Solve Equations with Fractions or Decimals

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 do not 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 without fractions. This process is called “clearing” the equation of fractions.

Let’s solve a similar equation, but this time use the method that eliminates the fractions.

How to Solve Equations with Fraction Coefficients

Try it.

Solve: \(\frac{1}{6}y-\frac{1}{3}=\frac{5}{6}\).

Solution

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

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 LCD of all fractions in the equation.
The LCD is 20.
Multiply both sides of the equation by 20.
Distribute.
Simplify, notice, no more fractions!
Combine like terms.
Divide by 3.
Simplify.
Check:
Let \(v=40\).

In the next example, we again have variables on both sides of the equation.

Condensed: the full section is in OpenStax Elementary Algebra 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 or percentages. But decimals can also be expressed as fractions. For example, \(0.3=\frac{3}{10}\) and \(0.17=\frac{17}{100}\). So, with an equation with decimals, we can use the same method we used to clear fractions, multiply both sides of the equation by the least common denominator.

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 from the equation.

Multiply both sides by 100.
Distribute.
Multiply, and now we have 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 money applications in the next chapter. Notice that we distribute the decimal before we clear all the decimals.

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 it yourself by substituting \(x=9\) into the original equation.

Key Concepts

• Strategy to Solve an Equation with Fraction Coefficients
  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.

Solve Equations with Fractions or Decimals

Solve Equations with Fraction Coefficients

In the following exercises, solve each equation with fraction coefficients.

Try it.

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

Try it.

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

Solution

\(x=1\)

Try it.

\(\frac{5}{6}y-\frac{2}{3}=-\frac{3}{2}\)

Try it.

\(\frac{5}{6}y-\frac{1}{3}=-\frac{7}{6}\)

Solution

\(y=-1\)

Try it.

\(\frac{1}{2}a+\frac{3}{8}=\frac{3}{4}\)

Try it.

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

Solution

\(b=-2\)

Try it.

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

Try it.

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

Solution

\(x=3\)

Try it.

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

Try it.

\(\frac{5}{6}n-\frac{1}{4}n-\frac{1}{2}n=-2\)

Solution

\(n=-24\)

Try it.

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

Try it.

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

Solution

\(x=-4\)

Try it.

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

Try it.

\(\frac{3}{2}z+\frac{1}{3}=z-\frac{2}{3}\)

Solution

\(z=-2\)

Try it.

\(\frac{1}{2}x-\frac{1}{4}=\frac{1}{12}x+\frac{1}{6}\)

Try it.

\(\frac{1}{2}a-\frac{1}{4}=\frac{1}{6}a+\frac{1}{12}\)

Solution

\(a=1\)

Try it.

\(\frac{1}{3}b+\frac{1}{5}=\frac{2}{5}b-\frac{3}{5}\)

Try it.

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

Solution

\(x=-6\)

Try it.

\(1=\frac{1}{6}(12x-6)\)

Try it.

\(1=\frac{1}{5}(15x-10)\)

Solution

\(x=1\)

Try it.

\(\frac{1}{4}(p-7)=\frac{1}{3}(p+5)\)

Try it.

\(\frac{1}{5}(q+3)=\frac{1}{2}(q-3)\)

Solution

\(q=7\)

Try it.

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

Try it.

\(\frac{1}{3}(x+5)=\frac{5}{6}\)

Solution

\(x=-\frac{5}{2}\)

Try it.

\(\frac{5q-8}{5}=\frac{2q}{10}\)

Try it.

\(\frac{4m+2}{6}=\frac{m}{3}\)

Solution

\(m=-1\)

Try it.

\(\frac{4n+8}{4}=\frac{n}{3}\)

Try it.

\(\frac{3p+6}{3}=\frac{p}{2}\)

Solution

\(p=-4\)

Try it.

\(\frac{u}{3}-4=\frac{u}{2}-3\)

Try it.

\(\frac{v}{10}+1=\frac{v}{4}-2\)

Solution

\(v=20\)

Try it.

\(\frac{c}{15}+1=\frac{c}{10}-1\)

Try it.

\(\frac{d}{6}+3=\frac{d}{8}+2\)

Solution

\(d=-24\)

Try it.

\(\frac{3x+4}{2}+1=\frac{5x+10}{8}\)

Try it.

\(\frac{10y-2}{3}+3=\frac{10y+1}{9}\)

Solution

\(y=-1\)

Try it.

\(\frac{7u-1}{4}-1=\frac{4u+8}{5}\)

Try it.

\(\frac{3v-6}{2}+5=\frac{11v-4}{5}\)

Solution

\(v=4\)

Solve Equations with Decimal Coefficients

In the following exercises, solve each equation with decimal coefficients.

Try it.

\(0.6y+3=9\)

Try it.

\(0.4y-4=2\)

Solution

\(y=15\)

Try it.

\(3.6j-2=5.2\)

Try it.

\(2.1k+3=7.2\)

Solution

\(k=2\)

Try it.

\(0.4x+0.6=0.5x-1.2\)

Try it.

\(0.7x+0.4=0.6x+2.4\)

Solution

\(x=20\)

Try it.

\(0.23x+1.47=0.37x-1.05\)

Try it.

\(0.48x+1.56=0.58x-0.64\)

Solution

\(x=22\)

Try it.

\(0.9x-1.25=0.75x+1.75\)

Try it.

\(1.2x-0.91=0.8x+2.29\)

Solution

\(x=8\)

Try it.

\(0.05n+0.10(n+8)=2.15\)

Try it.

\(0.05n+0.10(n+7)=3.55\)

Solution

\(n=19\)

Try it.

\(0.10d+0.25(d+5)=4.05\)

Try it.

\(0.10d+0.25(d+7)=5.25\)

Solution

\(d=10\)

Try it.

\(0.05(q-5)+0.25q=3.05\)

Try it.

\(0.05(q-8)+0.25q=4.10\)

Solution

\(q=15\)

Condensed: the full section is in OpenStax Elementary Algebra 2e.

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Übung (40)

Versuchen Sie zuerst jede auf Papier. Decken Sie die Antwort zur Kontrolle auf; überprüfte Aufgaben lassen sich im Löser Schritt für Schritt öffnen.

1. Multiply: \(8\cdot \frac{3}{8}.\)

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   Antwort aufdecken

   \(3\)

2. Find the LCD of \(\frac{5}{6}\) and \(\frac{1}{4}\).

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   Antwort aufdecken

   \(12\)

3. Multiply 4.78 by 100.

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   Antwort aufdecken

   \(478\)

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

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

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   Antwort aufdecken

   \(x=\frac{1}{2}\)

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

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   Antwort aufdecken

   \(x=-2\)

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

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   Antwort aufdecken

   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 LCD of all fractions in the equation.
   The LCD is 20.
   Multiply both sides of the equation by 20.
   Distribute.
   Simplify, notice, no more fractions!
   Combine like terms.
   Divide by 3.
   Simplify.
   Check:
   Let \(v=40\).

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

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   Antwort aufdecken

   \(x=12\)

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

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   Antwort aufdecken

   \(u=-12\)

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

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    Antwort aufdecken

    Find the LCD of all fractions in the equation. The LCD is 8.
    Multiply both sides by the LCD.
    Distribute.
    Simplify, no more fractions.
    Subtract \(3a\) from both sides.
    Simplify.
    Subtract 6 from both sides.
    Simplify.
    Divide by 5.
    Simplify.
    Check:
    Let \(a=-2\).

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

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    Antwort aufdecken

    \(x=-1\)

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

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    Antwort aufdecken

    \(c=-2\)

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

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    Antwort aufdecken

    Distribute.
    Simplify. Now there are no fractions.
    Subtract 1 from both sides.
    Simplify.
    Divide by 2.
    Simplify.
    Check:
    Let \(x=-3\).

14. Solve: \(-11=\frac{1}{2}(6p+2)\).

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    Antwort aufdecken

    \(p=-4\)

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

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    Antwort aufdecken

    \(q=2\)

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

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    Antwort aufdecken

    Distribute.
    Simplify.
    Multiply by the LCD, 4.
    Distribute.
    Simplify.
    Collect the variables to the left.
    Simplify.
    Collect the constants to the right.
    Simplify.
    Check:
    Let \(y=9\).
    Finish the check on your own.

17. Solve: \(\frac{1}{5}(n+3)=\frac{1}{4}(n+2)\).

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    Antwort aufdecken

    \(n=2\)

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

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    Antwort aufdecken

    \(m=-1\)

19. Solve: \(\frac{5x-3}{4}=\frac{x}{2}\).

    Schritt für Schritt durcharbeiten Tipp

    Antwort aufdecken

    Multiply by the LCD, 4.
    Simplify.
    Collect the variables to the right.
    Simplify.
    Divide.
    Simplify.
    Check:
    Let \(x=1\).

20. Solve: \(\frac{4y-7}{3}=\frac{y}{6}\).

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    Antwort aufdecken

    \(y=2\)

21. Solve: \(\frac{-2z-5}{4}=\frac{z}{8}\).

    Schritt für Schritt durcharbeiten Tipp

    Antwort aufdecken

    \(z=-2\)

22. Solve: \(\frac{a}{6}+2=\frac{a}{4}+3\).

    Schritt für Schritt durcharbeiten Tipp

    Antwort aufdecken

    Multiply by the LCD, 12.
    Distribute.
    Simplify.
    Collect the variables to the right.
    Simplify.
    Collect the constants to the left.
    Simplify.
    Check:
    Let \(a=-12\).

23. Solve: \(\frac{b}{10}+2=\frac{b}{4}+5\).

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    Antwort aufdecken

    \(b=-20\)

24. Solve: \(\frac{c}{6}+3=\frac{c}{3}+4\).

    Schritt für Schritt durcharbeiten Tipp

    Antwort aufdecken

    \(c=-6\)

25. Solve: \(\frac{4q+3}{2}+6=\frac{3q+5}{4}\).

    Schritt für Schritt durcharbeiten Tipp

    Antwort aufdecken

    Multiply by the LCD, 4.
    Distribute.
    Simplify.
    Collect the variables to the left.
    Simplify.
    Collect the constants to the right.
    Simplify.
    Divide by 5.
    Simplify.
    Check:
    Let \(q=-5\).
    Finish the check on your own.

26. Solve: \(\frac{3r+5}{6}+1=\frac{4r+3}{3}\).

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    Antwort aufdecken

    \(r=1\)

27. Solve: \(\frac{2s+3}{2}+1=\frac{3s+2}{4}\).

    Schritt für Schritt durcharbeiten Tipp

    Antwort aufdecken

    \(s=-8\)

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

    Schritt für Schritt durcharbeiten Tipp

    Antwort aufdecken

    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 from the equation.

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

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

    Schritt für Schritt durcharbeiten Tipp

    Antwort aufdecken

    \(h=12\)

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

    Schritt für Schritt durcharbeiten Tipp

    Antwort aufdecken

    \(k=-1\)

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

    Schritt für Schritt durcharbeiten Tipp

    Antwort aufdecken

    Distribute first.
    Combine like terms.
    To clear decimals, multiply by 100.
    Distribute.
    Subtract 15 from both sides.
    Simplify.
    Divide by 30.
    Simplify.
    Check it yourself by substituting \(x=9\) into the original equation.

32. Solve: \(0.25n+0.05(n+5)=2.95\).

    Schritt für Schritt durcharbeiten Tipp

    Antwort aufdecken

    \(n=9\)

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

    Schritt für Schritt durcharbeiten Tipp

    Antwort aufdecken

    \(d=16\)

34. \(\frac{1}{4}x-\frac{1}{2}=-\frac{3}{4}\)

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35. \(\frac{3}{4}x-\frac{1}{2}=\frac{1}{4}\)

    Schritt für Schritt durcharbeiten Tipp

    Antwort aufdecken

    \(x=1\)

36. \(\frac{5}{6}y-\frac{2}{3}=-\frac{3}{2}\)

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37. \(\frac{5}{6}y-\frac{1}{3}=-\frac{7}{6}\)

    Schritt für Schritt durcharbeiten Tipp

    Antwort aufdecken

    \(y=-1\)

38. \(\frac{1}{2}a+\frac{3}{8}=\frac{3}{4}\)

    Schritt für Schritt durcharbeiten Tipp

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

    Schritt für Schritt durcharbeiten Tipp

    Antwort aufdecken

    \(b=-2\)

40. \(2=\frac{1}{3}x-\frac{1}{2}x+\frac{2}{3}x\)

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Wie man: Solve Equations with Fractions or Decimals

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

Fragen, die die Leute stellen

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

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