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Study Guides > Prealgebra

Solving Equations By Clearing Decimals

Learning Outcomes

  • Determine the LCD of an equation that contains decimals
  • Solve equations with decimals that require several steps
  Some equations have decimals in them. This kind of equation will occur when we solve problems dealing with money and percent. But decimals are really another way to represent fractions. For example, 0.3=3100.3=\frac{3}{10} and 0.17=171000.17=\frac{17}{100}. So, when we have an equation with decimals, we can use the same process we used to clear fractions—multiply both sides of the equation by the least common denominator.

Example

Solve: 0.8x5=70.8x - 5=7. Solution: The only decimal in the equation is 0.80.8. Since 0.8=8100.8=\frac{8}{10}, the LCD is 1010. We can multiply both sides by 1010 to clear the decimal.
0.8x5=70.8x-5=7
Multiply both sides by the LCD. 10(0.8x5)=10(7)\color{red}{10}(0.8x-5)=\color{red}{10}(7)
Distribute. 10(0.8x)10(5)=10(7)10(0.8x)-10(5)=10(7)
Multiply, and notice, no more decimals! 8x50=708x-50=70
Add 50 to get all constants to the right. 8x50+50=70+508x-50\color{red}{+50}=70\color{red}{+50}
Simplify. 8x=1208x=120
Divide both sides by 88. 8x8=1208\frac{8x}{\color{red}{8}}=\frac{120}{\color{red}{8}}
Simplify. x=15x=15
Check: Let x=15x=15.
0.8(15)5=?70.8(\color{red}{15})-5\stackrel{\text{?}}{=}7 125=?712-5\stackrel{\text{?}}{=}7 7=77=7\quad\checkmark
     

Example

Solve: 0.06x+0.02=0.25x1.50.06x+0.02=0.25x - 1.5.

Answer:

Solution: Look at the decimals and think of the equivalent fractions. 0.06=6100,0.02=2100,0.25=25100,1.5=15100.06=\frac{6}{100},0.02=\frac{2}{100},0.25=\frac{25}{100},1.5=1\frac{5}{10} Notice, the LCD is 100100. By multiplying by the LCD we will clear the decimals.
0.06x+0.02=0.25x1.50.06x+0.02=0.25x-1.5
Multiply both sides by 100. 100(0.06x+0.02)=100(0.25x1.5)\color{red}{100}(0.06x+0.02)=\color{red}{100}(0.25x-1.5)
Distribute. 100(0.06x)+100(0.02)=100(0.25x)100(1.5)100(0.06x)+100(0.02)=100(0.25x)-100(1.5)
Multiply, and now no more decimals. 6x+2=25x1506x+2=25x-150
Collect the variables to the right. 6x6x+2=25x6x1506x\color{red}{-6x}+2=25x\color{red}{-6x}-150
Simplify. 2=19x1502=19x-150
Collect the constants to the left. 2+150=19x150+1502\color{red}{+150}=19x-150\color{red}{+150}
Simplify. 152=19x152=19x
Divide by 1919. 15219=19x19\frac{152}{\color{red}{19}}=\frac{19x}{\color{red}{19}}
Simplify. 8=x8=x
Check: Let x=8x=8.
0.06(8)+0.02=0.25(8)1.50.06(\color{red}{8})+0.02=0.25(\color{red}{8})-1.5 0.48+0.02=2.001.50.48+0.02=2.00-1.5 0.50=0.500.50=0.50\quad\checkmark

  In the following video we present another example of how to solve an equation that contains decimals and variable terms on both sides of the equal sign. https://youtu.be/pZWTJvua-P8 The next example uses an equation that is typical of the ones we will see in the money applications. Notice that we will distribute the decimal first before we clear all decimals in the equation.

Example

Solve: 0.25x+0.05(x+3)=2.850.25x+0.05\left(x+3\right)=2.85.

Answer:

Solution:
0.25x+0.05(x+3)=2.850.25x+0.05(x+3)=2.85
Distribute first. 0.25x+0.05x+0.15=2.850.25x+0.05x+0.15=2.85
Combine like terms. 0.30x+0.15=2.850.30x+0.15=2.85
To clear decimals, multiply by 100100. 100(0.30x+0.15)=100(2.85)\color{red}{100}(0.30x+0.15)=\color{red}{100}(2.85)
Distribute. 30x+15=28530x+15=285
Subtract 1515 from both sides. 30x+1515=2851530x+15\color{red}{-15}=285\color{red}{-15}
Simplify. 30x=27030x=270
Divide by 3030. 30x30=27030\frac{30x}{\color{red}{30}}=\frac{270}{\color{red}{30}}
Simplify. x=9x=9
Check: Let x=9x=9.
0.25x+0.05(x+3)=2.850.25x+0.05(x+3)=2.85 0.25(9)+0.05(9+3)=?2.850.25(\color{red}{9})+0.05(\color{red}{9}+3)\stackrel{\text{?}}{=}2.85 2.25+0.05(12)=?2.852.25+0.05(12)\stackrel{\text{?}}{=}2.85 2.85=2.852.85=2.85\quad\checkmark  

 

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