What is the value of x, if $5\left( {1 - \frac{x}{5}} \right) - (5 - x) - \frac{1}{{200}}{\rm{of (20 - x) = 0}}{\rm{.08}}$ ?

Question

Prepp Team · Accepted Answer

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.