Verified Answer :
Answer:
Mrs. Herman asked the class to write an expanded form for 412.638.
Nancy wrote the expanded form using fractions. Her response is like:
\((4\times100)+(1\times10)+(2\times1)+(6\times\frac{1}{10})+(3\times\frac{1}{100})+(8\times\frac{1}{1000} )\)
Charles wrote the expanded form using decimals. His response is like:
\((4\times100)+(1\times10)+(2\times1)+(6\times0.1)+(3\times0.01)+(8\times0.001)\)
The number 412.638 has six digits, three digits before decimal and three digits after the decimal.
Further explanation:
In number every digit has a place value. Place value represent by a digit in a number on the basis of its place in the number defined by place value chart.
From place value chart of decimal numbers, expand the number 412.638 in fraction as follows:
\(\begin{aligned}412.638&=4\times\left({\dfrac{{100}}{1}}\right)+1\times\left({\dfrac{{10}}{1}}\right)+2\times\left({\dfrac{1}{1}}\right)+6\times\left({\dfrac{1}{{10}}}\right)+3\times\left({\dfrac{1}{{100}}}\right)+8\times\left({\dfrac{1}{{1000}}}\right)\\&=\dfrac{{400}}{1}+\dfrac{{10}}{1}+\dfrac{2}{1}+\dfrac{3}{5}+\dfrac{3}{{100}}+\dfrac{1}{{125}}\\\end{aligned}\)
Therefore, the expanded form for 412.638 in fractions is \(\boxed{\dfrac{{400}}{1}+\dfrac{{10}}{1}+\dfrac{2}{1}+\dfrac{3}{5}+\dfrac{3}{{100}}+\dfrac{1}{{125}}}\).
So, Nancy write the number in the form \(\boxed{\dfrac{{400}}{1}+\dfrac{{10}}{1}+\dfrac{2}{1}+\dfrac{3}{5}+\dfrac{3}{{100}}+\dfrac{1}{{125}}}\).
From place value chart of decimal numbers, expand the number 412.638 as follows:
\(\begin{aligned}412.638&=4\times\left({100}\right)+1\times\left({10}\right)+2\times\left(1\right)+6\times\left({0.1}\right)+3\times\left({0.01}\right)+8\times\left({0.001}\right)\\&=400+10+2+0.6+0.03+0.008\\&=400.000+10.000+2.000+0.6.000+0.03.000+0.008.000\\\end{aligned}\)
Therefore, the expanded form for 412.638 in decimals is \(\boxed{400.000+10.000+2.000+0.600+0.030+0.008}\).
So, Charles write the number in the form \(\boxed{400.000+10.000+2.000+0.600+0.030+0.008}\).
Thus, the expanded form for 412.638 in fractions is \(\boxed{\dfrac{{400}}{1}+\dfrac{{10}}{1}+\dfrac{2}{1}+\dfrac{1}{6}+\dfrac{1}{{30}}+\dfrac{1}{{800}}}\) and in decimals is \(\boxed{400.000+10.000+2.000+0.600+0.030+0.008}\).
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Answer details:
Grade: Middle School
Subject: Mathematics
Chapter: Decimal system
Keywords: expanded form, fraction form, decimal form, expand, place value chart, Charles, Nancy, number.