Verify if (3·8+5)/(4+3·8) = 5/4: Fraction Simplification Check

Determine whether the simplification below is correct:

$\frac{3\cdot8+5}{4+3\cdot8}=\frac{5}{4}$

In this question we are asked whether we can "reduce" the expression on the left side.

In order to achieve this, first, let's numerically simplify the expression on the left side and calculate the value of the fraction in order to see if we obtain the same fraction (or its equivalent) that is on the right side:

$\frac{3\cdot8+5}{4+3\cdot8} =\\ \frac{24+5}{4+24}=\\ \frac{29}{28}$

as shown below:

$\frac{3\cdot8+5}{4+3\cdot8}\stackrel{?}{\rightarrow }\frac{5}{4} \\ \downarrow\\ \frac{29}{28}\stackrel{?}{\rightarrow }\frac{5}{4}$

Now we need to determine if the fractions are equivalent. For this we note that the denominator of the fraction on the right side, the number 28, is a whole multiple of the denominator of the fraction on the left side, 4, this whole multiple is the number 7 given that:

$4\cdot7=28$

Therefore we will expand the fraction on the right side, this we will do by multiplying both the numerator and denominator by 7. This is an operation that will not change the value of the fraction, given that it is equivalent to multiplying the fraction by 1. An operation that never changes the value of the number, as will be demonstrated in the following calculation:

$\frac{5}{4} =\\ \frac{5}{4} \cdot1=\\ \frac{5}{4} \cdot\frac{7}{7}=\\ \frac{5\cdot7}{4\cdot7}=\\ \frac{35}{28}$

We therefore obtain the following:

$\frac{29}{28}\stackrel{?}{\rightarrow }\frac{5}{4} \\ \downarrow\\ \frac{29}{28}\stackrel{?}{\rightarrow }\frac{35}{28}$

Now that the denominators of the fractions on the right and left sides are identical, we can determine that the fractions are different from each other since:

$29\neq35$

and therefore the expression on the left and the expression on the right are not identical,

Meaning:

$\frac{3\cdot8+5}{4+3\cdot8}=\frac{29}{30}\stackrel{!}{\not{\rightarrow} }\frac{5}{4}$

In other words - the reduction operation of the multiplication factor $3\cdot8$ is incorrect.

Therefore the correct answer is answer B.