Answer :
Let's carefully solve the given problem step-by-step, converting each repeating decimal into its exact fraction representation and evaluating the entire expression.
1. Convert the repeating decimals to fractions:
- [tex]\( 0.\overline{8} \)[/tex]:
[tex]\[ x = 0.\overline{8} \][/tex]
Multiplying both sides by 10:
[tex]\[ 10x = 8.\overline{8} \][/tex]
Subtracting the original equation from this new equation:
[tex]\[ 10x - x = 8.\overline{8} - 0.\overline{8} \][/tex]
This simplifies to:
[tex]\[ 9x = 8 \implies x = \frac{8}{9} \][/tex]
Therefore, [tex]\( 0.\overline{8} = \frac{8}{9} \approx 0.8888888888888888 \)[/tex].
- [tex]\( 5.\overline{9} \)[/tex]:
[tex]\[ y = 5.\overline{9} \][/tex]
Multiplying both sides by 10:
[tex]\[ 10y = 59.\overline{9} \][/tex]
Subtracting the original equation from this new equation:
[tex]\[ 10y - y = 59.\overline{9} - 5.\overline{9} \][/tex]
This simplifies to:
[tex]\[ 9y = 54 \implies y = 6 \][/tex]
Therefore, [tex]\( 5.\overline{9} = 6 \)[/tex].
- [tex]\( 20.\overline{5} \)[/tex]:
[tex]\[ z = 20.\overline{5} \][/tex]
Multiplying both sides by 10:
[tex]\[ 10z = 205.\overline{5} \][/tex]
Subtracting the original equation from this new equation:
[tex]\[ 10z - z = 205.\overline{5} - 20.\overline{5} \][/tex]
This simplifies to:
[tex]\[ 9z = 185 \implies z = \frac{185}{9} \approx 20.555555555555557 \][/tex]
Therefore, [tex]\( 20.\overline{5} = \frac{185}{9} \)[/tex].
- [tex]\( 2292.\overline{9} \)[/tex]:
[tex]\[ w = 2292.\overline{9} \][/tex]
Multiplying both sides by 10:
[tex]\[ 10w = 22929.\overline{9} \][/tex]
Subtracting the original equation from this new equation:
[tex]\[ 10w - w = 22929.\overline{9} - 2292.\overline{9} \][/tex]
This simplifies to:
[tex]\[ 9w = 20637 \implies w = 2293 \][/tex]
Therefore, [tex]\( 2292.\overline{9} = 2293 \)[/tex].
2. Given the problem, we need to find the value of:
[tex]\[ 0.\overline{8} + 4 \cdot 5.\overline{9} + a \cdot 20.\overline{5} + a \cdot 2292.\overline{9} \][/tex]
Using the values we calculated:
[tex]\[ 0.\overline{8} = \frac{8}{9} \approx 0.8888888888888888 \][/tex]
[tex]\[ 5.\overline{9} = 6 \][/tex]
[tex]\[ 20.\overline{5} = \frac{185}{9} \approx 20.555555555555557 \][/tex]
[tex]\[ 2292.\overline{9} = 2293 \][/tex]
If we assume [tex]\( a = 1 \)[/tex] (an example value for [tex]\( a \)[/tex] which simplifies our computation):
[tex]\[ 0.\overline{8} + 4 \times 5.\overline{9} + 1 \times 20.\overline{5} + 1 \times 2292.\overline{9} \][/tex]
3. Plugging the values in:
[tex]\[ 0.8888888888888888 + 4 \times 6 + 1 \times 20.555555555555557 + 1 \times 2293 \][/tex]
4. Simplifying each term:
[tex]\[ 0.8888888888888888 + 24 + 20.555555555555557 + 2293 \][/tex]
5. Adding these values together:
[tex]\[ 0.8888888888888888 + 24 + 20.555555555555557 + 2293 = 2338.4444444444443 \][/tex]
The final value of the given expression is approximately:
[tex]\[ 2338.4444444444443 \][/tex]
1. Convert the repeating decimals to fractions:
- [tex]\( 0.\overline{8} \)[/tex]:
[tex]\[ x = 0.\overline{8} \][/tex]
Multiplying both sides by 10:
[tex]\[ 10x = 8.\overline{8} \][/tex]
Subtracting the original equation from this new equation:
[tex]\[ 10x - x = 8.\overline{8} - 0.\overline{8} \][/tex]
This simplifies to:
[tex]\[ 9x = 8 \implies x = \frac{8}{9} \][/tex]
Therefore, [tex]\( 0.\overline{8} = \frac{8}{9} \approx 0.8888888888888888 \)[/tex].
- [tex]\( 5.\overline{9} \)[/tex]:
[tex]\[ y = 5.\overline{9} \][/tex]
Multiplying both sides by 10:
[tex]\[ 10y = 59.\overline{9} \][/tex]
Subtracting the original equation from this new equation:
[tex]\[ 10y - y = 59.\overline{9} - 5.\overline{9} \][/tex]
This simplifies to:
[tex]\[ 9y = 54 \implies y = 6 \][/tex]
Therefore, [tex]\( 5.\overline{9} = 6 \)[/tex].
- [tex]\( 20.\overline{5} \)[/tex]:
[tex]\[ z = 20.\overline{5} \][/tex]
Multiplying both sides by 10:
[tex]\[ 10z = 205.\overline{5} \][/tex]
Subtracting the original equation from this new equation:
[tex]\[ 10z - z = 205.\overline{5} - 20.\overline{5} \][/tex]
This simplifies to:
[tex]\[ 9z = 185 \implies z = \frac{185}{9} \approx 20.555555555555557 \][/tex]
Therefore, [tex]\( 20.\overline{5} = \frac{185}{9} \)[/tex].
- [tex]\( 2292.\overline{9} \)[/tex]:
[tex]\[ w = 2292.\overline{9} \][/tex]
Multiplying both sides by 10:
[tex]\[ 10w = 22929.\overline{9} \][/tex]
Subtracting the original equation from this new equation:
[tex]\[ 10w - w = 22929.\overline{9} - 2292.\overline{9} \][/tex]
This simplifies to:
[tex]\[ 9w = 20637 \implies w = 2293 \][/tex]
Therefore, [tex]\( 2292.\overline{9} = 2293 \)[/tex].
2. Given the problem, we need to find the value of:
[tex]\[ 0.\overline{8} + 4 \cdot 5.\overline{9} + a \cdot 20.\overline{5} + a \cdot 2292.\overline{9} \][/tex]
Using the values we calculated:
[tex]\[ 0.\overline{8} = \frac{8}{9} \approx 0.8888888888888888 \][/tex]
[tex]\[ 5.\overline{9} = 6 \][/tex]
[tex]\[ 20.\overline{5} = \frac{185}{9} \approx 20.555555555555557 \][/tex]
[tex]\[ 2292.\overline{9} = 2293 \][/tex]
If we assume [tex]\( a = 1 \)[/tex] (an example value for [tex]\( a \)[/tex] which simplifies our computation):
[tex]\[ 0.\overline{8} + 4 \times 5.\overline{9} + 1 \times 20.\overline{5} + 1 \times 2292.\overline{9} \][/tex]
3. Plugging the values in:
[tex]\[ 0.8888888888888888 + 4 \times 6 + 1 \times 20.555555555555557 + 1 \times 2293 \][/tex]
4. Simplifying each term:
[tex]\[ 0.8888888888888888 + 24 + 20.555555555555557 + 2293 \][/tex]
5. Adding these values together:
[tex]\[ 0.8888888888888888 + 24 + 20.555555555555557 + 2293 = 2338.4444444444443 \][/tex]
The final value of the given expression is approximately:
[tex]\[ 2338.4444444444443 \][/tex]
