The Commutative property: Using fractions

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:

$5.2+9.8=15$

$15+5=20$

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 $-1.5$.

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 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 $1.2$

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:

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

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:

$3.4+2.6=6$

$6+2.36=8.36$

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:

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