Answer :
Sure, let's break down the problem step by step:
1. Convert the decimals to fractions:
- [tex]\(0.6\)[/tex] is a terminating decimal and can be represented as [tex]\(\frac{6}{10}\)[/tex] which simplifies to [tex]\(\frac{3}{5}\)[/tex].
- [tex]\(0.\overline{7}\)[/tex] is a repeating decimal. To convert [tex]\(0.\overline{7}\)[/tex] to a fraction:
[tex]\[ x = 0.\overline{7} \][/tex]
Multiply both sides by 10:
[tex]\[ 10x = 7.\overline{7} \][/tex]
Subtract the original [tex]\(x\)[/tex] from this equation:
[tex]\[ 10x - x = 7.\overline{7} - 0.\overline{7} \][/tex]
[tex]\[ 9x = 7 \][/tex]
Therefore,
[tex]\[ x = \frac{7}{9} \][/tex]
- [tex]\(0.4\overline{7}\)[/tex]: First represent it as:
[tex]\[ x = 0.4\overline{7} \][/tex]
Multiply both sides by 10:
[tex]\[ 10x = 4.\overline{7} \][/tex]
Multiply both sides by 10 again:
[tex]\[ 100x = 47.\overline{7} \][/tex]
Subtract the first multiplied version from the second:
[tex]\[ 100x - 10x = 47.\overline{7} - 4.\overline{7} \][/tex]
[tex]\[ 90x = 43 \][/tex]
Therefore,
[tex]\[ x = \frac{43}{90} \][/tex]
2. Set up the initial expression in fractional form:
[tex]\[ \frac{3}{5} \div \frac{7}{9} + \frac{43}{90} \][/tex]
3. Perform division of fractions:
Division of [tex]\(\frac{3}{5}\)[/tex] by [tex]\(\frac{7}{9}\)[/tex] can be done by multiplying [tex]\(\frac{3}{5}\)[/tex] by the reciprocal of [tex]\(\frac{7}{9}\)[/tex]:
[tex]\[ \frac{3}{5} \div \frac{7}{9} = \frac{3}{5} \times \frac{9}{7} = \frac{3 \cdot 9}{5 \cdot 7} = \frac{27}{35} \][/tex]
4. Add the fractions [tex]\(\frac{27}{35}\)[/tex] and [tex]\(\frac{43}{90}\)[/tex]:
To add these fractions, we need a common denominator. The least common multiple (LCM) of 35 and 90 is 630.
Convert both fractions to have the same denominator:
[tex]\[ \frac{27}{35} = \frac{27 \times 18}{35 \times 18} = \frac{486}{630} \][/tex]
[tex]\[ \frac{43}{90} = \frac{43 \times 7}{90 \times 7} = \frac{301}{630} \][/tex]
Now we can add them:
[tex]\[ \frac{486}{630} + \frac{301}{630} = \frac{486 + 301}{630} = \frac{787}{630} \][/tex]
5. Simplify the resulting fraction:
Find the greatest common divisor (GCD) of 787 and 630. The GCD of 787 and 630 is 1, hence the fraction [tex]\(\frac{787}{630}\)[/tex] is already in its simplest form.
So, the expression [tex]\(0.6 \div 0. \overline{7} + 0.4 \overline{7}\)[/tex] in the form [tex]\(\frac{p}{q}\)[/tex] is:
[tex]\[ \frac{787}{630} \][/tex]
1. Convert the decimals to fractions:
- [tex]\(0.6\)[/tex] is a terminating decimal and can be represented as [tex]\(\frac{6}{10}\)[/tex] which simplifies to [tex]\(\frac{3}{5}\)[/tex].
- [tex]\(0.\overline{7}\)[/tex] is a repeating decimal. To convert [tex]\(0.\overline{7}\)[/tex] to a fraction:
[tex]\[ x = 0.\overline{7} \][/tex]
Multiply both sides by 10:
[tex]\[ 10x = 7.\overline{7} \][/tex]
Subtract the original [tex]\(x\)[/tex] from this equation:
[tex]\[ 10x - x = 7.\overline{7} - 0.\overline{7} \][/tex]
[tex]\[ 9x = 7 \][/tex]
Therefore,
[tex]\[ x = \frac{7}{9} \][/tex]
- [tex]\(0.4\overline{7}\)[/tex]: First represent it as:
[tex]\[ x = 0.4\overline{7} \][/tex]
Multiply both sides by 10:
[tex]\[ 10x = 4.\overline{7} \][/tex]
Multiply both sides by 10 again:
[tex]\[ 100x = 47.\overline{7} \][/tex]
Subtract the first multiplied version from the second:
[tex]\[ 100x - 10x = 47.\overline{7} - 4.\overline{7} \][/tex]
[tex]\[ 90x = 43 \][/tex]
Therefore,
[tex]\[ x = \frac{43}{90} \][/tex]
2. Set up the initial expression in fractional form:
[tex]\[ \frac{3}{5} \div \frac{7}{9} + \frac{43}{90} \][/tex]
3. Perform division of fractions:
Division of [tex]\(\frac{3}{5}\)[/tex] by [tex]\(\frac{7}{9}\)[/tex] can be done by multiplying [tex]\(\frac{3}{5}\)[/tex] by the reciprocal of [tex]\(\frac{7}{9}\)[/tex]:
[tex]\[ \frac{3}{5} \div \frac{7}{9} = \frac{3}{5} \times \frac{9}{7} = \frac{3 \cdot 9}{5 \cdot 7} = \frac{27}{35} \][/tex]
4. Add the fractions [tex]\(\frac{27}{35}\)[/tex] and [tex]\(\frac{43}{90}\)[/tex]:
To add these fractions, we need a common denominator. The least common multiple (LCM) of 35 and 90 is 630.
Convert both fractions to have the same denominator:
[tex]\[ \frac{27}{35} = \frac{27 \times 18}{35 \times 18} = \frac{486}{630} \][/tex]
[tex]\[ \frac{43}{90} = \frac{43 \times 7}{90 \times 7} = \frac{301}{630} \][/tex]
Now we can add them:
[tex]\[ \frac{486}{630} + \frac{301}{630} = \frac{486 + 301}{630} = \frac{787}{630} \][/tex]
5. Simplify the resulting fraction:
Find the greatest common divisor (GCD) of 787 and 630. The GCD of 787 and 630 is 1, hence the fraction [tex]\(\frac{787}{630}\)[/tex] is already in its simplest form.
So, the expression [tex]\(0.6 \div 0. \overline{7} + 0.4 \overline{7}\)[/tex] in the form [tex]\(\frac{p}{q}\)[/tex] is:
[tex]\[ \frac{787}{630} \][/tex]
