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0.31 | 0.308 | |
Convert to fractions. | $\frac{31}{100}$ | $\frac{308}{1000}$ |
We need a common denominator to compare them. | ||
$\frac{310}{1000}$ | $\frac{308}{1000}$ |
Because 310>308, we know that $\frac{310}{1000}>\frac{308}{1000}.$ Therefore, 0.31>0.308.
Notice what we did in converting 0.31 to a fraction—we started with the fraction $\frac{31}{100}$ and ended with the equivalent fraction $\frac{310}{1000}.$ Converting $\frac{310}{1000}$ back to a decimal gives 0.310. So 0.31 is equivalent to 0.310. Writing zeros at the end of a decimal does not change its value!
We say 0.31 and 0.310 are equivalent decimals .
Two decimals are equivalent if they convert to equivalent fractions.
We use equivalent decimals when we order decimals.
The steps we take to order decimals are summarized here.
Order $0.64\_\_\_0.6$ using $<$ or $>.$
$\begin{array}{cccccc}\begin{array}{c}\text{Write the numbers one under the other,}\hfill \\ \text{lining up the decimal points}.\hfill \end{array}\hfill & & & & & \begin{array}{c}\phantom{\rule{1.4em}{0ex}}0.64\hfill \\ \phantom{\rule{1.4em}{0ex}}0.6\hfill \end{array}\hfill \\ \\ \\ \begin{array}{c}\text{Add a zero to 0.6 to make it a decimal}\hfill \\ \text{with 2 decimal places.}\hfill \end{array}\hfill & & & & & \hfill \begin{array}{c}\hfill 0.64\hfill \\ \hfill 0.60\hfill \end{array}\hfill \\ \text{Now they are both hundredths.}\hfill & & & & & \\ \\ \\ \text{64 is greater than 60.}\hfill & & & & & \hfill 64>60\hfill \\ \\ \\ \text{64 hundredths is greater than 60 hundredths.}\hfill & & & & & \hfill 0.64>0.60\hfill \\ \\ \\ & & & & & \hfill 0.64>0.6\hfill \end{array}$
Order each of the following pairs of numbers, using $<\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}>\text{:}\phantom{\rule{0.2em}{0ex}}0.42\_\_\_0.4.$
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Order each of the following pairs of numbers, using $<\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}>\text{:}\phantom{\rule{0.2em}{0ex}}0.18\_\_\_0.1.$
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Order $0.83\_\_\_0.803$ using $<$ or $>.$
$\begin{array}{cccccc}& & & & & 0.83\_\_\_0.803\hfill \\ \\ \\ \begin{array}{c}\text{Write the numbers one under the other,}\hfill \\ \text{lining up the decimals.}\hfill \end{array}\hfill & & & & & \begin{array}{c}0.83\hfill \\ 0.803\hfill \end{array}\hfill \\ \\ \\ \begin{array}{c}\text{They do not have the same number of}\hfill \\ \text{digits.}\hfill \end{array}\hfill & & & & & \begin{array}{c}0.830\hfill \\ 0.803\hfill \end{array}\hfill \\ \text{Write one zero at the end of}\phantom{\rule{0.2em}{0ex}}0.83.\hfill & & & & & \\ \\ \\ \begin{array}{c}\text{Since}\phantom{\rule{0.2em}{0ex}}830>803,830\phantom{\rule{0.2em}{0ex}}\text{thousandths is}\hfill \\ \text{greater than 803 thousandths.}\hfill \end{array}\hfill & & & & & 0.830>0.803\hfill \\ \\ \\ & & & & & 0.83>0.803\hfill \end{array}$
Order the following pair of numbers, using $<\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}>\text{:}\phantom{\rule{0.2em}{0ex}}0.76\_\_\_0.706.$
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Order the following pair of numbers, using $<\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}>\text{:}\phantom{\rule{0.2em}{0ex}}0.305\_\_\_0.35.$
<
When we order negative decimals, it is important to remember how to order negative integers. Recall that larger numbers are to the right on the number line. For example, because $\mathrm{-2}$ lies to the right of $\mathrm{-3}$ on the number line, we know that $\mathrm{-2}>\mathrm{-3}.$ Similarly, smaller numbers lie to the left on the number line. For example, because $\mathrm{-9}$ lies to the left of $\mathrm{-6}$ on the number line, we know that $\mathrm{-9}<\mathrm{-6}.$ See [link] .
If we zoomed in on the interval between 0 and $\mathrm{-1},$ as shown in [link] , we would see in the same way that $\mathrm{-0.2}>\mathrm{-0.3}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}-0.9<\mathrm{-0.6}.$
Use $<$ or $>$ to order $\mathrm{-0.1}\_\_\_\mathrm{-0.8}.$
$\begin{array}{cccccc}& & & & & \mathrm{-0.1}\_\_\_\mathrm{-0.8}\hfill \\ \\ \\ \begin{array}{c}\text{Write the numbers one under the other, lining up the}\hfill \\ \text{decimal points.}\hfill \end{array}\hfill & & & & & \begin{array}{c}-0.1\hfill \\ -0.8\hfill \end{array}\hfill \\ \text{They have the same number of digits.}\hfill & & & & & \\ \\ \\ \text{Since}\phantom{\rule{0.2em}{0ex}}\mathrm{-1}>\mathrm{-8},\mathrm{-1}\phantom{\rule{0.2em}{0ex}}\text{tenth is greater than}\phantom{\rule{0.2em}{0ex}}\mathrm{-8}\phantom{\rule{0.2em}{0ex}}\text{tenths.}\hfill & & & & & \mathrm{-0.1}>\mathrm{-0.8}\hfill \end{array}$
Order the following pair of numbers, using<or>: $\mathrm{-0.3}\_\_\_\mathrm{-0.5}.$
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Order the following pair of numbers, using<or>: $\mathrm{-0.6}\_\_\_\mathrm{-0.7}.$
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