How to solve equations that include decimal numbers
Approach to solving decimal equations
In this lesson you’ll learn to solve equations with decimals by multiplying by powers of ???10???.
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Remember your place values.
???0.1??? has a ???1??? in the tenths place
???0.02??? has a ???2??? in the hundredths place
???0.003??? has a ???3??? in the thousandths place
???0.0004??? has a ???4??? in the ten-thousandths place
When you multiply by…
???10??? move the decimal point one spot to the right
???100??? move the decimal point two spots to the right
???1,000??? move the decimal point three spots to the right
???10,000??? move the decimal point four spots to the right
Notice we move the decimal point to the right one place for each ???0??? in the multiple of ???10???.
One method of solving an equation with decimals is to multiply the equation by a power of ???10???. In order to solve an equation involving decimals, the first thing we’ll try to do is get rid of the decimals by changing them to whole numbers.
How to solve equations with decimals by eliminating them
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Multiplying through the equation by 10, 100, 1000, etc.
Example
Solve the decimal equation.
???0.3x+5=11???
In order to solve an equation involving decimals, the first thing we’ll try to do is get rid of the decimals by changing them to whole numbers. Since the only decimal in our equation ends in the ???10???ths place, we’ll need to multiply by ???10??? in order to change it to a whole number.
???0.3x+5=11???
???(0.3x+5=11)10???
???0.3x(10)+5(10)=11(10)???
???3x+50=110???
???3x+50-50=110-50???
???3x=60???
???\frac{3x}{3}=\frac{60}{3}???
???x=20???
Let’s do another one.
In order to solve an equation involving decimals, the first thing we’ll try to do is get rid of the decimals by changing them to whole numbers.
Example
Solve for the variable.
???6a+5a=-1.1???
In order to solve an equation involving decimals, the first thing we’ll try to do is get rid of the decimal by changing it to an integer. Since the decimal in our equation ends in the ???10???ths place, we’ll need to multiply by ???10??? in order to change it to an integer.
???6a+5a=-1.1???
???(6a+5a=-1.1)10???
???6a(10)+5a(10)=-1.1(10)???
???60a+50a=-11???
???110a=-11???
???\frac{110a}{110}=\frac{-11}{110}???
???a=\frac{-1}{10}???
???a=-0.1???
Let’s look at another example.
Example
Solve for the variable.
???0.4n+3.9=5.78???
In order to solve an equation involving decimals, the first thing we’ll try to do is get rid of the decimals by changing them to whole numbers. Since the longest decimal in our equation ends in the ???100???ths place, we’ll need to multiply by ???100??? in order to change them all to whole numbers. This means we’ll move each decimal point two spots to the right.
???0.4n+3.9=5.78???
???(0.4n+3.9=5.78)100???
???0.4n(100)+3.9(100)=5.78(100)???
???40n+390=578???
???40n+390-390=578-390???
???40n=188???
???\frac{40n}{40}=\frac{188}{40}???
???n=\frac{47}{10}???
???n=4.7???
