Answer :
To solve the given problem, we need to subtract the repeating decimal [tex]\(7.\overline{2}\)[/tex] from the decimal [tex]\(2.8\)[/tex].
1. First, we express [tex]\(7.\overline{2}\)[/tex] as a fraction. Consider the repeating decimal:
[tex]\[ x = 7.2222 \ldots \][/tex]
Multiplying both sides by 10 to shift the decimal point:
[tex]\[ 10x = 72.2222 \ldots \][/tex]
Subtracting the original [tex]\(x = 7.2222 \ldots\)[/tex] from this result:
[tex]\[ 10x - x = 72.2222 \ldots - 7.2222 \ldots \][/tex]
[tex]\[ 9x = 65 \][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{65}{9} \][/tex]
Therefore:
[tex]\[ 7.\overline{2} = \frac{65}{9} \][/tex]
2. Convert [tex]\(\frac{65}{9}\)[/tex] to its decimal form:
[tex]\[ \frac{65}{9} \approx 7.2222\overline{2} \][/tex]
3. Next, perform the subtraction:
[tex]\[ 2.8 - 7.2222\overline{2} \][/tex]
Given the above computations:
[tex]\[ 2.8 - 7.2222\overline{2} \approx -4.4222\overline{2} \][/tex]
So, the answer to [tex]\( 2.8 - 7.\overline{2} \)[/tex] is approximately [tex]\(-4.4222222222222225\)[/tex].
1. First, we express [tex]\(7.\overline{2}\)[/tex] as a fraction. Consider the repeating decimal:
[tex]\[ x = 7.2222 \ldots \][/tex]
Multiplying both sides by 10 to shift the decimal point:
[tex]\[ 10x = 72.2222 \ldots \][/tex]
Subtracting the original [tex]\(x = 7.2222 \ldots\)[/tex] from this result:
[tex]\[ 10x - x = 72.2222 \ldots - 7.2222 \ldots \][/tex]
[tex]\[ 9x = 65 \][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{65}{9} \][/tex]
Therefore:
[tex]\[ 7.\overline{2} = \frac{65}{9} \][/tex]
2. Convert [tex]\(\frac{65}{9}\)[/tex] to its decimal form:
[tex]\[ \frac{65}{9} \approx 7.2222\overline{2} \][/tex]
3. Next, perform the subtraction:
[tex]\[ 2.8 - 7.2222\overline{2} \][/tex]
Given the above computations:
[tex]\[ 2.8 - 7.2222\overline{2} \approx -4.4222\overline{2} \][/tex]
So, the answer to [tex]\( 2.8 - 7.\overline{2} \)[/tex] is approximately [tex]\(-4.4222222222222225\)[/tex].
