Question

How do you order from least to greatest $\left\{ 5,4,\dfrac{537}{100},5.09,\sqrt{29} \right\}$?

Answer 1

Hint: In the above question, we are given a set of five numbers, some of which are written in the form of decimal, one is in the fractional form, and the last one is in the form of a radical. We have been asked to order them from least to greatest, so we have to compare the given numbers. For this, we have to write all the five numbers in the form of decimal. The fraction $\dfrac{537}{100}$ can be converted to decimal by dividing $537$ by $100$. And the radical $\sqrt{29}$ can be converted to the decimal by using the long division method to obtain the square root. Then finally we can easily compare the decimal numbers obtained which can be arranged in the required order.

Complete step by step solution:
The set of numbers given in the question is $\left\{ 5,4,\dfrac{537}{100},5.09,\sqrt{29} \right\}$.
We can observe that in the given set, apart from having decimal numbers, we also have a fractional number and a radical. For arranging them from least to greatest, we have to write them all in the decimal form. Firstly we consider the fractional number $\dfrac{537}{100}$. For converting it in the decimal, we divide $537$ by $100$ from which we get $5.37$.
Now, we consider the radical $\sqrt{29}$. For writing it in the decimal form, we use the long division method
\[5\overset{5.38}{\overline{\left){\begin{align}
  & \overline{29}.\overline{00} \\
 & \underline{25} \\
 & 103\overline{\left){\begin{align}
  & 400 \\
 & \underline{309} \\
 & 1068\overline{\left){\begin{align}
  & 9100 \\
 & \underline{8544} \\
 & 556 \\
\end{align}}\right.} \\
\end{align}}\right.} \\
\end{align}}\right.}}\]
From the above long division, we got the approximate decimal expansion of $\sqrt{29}$ as $5.38$. So we can now write the given set as $\left\{ 5,4,5.37,5.09,5.38 \right\}$. According to the question, we have to arrange the numbers from greatest to least. We can see that the least number is $4$, then \[5\], then \[5.09\], then $5.37$ and then finally $5.38$. So we can arrange the numbers as $\left\{ 4,5,5.09,5.37,5.38 \right\}$. Writing all the numbers in their original forms, we finally get the required arrangement as $\left\{ 4,5,5.09,\dfrac{537}{100},\sqrt{29} \right\}$.
Hence, we have ordered the given set of numbers from least to greatest.

Note: We must not forget to write the numbers in the form in which they were given in the question. Also, we must obtain the square root correct to two decimal places since we can observe that the numbers are given up to two decimal places.