What is the value of x, if \(5\left( {1 - \frac{x}{5}} \right) - (5 - x) - \frac{1}{{200}}{\rm{of (20 - x) = 0}}{\rm{.08}}\) ?
36
Solving the Algebraic Equation to Find the Value of x
The problem asks us to find the value of \(x\) that satisfies the given equation:
\[5\left( {1 - \frac{x}{5}} \right) - (5 - x) - \frac{1}{{200}}{\rm{of (20 - x) = 0}}{\rm{.08}}\]
To solve for \(x\), we need to simplify the equation and isolate the term containing \(x\).
Step-by-Step Solution Process
Let's break down the equation and simplify each part.
Step 1: Simplify the terms within the equation.
- Simplify the first term: \(5\left( {1 - \frac{x}{5}} \right)\) \[5 \times 1 - 5 \times \frac{x}{5} = 5 - x\]
- Simplify the second term: \(-(5 - x)\) \[-1 \times 5 - 1 \times (-x) = -5 + x\]
- Simplify the third term: \(-\frac{1}{{200}}{\rm{of (20 - x)}}\). Remember 'of' means multiplication. \[-\frac{1}{{200}} \times (20 - x) = -\left(\frac{1}{200} \times 20 - \frac{1}{200} \times x\right) = -\left(\frac{20}{200} - \frac{x}{200}\right)\] \[= -\left(\frac{1}{10} - \frac{x}{200}\right) = -\frac{1}{10} + \frac{x}{200}\] We can also write this in decimal form: \(-\left(0.1 - 0.005x\right) = -0.1 + 0.005x\).
- The right side of the equation is \(0.08\).
Step 2: Substitute the simplified terms back into the original equation.
The equation becomes:
\[(5 - x) + (-5 + x) + \left(-0.1 + 0.005x\right) = 0.08\]
Step 3: Combine like terms on the left side of the equation.
Combine the terms with \(x\):
\(-x + x + 0.005x = ( -1 + 1 + 0.005)x = 0.005x\)
Combine the constant terms:
\(5 - 5 - 0.1 = 0 - 0.1 = -0.1\)
So, the simplified equation is:
\[0.005x - 0.1 = 0.08\]
Step 4: Isolate the term with \(x\).
Add \(0.1\) to both sides of the equation:
\[0.005x - 0.1 + 0.1 = 0.08 + 0.1\]
\[0.005x = 0.18\]
Step 5: Solve for \(x\).
Divide both sides by \(0.005\):
\[x = \frac{0.18}{0.005}\]
To perform the division, we can multiply the numerator and the denominator by 1000 to remove the decimals:
\[x = \frac{0.18 \times 1000}{0.005 \times 1000} = \frac{180}{5}\]
Now, divide 180 by 5:
\[x = 36\]
The value of \(x\) that satisfies the given equation is 36.
Final Answer Check
Let's quickly check if \(x=36\) works in the original equation. This step is optional but helpful.
\(5\left( {1 - \frac{36}{5}} \right) - (5 - 36) - \frac{1}{{200}}{\rm{(20 - 36)}}\)
\(= 5\left( {1 - 7.2} \right) - (-31) - \frac{1}{{200}}{(-16)}\)
\(= 5\left( {-6.2} \right) + 31 - (-\frac{16}{200})\)
\(= -31 + 31 - (-\frac{2}{25})\)
\(= 0 + \frac{2}{25}\)
\(= \frac{8}{100} = 0.08\)
The left side equals \(0.08\), which matches the right side of the original equation. So, the value \(x=36\) is correct.
Revision Table: Key Steps in Solving Linear Equations
Understanding the general approach to solving linear equations is crucial.
| Step | Action | Purpose |
|---|---|---|
| 1 | Simplify expressions | Remove parentheses by distributing multiplication or signs. |
| 2 | Combine like terms | Group terms with variables and constant terms separately on each side. |
| 3 | Isolate variable term | Move all constant terms to one side of the equation. |
| 4 | Solve for the variable | Divide or multiply to get the variable by itself. |
| 5 | Verify solution (Optional) | Substitute the found value back into the original equation. |
Additional Information: Understanding 'of' in Mathematical Expressions
In mathematics, the word 'of' often indicates multiplication, especially when dealing with fractions or percentages. For example, 'half of 10' means \(\frac{1}{2} \times 10\), and '10% of 50' means \(\frac{10}{100} \times 50\). In the given equation, \(\frac{1}{200}{\rm{of (20 - x)}}\) means \(\frac{1}{200} \times (20 - x)\).
Understanding these basic conventions helps in correctly translating word problems and expressions into mathematical equations that can be solved.
Similar Questions
What is the value of \(\frac{{5.6 \times 0.36 + 0.42 \times 3.2}}{{0.8 \times 2.1}}?\)
Find the value of (1.6) 3 - (0.9) 3 - (0.7) 3.
The value of \(11.\overline{4}\) + \(22.5\overline{67}\) – \(33.5\overline{9}\) is:
The value of \(0.5\bar 6 - 0.7\overline {23} + 0.3\bar 9 \times 0.\bar 7\) is:
If N = 0.369369369369… and M = 0.531531531531…, then what is the value of (1/N) + (1/M)?
If \({\rm{A}} = {\rm{}}\frac{{0.216{\rm{\;}} + {\rm{\;}}0.008}}{{0.36{\rm{\;}} + {\rm{\;}}0.04{\rm{\;}} - {\rm{\;}}0.12}}\) and \({\rm{B}} = {\rm{}}\frac{{0.729{\rm{\;}} - {\rm{\;}}0.027}}{{0.81{\rm{\;}} + {\rm{\;}}0.09{\rm{\;}} + {\rm{\;}}0.27}}\) , then what is the value of (A 2+ B 2) 2?
If A = 0.142857142857 _____ and B = 0.16666 _____, then what is the value of (A + B)/AB?
The value of \(\frac{48.3\times[(4.95)^2+4.95\times13.25]}{[(12.55)^2-(5.65)^2]\times19.8} \) is:
If A = 0.abcabc _____, then by what number A should be multiplied so as to get an integeral value?
The value of \(0.4\bar 7\; + \;0.5\overline {03} - 0.3\bar 9\; \times \;0.\bar 8\) is:
Important Questions from Decimals
What is the result when 0.129129129… is converted to a fraction?
Which of the following statement(s) is/are correct?
I. (3/11) > 0.3
II. (7/8) > 0.86
The value of \(1.\overline{3}+0.\overline{69}-0.5\overline{23}\) is equal to:
The value of \(0.\bar 4 + 0.5 \bar 9 - 0.4 \overline{23}\) is equal to:
If 19 × 23 = 437, then find the value of (190 × 0.023).
