Math Help for Section 3.2, Page 114

Equations
Involving Decimals
To clear an equation of decimals, you must multiply each side of the
equation by a power of 10. When you multiply a decimal by a power of 10
such as 10, 100, or 1000, the decimal point is moved according to the
number of zeros. For instance, consider the following
products.

    $\eqalign{ 2.718 \bullet&
1\color{red}0\color{black} =
27.18&{\small\color{red}\quad\quad\text{Move decimal 1
place to the right.}} \cr 2.718 \bullet&
1\color{red}00\color{black} =
271.8&{\small\color{red}\quad\quad\text{Move decimal 2
places to the right.}} \cr 2.718 \bullet&
1\color{red}000\color{black} =
2718&{\small\color{red}\quad\quad\text{Move decimal 3
places to the right.}}} $

Example
5: Tip

Here are some of the alternate ways to solve the equation mentioned in
the study tip on page 114.

Clear the equation of decimals first by multiplying each side
by 100:

$\eqalign{0.3x + 0.2\left( {10 – x}
\right) =& 0.15\left( {30}
\right)&{\small\color{red}\quad\quad\text{Write original
equation.}}  \cr \color{red}100\color{black}\left[ {0.3x +
0.2\left( {10-x} \right)} \right]
=& \color{red}100\color{black}\left[ {0.15\left( {30}
\right)} \right]&{\small\color{red}\quad\quad\text{Multiply
each side by 100.}}  \cr 30x + 20\left( {10 – x} \right)
=& 450 &{\small\color{red}\quad\quad\text{Distributive Property}}  \cr 30x + 200 –
20x =& 450 &{\small\color{red}\quad\quad\text{Distributive Property}}  \cr 10x + 200 =&
450&{\small\color{red}\quad\quad\text{Combine like
terms.}}  \cr 10x =&
250&{\small\color{red}\quad\quad\text{Subtract 200 from
each side.}}  \cr x =&
25&{\small\color{red}\quad\quad\text{Divide each side by
10.}} \cr} $

Keep the decimals in the equation:

    $\eqalign{
0.3x + 0.2\left({10 – x} \right) & =
0.15\left({30} \right)&{\small\color{red}\quad\quad\text{Write
original equation.}} \cr 0.3x + 2 – 0.2x & =
4.5&{\small\color{red}\quad\quad\text{Distributive Property}} \cr 0.1x + 2 & =
4.5&{\small\color{red}\quad\quad\text{Combine like terms.}}
\cr
0.1x & = 2.5&{\small\color{red}\quad\quad\text{Subtract 2 from each side.}} \cr
x & = 25&{\small\color{red}\quad\quad\text{Divide each side by
0.1.}} \cr} $

The solution is $x=25$ regardless of the method used to solve the
equation.