Chapter 3.1, Problem 43PPS

To determine

To replace $•$ in the sentence $-5.\overline{42}•-5\frac{3}{7}$ with $<,>,\text{or }=$ to make a true sentence.

Answer to Problem 43PPS

  $-5.\overline{42}>-5\frac{3}{7}$

Explanation of Solution

Given:

The sentence, $-5.\overline{42}•-5\frac{3}{7}$ .

Concept Used:

To convert a fraction into an equivalent decimal, use long division method and divide the numerator by the denominator.

Calculation:

In order to checkwhether the fraction on the left of $•$ is less than, greater than or equal to the number on the right of $•$ , first convert the fractions into equivalent decimals and then compare the numbers.

So, to convert the rightfraction into decimal, leave the whole part as it is and divide the numerator by the denominator of fractional part using long division as shown below,

  $7\begin{array}{c}\hfill 0.4285\\ \hfill \overline{)\begin{array}{l}3.0000\\ \underset{\_}{0}\\ 30\\ \underset{\_}{28}\\ 20\\ \underset{\_}{14}\\ 60\\ \underset{\_}{56}\\ 40\\ \underset{\_}{35}\\ 5\end{array}}\end{array}$

So, the second fraction in decimal form can be written as $-5.4285$ . While the first repeating decimal in expanded form can be written as $-5.4242$ .

So, in decimal form the given sentence can be written as $-5.4242•-5.4285$ .

Observe that on the number line as we move from left to right, the value of numbers increase. So, the decimal number to the left of the symbol $•$ is greater than the decimal number to the right. So, replace the symbol by $>$ sign.

Thus, the true sentence is $-5.\overline{42}>-5\frac{3}{7}$ .