$\begin{array}{l}x=0.7 \\ x \cdot 10^1=0 . \overline{7} \cdot 10^1 \\ 10 x=7.7 \\ 10 x-x=7.7 - 0.7\end{array}$
Answer (2)
The problem analyzes a series of equations involving x = 0.7 and aims to understand the relationship between decimal and repeating decimal representations.
It checks the validity of each equation by substituting x = 0.7 and x = 0. 7 .
It identifies inaccuracies in the given equations, particularly in the equation 10 x − x = 7.7 c d o t 0.7 .
It concludes that the equations attempt to relate 0.7 with 0. 7 and derive a fractional representation, but contain errors. There is no final boxed answer as the problem is an analysis of given equations rather than solving for a numerical value.
Explanation
Analyzing the Equations We are given a series of equations involving x = 0.7 and are asked to analyze them. Let's break down each equation and see what it implies.
First Equation The first equation is x = 0.7 . This is a simple assignment of a value to the variable x .
Second Equation The second equation is x ⋅ 1 0 1 = 0. 7 ⋅ 1 0 1 . This can be rewritten as 10 x = 10 ⋅ 0. 7 . Since 0. 7 is a repeating decimal, it's equal to 9 7 . Therefore, 10 ⋅ 0. 7 = 10 ⋅ 9 7 = 9 70 = 7. 7 .
Third Equation The third equation is 10 x = 7.7 . Substituting x = 0.7 into this equation, we get 10 ( 0.7 ) = 7 , which is not equal to 7.7 . However, if x = 0.777... = 0. 7 = 9 7 , then 10 x = 10 ⋅ 9 7 = 9 70 = 7. 7 ≈ 7.777... . So, if we interpret x as 0. 7 , then 10 x = 7. 7 .
Fourth Equation The fourth equation is 10 x − x = 7.7 ⋅ 0.7 . The left side simplifies to 9 x . The right side is 7.7 ⋅ 0.7 = 5.39 . If x = 0.7 , then 9 x = 9 ( 0.7 ) = 6.3 , which is not equal to 5.39 . However, if we consider x = 0. 7 = 9 7 , then 9 x = 9 ⋅ 9 7 = 7 . This is also not equal to 5.39 .
Summary The equations seem to be exploring the relationship between 0.7 and 0. 7 , and how multiplying by 10 and subtracting might lead to a fractional representation. However, there are some inaccuracies in the given equations.
Conclusion The given equations are attempting to relate x = 0.7 with the repeating decimal 0. 7 and derive a fractional representation. However, the equation 10 x − x = 7.7 c d o t 0.7 is incorrect as 7.7 c d o t 0.7 = 5.39 and 10 x − x = 9 x = 9 ( 0.7 ) = 6.3 . If the intention was to represent x = 0. 7 = 9 7 , then 10 x = 7. 7 and 10 x − x = 7. 7 − 0. 7 = 7 .
Examples
Understanding repeating decimals and their fractional representations is crucial in various fields, such as computer science, where numbers are often represented in binary format. Converting repeating decimals to fractions allows for precise calculations and avoids rounding errors. For example, in financial calculations, even small discrepancies can lead to significant differences over time, making accurate representation of numbers essential.
The analysis of the equations reveals that while 0.7 and 0. 7 may seem related, they differ in value which is evidenced by the discrepancies found in the calculations. The equations attempt to connect these values, but several inaccuracies emerge during the analysis, especially in terms of multiplication and operations. Understanding the distinct properties of repeating decimals is crucial in this context.
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