Back to homeMathematics · 3rd Year · Change subject
Start for free
3rd Year · Mathematics
Applied Mathematics Gaps for Junior Cycle: Trigonometry, Probability, Arithmetic, Sets and Algebra · Linear Equations with Rational Coefficients

Solving linear equations with fractions and decimals

Orla, Pádraig and Eoin win a prize and split it: one third for Orla, one quarter for Pádraig, and the last €100 for Eoin. How big was the prize? Fractions turn up in equations all the time, and they look scary until you learn one trick: get rid of them. By the end of this page you will be able to answer that question.

Balancing with fractions

An equation is a balance. Whatever you do to one side, you must do to the other. That rule does not change when the numbers are fractions or decimals.

A balance with x over 3 on the left and 5 on the right; multiplying both sides by 3 gives x equals 15. x ÷ 3 5 Multiply both sides by 3: x = 15

To undo "÷ 3" we multiply by 3. To undo "× 2" we divide by 2. To undo "+ ½" we subtract ½.

Decimals work the same way. In $0.5x - 2 = 3$, add 2 to get $0.5x = 5$, then divide by 0.5 to get $x = 10$.

Practice★☆☆
Solve each equation. a) $\frac{x}{3} = 5$ b) $\frac{2x}{5} = 8$ c) $x + \frac{1}{2} = \frac{3}{4}$

Solve each equation.

  1. a) $\frac{x}{3} = 5$
  2. b) $\frac{2x}{5} = 8$
  3. c) $x + \frac{1}{2} = \frac{3}{4}$
Solve
Practice★☆☆
Solve each equation. a) $\frac{x}{2} + 3 = 7$ b) $\frac{x - 1}{4} = 2$ c) $\frac{3x}{4} = \frac{9}{2}$

Solve each equation.

  1. a) $\frac{x}{2} + 3 = 7$
  2. b) $\frac{x - 1}{4} = 2$
  3. c) $\frac{3x}{4} = \frac{9}{2}$
Solve
Practice★☆☆
Solve each equation, giving the answer as a fraction or decimal. a) $2x + 1 = 6$ b) $0.5x - 2 = 3$ c) $4x - 3 = 2x + 5$

Solve each equation, giving the answer as a fraction or decimal.

  1. a) $2x + 1 = 6$
  2. b) $0.5x - 2 = 3$
  3. c) $4x - 3 = 2x + 5$
Solve

Clearing denominators

When there are several fractions, multiply every term by a number that all the denominators divide into: the lowest common multiple (LCM) of the denominators. All the fractions vanish at once.

Practice★★☆
Solve each equation. a) $\frac{x}{2} + \frac{x}{3} = 10$ b) $\frac{2x - 1}{3} = \frac{x + 2}{2}$ c) $\frac{x + 3}{4} - \frac{x - 1}{6} = 1$

Solve each equation.

  1. a) $\frac{x}{2} + \frac{x}{3} = 10$
  2. b) $\frac{2x - 1}{3} = \frac{x + 2}{2}$
  3. c) $\frac{x + 3}{4} - \frac{x - 1}{6} = 1$
Solve

Brackets and decimals

Fractions and decimals can sit outside brackets too. Expand first, or clear the denominators first. Both work.

You can leave an answer as a fraction or a decimal, whichever is exact. If the decimal does not terminate (like $\frac{1}{3}$), keep the fraction.

Practice★★☆
Solve each equation. a) $3(x - \frac{1}{2}) = 2x + 4$ b) $\frac{1}{2}(x + 4) = \frac{1}{3}(x + 9)$ c) $0.2x + 1.5 = 0.5x - 0.3$

Solve each equation.

  1. a) $3(x - \frac{1}{2}) = 2x + 4$
  2. b) $\frac{1}{2}(x + 4) = \frac{1}{3}(x + 9)$
  3. c) $0.2x + 1.5 = 0.5x - 0.3$
Solve
Practice★★★
Solve each equation. a) $\frac{3x - 2}{4} - \frac{2x + 1}{3} = \frac{1}{6}$ b) $\frac{5}{2}(x - 1) - \frac{3}{4}(2x + 3) = \frac{1}{2}$

Solve each equation.

  1. a) $\frac{3x - 2}{4} - \frac{2x + 1}{3} = \frac{1}{6}$
  2. b) $\frac{5}{2}(x - 1) - \frac{3}{4}(2x + 3) = \frac{1}{2}$
Solve

Word problems

To solve a problem in words, follow four steps: let a letter stand for the unknown, form an equation, solve it, then check the answer in the original words and ask whether it is sensible.

Practice★★☆
Three friends share a prize. Orla gets $\frac{1}{3}$ of the total, Pádraig gets $\frac{1}{4}$ of the total and Eoin gets the remaining €100. a) Form an equation for the…

Three friends share a prize. Orla gets $\frac{1}{3}$ of the total, Pádraig gets $\frac{1}{4}$ of the total and Eoin gets the remaining €100.

  1. a) Form an equation for the total prize $T$.
  2. b) Solve it.
  3. c) Find how much Orla and Pádraig each receive.
Solve
Practice★★★
A number is such that one third of it, increased by half of the number that is 4 more than it, equals 11. a) Form an equation. b) Solve it and check your answer. c)…

A number is such that one third of it, increased by half of the number that is 4 more than it, equals 11.

  1. a) Form an equation.
  2. b) Solve it and check your answer.
  3. c) Explain how you checked your answer.
Solve
Practice★★★
A cyclist travels from Athlone to a town $d$ km away. Going there at $20 \text{ km/h}$ and returning at $15 \text{ km/h}$, the total time is $3\frac{1}{2}$ hours. a)…

A cyclist travels from Athlone to a town $d$ km away. Going there at $20 \text{ km/h}$ and returning at $15 \text{ km/h}$, the total time is $3\frac{1}{2}$ hours.

  1. a) Form an equation in $d$ using time = distance ÷ speed.
  2. b) Solve it and find $d$.
  3. c) Find the average speed for the whole journey.
Solve
Let's practice! 9 exercises in this section
★☆☆ Solve each equation. a) $\frac{x}{3} = 5$ b) $\frac{2x}{5} = 8$ c)…★☆☆ Solve each equation. a) $\frac{x}{2} + 3 = 7$ b) $\frac{x - 1}{4} = 2$ c)…★☆☆ Solve each equation, giving the answer as a fraction or decimal. a) $2x + 1 = 6$ b)…

Summary

  • An equation is a balance: do the same to every term on both sides.

  • To remove fractions, multiply every term by the LCM of the denominators.

  • For decimals, multiply by 10 or 100 to get whole numbers, or convert to fractions.

  • Put brackets round numerators and take care with minus signs when expanding.

  • For word problems: define the unknown, form the equation, solve, and check in the original words.

  • Average speed = total distance ÷ total time.

See all the exercises in this section