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Solve Equations with Fraction or Decimal Coefficients

Solve equations with fraction coefficients

Solve Equations with Fraction Coefficients

Let’s use the General Strategy for Solving Linear Equations introduced earlier to solve the equation \(\frac{1}{8}\ x+\frac{1}{2}=\frac{1}{4}.\)


To isolate the \(x\) term, subtract \(\frac{1}{2}\) from both sides.
Simplify the left side.
Change the constants to equivalent fractions with the LCD.
Subtract.
Multiply both sides by the reciprocal of \(\frac{1}{8}\).
Simplify.

This method worked fine, but many students don’t feel very confident when they see all those fractions. So we are going to show an alternate method to solve equations with fractions. This alternate method eliminates the fractions.

We will apply the Multiplication Property of Equality and multiply both sides of an equation by the least common denominator of all the fractions in the equation. The result of this operation will be a new equation, equivalent to the first, but with no fractions. This process is called clearing the equation of fractions. Let’s solve the same equation again, but this time use the method that clears the fractions.

Example

Try it.

Solve: \(\frac{1}{8}\ x+\frac{1}{2}=\frac{1}{4}.\)

Solution

Find the least common denominator of all the fractions in the equation.
Multiply both sides of the equation by that LCD, 8. This clears the fractions.
Use the Distributive Property.
Simplify — and notice, no more fractions!
Solve using the General Strategy for Solving Linear Equations.
Simplify.
Check: Let \(x=-2\)

Notice in that once we cleared the equation of fractions, the equation was like those we solved earlier in this chapter. We changed the problem to one we already knew how to solve! We then used the General Strategy for Solving Linear Equations.

Example

Try it.

Solve: \(7=\frac{1}{2}\ x+\frac{3}{4}\ x-\frac{2}{3}\ x.\)

Solution

We want to clear the fractions by multiplying both sides of the equation by the LCD of all the fractions in the equation.

Find the least common denominator of all the fractions in the equation.
Multiply both sides of the equation by 12.
Distribute.
Simplify — and notice, no more fractions!
Combine like terms.
Divide by 7.
Simplify.
Check: Let \(x=12.\)

Condensed — the full section is in OpenStax Prealgebra 2e.

Solve Equations with Decimal Coefficients

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, \(0.3=\frac{3}{10}\) and \(0.17=\frac{17}{100}.\) 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

Try it.

Solve: \(0.8x-5=7.\)

Solution

The only decimal in the equation is \(0.8.\) Since \(0.8=\frac{8}{10},\) the LCD is \(10.\) We can multiply both sides by \(10\) to clear the decimal.


Multiply both sides by the LCD.
Distribute.
Multiply, and notice, no more decimals!
Add 50 to get all constants to the right.
Simplify.
Divide both sides by 8.
Simplify.
Check: Let \(x=15.\)

Example

Try it.

Solve: \(0.06x+0.02=0.25x-1.5.\)

Solution

Look at the decimals and think of the equivalent fractions.

\(0.06=\frac{6}{100},\ 0.02=\frac{2}{100},\ 0.25=\frac{25}{100},\ 1.5=1\frac{5}{10}\)

Notice, the LCD is \(100.\)

By multiplying by the LCD we will clear the decimals.


Multiply both sides by 100.
Distribute.
Multiply, and now no more decimals.
Collect the variables to the right.
Simplify.
Collect the constants to the left.
Simplify.
Divide by 19.
Simplify.
Check: Let \(x=8.\)

The next example uses an equation that is typical of the ones we will see in the money applications in the next chapter. Notice that we will distribute the decimal first before we clear all decimals in the equation.

Example

Try it.

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

Solution


Distribute first.
Combine like terms.
To clear decimals, multiply by 100.
Distribute.
Subtract 15 from both sides.
Simplify.
Divide by 30.
Simplify.
Check: Let \(x=9.\)

Key Concepts

Solve equations with fraction coefficients by clearing the fractions.
1. Find the least common denominator of all the fractions in the equation.
2. Multiply both sides of the equation by that LCD. This clears the fractions.
3. Solve using the General Strategy for Solving Linear Equations.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Multiply: \(8\cdot \frac{3}{8}.\)
If you missed this problem, review

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\(3\)
Find the LCD of \(\frac{5}{6}\ \text{and}\ \frac{1}{4}.\)
If you missed this problem, review

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\(12\)
Multiply: \(4.78\) by \(100.\)
If you missed this problem, review

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\(478\)

Solve: \(\frac{1}{8}\ x+\frac{1}{2}=\frac{1}{4}.\)

Mengungkapkan jawabannya

Find the least common denominator of all the fractions in the equation.
Multiply both sides of the equation by that LCD, 8. This clears the fractions.
Use the Distributive Property.
Simplify — and notice, no more fractions!
Solve using the General Strategy for Solving Linear Equations.
Simplify.
Check: Let \(x=-2\)

Solve: \(\frac{1}{4}\ x+\frac{1}{2}=\frac{5}{8}.\)

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\(x=\frac{1}{2}\)
Solve: \(\frac{1}{6}\ y-\frac{1}{3}=\frac{1}{6}.\)

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y = 3

Solve: \(7=\frac{1}{2}\ x+\frac{3}{4}\ x-\frac{2}{3}\ x.\)

Mengungkapkan jawabannya

We want to clear the fractions by multiplying both sides of the equation by the LCD of all the fractions in the equation.

Find the least common denominator of all the fractions in the equation.
Multiply both sides of the equation by 12.
Distribute.
Simplify — and notice, no more fractions!
Combine like terms.
Divide by 7.
Simplify.
Check: Let \(x=12.\)

Solve: \(6=\frac{1}{2}\ v+\frac{2}{5}\ v-\frac{3}{4}\ v.\)

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v = 40
Solve: \(-1=\frac{1}{2}\ u+\frac{1}{4}\ u-\frac{2}{3}\ u.\)

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u = −12

Solve: \(x+\frac{1}{3}=\frac{1}{6}\ x-\frac{1}{2}.\)

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Find the LCD of all the fractions in the equation.
Multiply both sides by the LCD.
Distribute.
Simplify — no more fractions!
Subtract \(x\) from both sides.
Simplify.
Subtract 2 from both sides.
Simplify.
Divide by 5.
Simplify.
Check: Substitute \(x=-1.\)

Solve: \(a+\frac{3}{4}=\frac{3}{8}\ a-\frac{1}{2}.\)

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a = −2
Solve: \(c+\frac{3}{4}=\frac{1}{2}\ c-\frac{1}{4}.\)

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c = −2
Solve: \(1=\frac{1}{2}(4x+2).\)

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Distribute.
Simplify. Now there are no fractions to clear!
Subtract 1 from both sides.
Simplify.
Divide by 2.
Simplify.
Check: Let \(x=0.\)
Solve: \(-11=\frac{1}{2}(6p+2).\)

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p = −4
Solve: \(8=\frac{1}{3}(9q+6).\)

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q = 2
Solve: \(\frac{1}{2}(y-5)=\frac{1}{4}(y-1).\)

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Distribute.
Simplify.
Multiply by the LCD, 4.
Distribute.
Simplify.
Collect the \(y\) terms to the left.
Simplify.
Collect the constants to the right.
Simplify.
Check: Substitute \(9\) for \(y.\)
Solve: \(\frac{1}{5}(n+3)=\frac{1}{4}(n+2).\)

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n = 2
Solve: \(\frac{1}{2}(m-3)=\frac{1}{4}(m-7).\)

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m = −1
Solve: \(0.8x-5=7.\)

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The only decimal in the equation is \(0.8.\) Since \(0.8=\frac{8}{10},\) the LCD is \(10.\) We can multiply both sides by \(10\) to clear the decimal.
Multiply both sides by the LCD.
Distribute.
Multiply, and notice, no more decimals!
Add 50 to get all constants to the right.
Simplify.
Divide both sides by 8.
Simplify.
Check: Let \(x=15.\)
Solve: \(0.6x-1=11.\)

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x = 20
Solve: \(1.2x-3=9.\)

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x = 10

Solve: \(0.06x+0.02=0.25x-1.5.\)

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Look at the decimals and think of the equivalent fractions.

\(0.06=\frac{6}{100},\ 0.02=\frac{2}{100},\ 0.25=\frac{25}{100},\ 1.5=1\frac{5}{10}\)

Notice, the LCD is \(100.\)

By multiplying by the LCD we will clear the decimals.


Multiply both sides by 100.
Distribute.
Multiply, and now no more decimals.
Collect the variables to the right.
Simplify.
Collect the constants to the left.
Simplify.
Divide by 19.
Simplify.
Check: Let \(x=8.\)

Solve: \(0.14h+0.12=0.35h-2.4.\)

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h = 12
Solve: \(0.65k-0.1=0.4k-0.35.\)

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k = −1
Solve: \(0.25x+0.05(x+3)=2.85.\)

Mengungkapkan jawabannya

Distribute first.
Combine like terms.
To clear decimals, multiply by 100.
Distribute.
Subtract 15 from both sides.
Simplify.
Divide by 30.
Simplify.
Check: Let \(x=9.\)
Solve: \(0.25n+0.05(n+5)=2.95.\)

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n = 9
Solve: \(0.10d+0.05(d-5)=2.15.\)

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d = 16
\(\frac{1}{4}\ x-\frac{1}{2}=-\frac{3}{4}\)

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x = −1
\(\frac{3}{4}\ x-\frac{1}{2}=\frac{1}{4}\)
\(\frac{5}{6}\ y-\frac{2}{3}=-\frac{3}{2}\)

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y = −1
\(\frac{5}{6}\ y-\frac{1}{3}=-\frac{7}{6}\)
\(\frac{1}{2}\ a+\frac{3}{8}=\frac{3}{4}\)

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\(a=\frac{3}{4}\)
\(\frac{5}{8}\ b+\frac{1}{2}=-\frac{3}{4}\)
\(2=\frac{1}{3}\ x-\frac{1}{2}\ x+\frac{2}{3}\ x\)

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x = 4
\(2=\frac{3}{5}\ x-\frac{1}{3}\ x+\frac{2}{5}\ x\)
\(\frac{1}{4}\ m-\frac{4}{5}\ m+\frac{1}{2}\ m=-1\)

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m = 20
\(\frac{5}{6}\ n-\frac{1}{4}\ n-\frac{1}{2}\ n=-2\)
\(x+\frac{1}{2}=\frac{2}{3}\ x-\frac{1}{2}\)

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x = −3
\(x+\frac{3}{4}=\frac{1}{2}\ x-\frac{5}{4}\)
\(\frac{1}{3}\ w+\frac{5}{4}=w-\frac{1}{4}\)

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\(w=\frac{9}{4}\)

Symbols used here

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\approx

approximately equal
Equal to the precision shown, not exactly.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Solve Equations with Fraction or Decimal Coefficients

Solve equations with fraction coefficients
Solve equations with decimal coefficients
Find the least common denominator of
Multiply both sides of the equation by that LCD. This clears the fractions.
Solve using the General Strategy for Solving Linear Equations.
Find the least common denominator of
Multiply both sides of the equation by that LCD. This clears the fractions.
Solve using the General Strategy for Solving Linear Equations.

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

Cobalah sendiri

Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

Lebih dalam Arithmetic

Order of operations Fractions Percentages Powers and roots


Distribute.
Simplify. Now there are no fractions to clear!
Subtract 1 from both sides.
Simplify.
Divide by 2.
Simplify.
Check: Let \(x=0.\)


Distribute.
Simplify.
Multiply by the LCD, 4.
Distribute.
Simplify.
Collect the \(y\) terms to the left.
Simplify.
Collect the constants to the right.
Simplify.
Check: Substitute \(9\) for \(y.\)