The Commutative property: Using fractions

To solve the exercise:

$0.8+\frac{2}{10}-\frac{3}{2}\times\frac{4}{2}+\frac{1}{2}=$

We will use 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))

Step-by-step Solution:

• First, simplify the fractions and perform multiplication:

  $\frac{3}{2} \times \frac{4}{2}$

  Multiply the numerators and the denominators:

  $\frac{3 \times 4}{2 \times 2} = \frac{12}{4}$

  Simplify the fraction:

  $\frac{12}{4} = 3$

• Now, substitute back into the expression:

  $0.8 + \frac{2}{10} - 3 + \frac{1}{2}$

• Convert the decimals and fractions to have a common base for easy calculation:

  $0.8 = \frac{8}{10}$ and simplify $\frac{2}{10} = \frac{1}{5} = \frac{2}{10}$ as it is.

  The fraction $\frac{1}{2} = \frac{5}{10}$.

• Now simplify without changing the structure of the original question:

  $\frac{8}{10} + \frac{2}{10} - 3 + \frac{5}{10}$

• Add the fractions:

  $\frac{8}{10} + \frac{2}{10} + \frac{5}{10} = \frac{15}{10} = 1.5$

• Finally, subtract the whole number:

  $1.5 - 3$

  The result is:

  $1.5 - 3 = -1.5$

Thus, the final result is -1.5, indicating a typo or misrepresentation in the given correct answer statement 1.5-. However, following the provided order of operations correctly, the computed solution is $-1.5$.

Given that this is an exercise with only addition operation, we can change the order of the numbers.

We organize the exercise in a way that we can obtain a pair that gives us an integer.

Keep in mind that there is a pair of fractions that if we add them we will obtain an integer:

$4\frac{2}{3}+3\frac{1}{3}+2\frac{2}{7}+1\frac{3}{7}=$

We solve the exercise from left to right:

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

$4+3=7$

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

$7+1=8$

Now we obtain the exercise:

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

We leave the 8 aside and add the rest of the exercise:

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

$2+1=3$

$\frac{2}{7}+\frac{3}{7}=\frac{5}{7}$

Now we obtain the exercise:

$8+3+\frac{5}{7}=11\frac{5}{7}$

To make things simpler, we use the substitution property and arrange the exercise in the following manner:

$5.2+12.4-7.4+3.2+6.6=$

Keep in mind that the subtraction operation here gives us a whole number:

$12.4-7.4=5$

Now we obtain the exercise

$5.2+5+3.2+6.6=$

We solve the addition exercise:

$3.2+6.6=9.8$

And we obtain the exercise:

$5.2+5+9.8=$

We arrange the exercise using the substitution property to make fiding the solution simpler:

$5.2+9.8+5=$

We solve the exercise from left to right:

$5.2+9.8=15$

$15+5=20$

Let's break down the given expression step by step by following the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

The expression given is:

$\frac{2}{4}+\frac{2}{5}\times\frac{5}{4}-0.2+0.4=$

1. Simplifying Fractions: Simplify the fraction $\frac{2}{4}$:

• $\frac{2}{4} = \frac{1}{2}$

So, the expression becomes:

$\frac{1}{2}+\frac{2}{5}\times\frac{5}{4}-0.2+0.4$

2. Multiplication of Fractions: Perform the multiplication before heading into addition and subtraction.

• $\frac{2}{5} \times \frac{5}{4} = \frac{2 \times 5}{5 \times 4} = \frac{10}{20} = \frac{1}{2}$

The expression now looks like:

$\frac{1}{2} + \frac{1}{2} - 0.2 + 0.4$

3. Addition of Fractions and Decimals: Add$\frac{1}{2}$and $\frac{1}{2}$:

• $\frac{1}{2} + \frac{1}{2} = 1$

With this result, add to the decimals:

$1 - 0.2 + 0.4$

4. Perform Addition and Subtraction of Decimals: Let's calculate from left to right:

• $1 - 0.2 = 0.8$

• $0.8 + 0.4 = 1.2$

Thus, the result of $\frac{2}{4}+\frac{2}{5}\times\frac{5}{4}-0.2+0.4$ is $1.2$

Multiply each fraction as follows:

Multiply the whole number by the denominator of the fraction and add the number in the numerator of the fraction.

That is:

$4\frac{1}{4}=\frac{4\times4+1}{4}=\frac{16+1}{4}=\frac{17}{4}$

$3\frac{4}{9}=\frac{9\times3+4}{9}=\frac{27+4}{9}=\frac{31}{9}$

$3\frac{1}{17}=\frac{17\times3+1}{17}=\frac{51+1}{17}=\frac{52}{17}$

Now we get the exercise:

$\frac{17}{4}\times\frac{31}{9}\times\frac{52}{17}=$

We simplify the 17 and get:

$\frac{31\times52}{4\times9}=\frac{52}{4}\times\frac{31}{9}=13\times\frac{31}{9}=\frac{403}{9}=44\frac{7}{9}$

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,

Therefore, we'll start by simplifying the expressions in parentheses first:
$(\frac{1}{4}+\frac{7}{4}-\frac{5}{4}-\frac{1}{4})\cdot10:7:5=\\ \frac{1+7-5-1}{4}\cdot10:7:5 =\\ \frac{2}{4}\cdot10:7:5 = \\ \frac{1}{2}\cdot10:7:5$

We calculated the expression inside the parentheses by adding the fractions, which we did by creating one fraction using the common denominator (4) which in this case is the denominator in all fractions, so we only added/subtracted the numerators (according to the fraction sign), then we reduced the resulting fraction,

We'll continue and note that between multiplication and division operations there is no defined precedence for either operation, therefore we'll calculate the result of the expression obtained in the last step step by step from left to right (which is the regular order in arithmetic operations), meaning we'll first perform the multiplication operation, which is the first from the left, and then we'll perform the division operation that comes after it, and so on:

$\frac{1}{2}\cdot10:7:5 =\\ \frac{1\cdot10}{2}:7:5 =\\ \frac{10}{2}:7:5 =\\ 5:7:5 =\\ \frac{5}{7}:5$

In the first step, we performed the multiplication of the fraction by the whole number, remembering that multiplying by a fraction means multiplying by the fraction's numerator, then we simplified the resulting fraction and reduced it (effectively performing the division operation that results from it), in the final step we wrote the division operation as a simple fraction, since this division operation yields a non-whole result,

We'll continue and to perform the final division operation, we'll remember that dividing by a number is the same as multiplying by its reciprocal, and therefore we'll replace the division operation with multiplication by the reciprocal:

$\frac{5}{7}:5 =\\ \frac{5}{7}\cdot\frac{1}{5}$

In this case we preferred to multiply by the reciprocal because the divisor in the expression is a fraction and it's more convenient to perform multiplication between fractions,

We will then perform the multiplication between the fractions we got in the last step, while remembering that multiplication between fractions is performed by multiplying numerator by numerator and denominator by denominator while maintaining the fraction line, then we'll simplify the resulting expression by reducing it:

$\frac{5}{7}\cdot\frac{1}{5} =\\ \frac{5\cdot1}{7\cdot5}=\\ \frac{5}{35}=\\ \frac{1}{7}$

Let's summarize the solution steps, we got that:

$(\frac{1}{4}+\frac{7}{4}-\frac{5}{4}-\frac{1}{4})\cdot10:7:5=\\ \frac{1+7-5-1}{4}\cdot10:7:5 =\\ \frac{1}{2}\cdot10:7:5 =\\ 5:7:5 =\\ \frac{5}{7}\cdot\frac{1}{5} =\\ \frac{1}{7}$

Therefore the correct answer is answer B.

Let's consider the number of kilometers Damian ran each day

On the second day it is written "round trip", that is, twice.

Therefore, we write the following exercise:

$3.4+2\times1.18+2.6=$

We solve the multiplication exercise:

$3.4+2.36+2.6=$

We arrange the exercise using the commutative property, to make it more convenient to solve:

$3.4+2.6+2.36=$

We solve the exercise from left to right:

$3.4+2.6=6$

$6+2.36=8.36$

We rewrite the decimal fraction in the form of a mixed fraction:

$5.25=5\frac{1}{4}$

Now we get the exercise:

$5\frac{1}{4}\cdot\frac{2}{3}\cdot\frac{7}{21}=$

We write the mixed fraction as a simple fraction:

$5\frac{1}{4}=\frac{4\times5+1}{4}=\frac{20+1}{4}=\frac{21}{4}$

Now we obtain:

$\frac{21}{4}\cdot\frac{2}{3}\cdot\frac{7}{21}=$

We simplify the 21 and obtain:

$\frac{7}{4}\cdot\frac{2}{3}=\frac{14}{12}=\frac{12+2}{12}=1\frac{2}{12}=1\frac{1}{6}$

According to the order of operations, we will first solve the expression in parentheses.

Note that since the denominators are not common, we will look for a number that is both divisible by 2 and 3. That is 6.

We will multiply one-third by 2 and one-half by 3, now we will get the expression:

$\frac{1}{4}\times(\frac{2+3}{6})=$

Let's solve the numerator of the fraction:

$\frac{1}{4}\times\frac{5}{6}=$

We will combine the fractions into a multiplication expression:

$\frac{1\times5}{4\times6}=\frac{5}{24}$