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Write as a fraction or mixed number. Simplify the answer if possible.
ⓐ $\phantom{\rule{0.2em}{0ex}}5.3$ ⓑ $\phantom{\rule{0.2em}{0ex}}6.07$ ⓒ $\phantom{\rule{0.2em}{0ex}}\mathrm{-0.234}$
Write as a fraction or mixed number. Simplify the answer if possible.
ⓐ $\phantom{\rule{0.2em}{0ex}}8.7$ ⓑ $\phantom{\rule{0.2em}{0ex}}1.03$ ⓒ $\phantom{\rule{0.2em}{0ex}}\mathrm{-0.024}$
Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line.
Locate $0.4$ on a number line.
The decimal $0.4$ is equivalent to $\frac{4}{10},$ so $0.4$ is located between $0$ and $1.$ On a number line, divide the interval between $0$ and $1$ into $10$ equal parts and place marks to separate the parts.
Label the marks
$0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0.$ We write
$0$ as
$0.0$ and
$1$ as
$1.0,$ so that the numbers are consistently in tenths. Finally, mark
$0.4$ on the number line.
Locate $\mathrm{-0.74}$ on a number line.
The decimal
$\mathrm{-0.74}$ is equivalent to
$-\frac{74}{100},$ so it is located between
$0$ and
$\mathrm{-1}.$ On a number line, mark off and label the hundredths in the interval between
$0$ and
$\mathrm{-1}$ (
$\mathrm{-0.10}$ ,
$\mathrm{-0.20}$ , etc.) and mark
$\mathrm{-0.74}$ between
$\mathrm{-0.70}$ and
$\mathrm{-0.80,}$ a little closer to
$\mathrm{-0.70}$ .
Which is larger, $0.04$ or $0.40?$
If you think of this as money, you know that $\text{\$0.40}$ (forty cents) is greater than $\text{\$0.04}$ (four cents). So,
In previous chapters, we used the number line to order numbers.
Where are $0.04$ and $0.40$ located on the number line?
We see that $0.40$ is to the right of $0.04.$ So we know $0.40>0.04.$
How does $0.31$ compare to $0.308?$ This doesn’t translate into money to make the comparison easy. But if we convert $0.31$ and $0.308$ to fractions, we can tell which is larger.
$0.31$ | $0.308$ | |
Convert to fractions. | $\frac{31}{100}$ | $\frac{308}{1000}$ |
We need a common denominator to compare them. | $\frac{308}{1000}$ | |
$\frac{310}{1000}$ | $\frac{308}{1000}$ |
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.
If two decimals have the same value, they are said to be equivalent decimals.
We say $0.31$ and $0.310$ are equivalent decimals.
Two decimals are equivalent decimals if they convert to equivalent fractions.
Remember, writing zeros at the end of a decimal does not change its value.
Order the following decimals using $<\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}\text{>:}$
ⓐ $\phantom{\rule{0.2em}{0ex}}0.64\phantom{\rule{0.2em}{0ex}}\_\_0.6$
ⓑ $\phantom{\rule{0.2em}{0ex}}0.83\phantom{\rule{0.2em}{0ex}}\_\_0.803$
ⓐ | |
$\phantom{\rule{0.2em}{0ex}}0.64\phantom{\rule{0.2em}{0ex}}\_\_0.6$ | |
Check to see if both numbers have the same number of decimal places. They do not, so write one zero at the right of 0.6. | $\phantom{\rule{0.2em}{0ex}}0.64\phantom{\rule{0.2em}{0ex}}\_\_0.60$ |
Compare the numbers to the right of the decimal point as if they were whole numbers. | $64>60$ |
Order the numbers using the appropriate inequality sign. |
$0.64>0.60$
$0.64>0.6$ |
ⓑ | |
$\phantom{\rule{0.2em}{0ex}}0.83\phantom{\rule{0.2em}{0ex}}\_\_0.803$ | |
Check to see if both numbers have the same number of decimal places. They do not, so write one zero at the right of 0.83. | $\phantom{\rule{0.2em}{0ex}}0.830\phantom{\rule{0.2em}{0ex}}\_\_0.803$ |
Compare the numbers to the right of the decimal point as if they were whole numbers. | $830>803$ |
Order the numbers using the appropriate inequality sign. |
$0.830>0.803$
$0.83>0.803$ |
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