Chapter 0.4, Problem 7E

To determine

To calculate: The order of set of rational numbers $3.8,3.06,3\frac{1}{6},3\frac{3}{4}$ from least to greatest.

Answer to Problem 7E

The correct order of rational numbers from least to greatest is $3.06<3\frac{1}{6}<3\frac{3}{4}<3.8$ .

Explanation of Solution

Given information:

The set of rational numbers $3.8,3.06,3\frac{1}{6},3\frac{3}{4}$ .

Formula used:

Rational numbers are of the form $\frac{p}{q}$ where p and q are integers and $\text{q}\ne \text{0}$ .

Mixed fractions of the form $m\frac{p}{q}$ can be converted into proper fractions as,

  $m\frac{p}{q}=\frac{\left(m\times q\right)+p}{q}$

Calculation:

Consider the provided statement“The set of rational numbers $3.8,3.06,3\frac{1}{6},3\frac{3}{4}$ .”

Recall that the mixed fractions can be converted into proper fractions as,

  $\text{m}\frac{\text{p}}{\text{q}}\text{=}\frac{\left(\text{m×q}\right)\text{+p}}{\text{q}}$

So, $3\frac{1}{6}$ will be written as,

  $\begin{array}{c}3\frac{1}{6}=\frac{\left(6\times 3\right)+1}{6}\\ 3\frac{1}{6}=\frac{19}{6}\end{array}$

And, $3\frac{3}{4}$ will be written as,

  $\begin{array}{c}3\frac{3}{4}=\frac{\left(3\times 4\right)+3}{4}\\ 3\frac{3}{4}=\frac{15}{4}\end{array}$

Now, convert $\frac{19}{6}$ and $\frac{15}{4}$ into decimal notation and then compare the set of rational numbers.

Decimal notation for $\frac{19}{6}$ is equivalent to $3.16$ .

Decimal notation for $\frac{15}{4}$ is $3.75$ .

Thus, the set of rational number $3.8,3.06,3\frac{1}{6},3\frac{3}{4}$ becomes $3.8,3.06,3.16,3.75$ .

Now, arrange this set of rational numbers from least to greatest as,

  $\begin{array}{l}3.06<3.16<3.75<3.8\\ 3.06<3\frac{1}{6}<3\frac{3}{4}<3.8\end{array}$

Hence, the correct order of rational numbers from least to greatest is $3.06<3\frac{1}{6}<3\frac{3}{4}<3.8$ .