations in arithmetic, multiplication and division take precedence over addition and subtraction.

We'll start with the division operation and write the fractions as decimal fractions, then as simple fractions:

$0.1:0.2=\frac{0.1}{0.2}=\frac{1}{2}$

In the next step, we'll write the decimal fraction 0.5 as a simple fraction:

$0.5=\frac{1}{2}$

Now let's solve the problem

$\frac{1}{2}-\frac{1}{2}=0$

According to the order of operations rules, we must first solve the expression within the parentheses:

$85+5=90$

We should obtain the following expression:

$90:10=9$

First, let's solve the numerator of the fraction:

$20-5=15$

Now let's solve the denominator of the fraction:

$7+3=10$

We get:

$\frac{15}{10}=1\frac{5}{10}=1\frac{1}{2}$

Given that, according to the rules of the order of operations, parentheses come first, we will first solve the exercise that appears within the parentheses.

$4\times2-\frac{9}{3}=$

We solve the multiplication exercise:

$4\times2=8$

We divide the fraction (numerator by denominator)$\frac{9}{3}=3$

And now the exercise obtained within the parentheses is$8-3=5$

Finally, we divide:$12:5=\frac{12}{5}$

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:

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

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:

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

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

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, $1\frac{5}{12}$.

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.