Year 10 Maths · Unit 2 · Lesson 9

Equations with Fractions

Clear fractions using the LCD, cross-multiply with confidence, and spot restrictions before you solve.

40 min 8 cards 130 XP
x/3 + 2 = 7
Think First
warm-up

Before we begin: Solve $\\frac{x}{4} = 5$ without a calculator. Then solve $\\frac{x}{4} + 3 = 5$. How are the two problems different? What extra step did you need?

Record your answer in your workbook.
1
The Big Idea
+5 XP to read

Equations with fractions look intimidating, but the strategy is simple: eliminate the fractions first. Multiply every term by the lowest common denominator (LCD). This turns a fraction equation into a regular linear equation that you already know how to solve.

$\\frac{x}{3} + 2 = 7$: multiply every term by 3. $x + 6 = 21$.

x/3 + 2 = 7 x + 6 = 21 x = 15
LCD clears fractions
2
Lesson Objectives
what you will learn
  • Find the lowest common denominator (LCD) of algebraic fractions
  • Clear fractions by multiplying every term by the LCD
  • Use cross-multiplication for single fractions on each side
  • Solve equations with algebraic fractions in the denominator
  • Identify and state restrictions on the variable
3
Key Vocabulary
terms to know
Lowest Common Denominator (LCD)The smallest number that all denominators divide into evenly. For 3 and 4, the LCD is 12.
Clearing fractionsMultiplying every term by the LCD to eliminate denominators.
Cross-multiplicationA shortcut for equations of the form a/b = c/d: multiply a by d and b by c.
Algebraic fractionA fraction where the numerator, denominator, or both contain variables.
RestrictionA value that the variable cannot take because it would make a denominator zero.
NumeratorThe top part of a fraction. In 5/(x-2), the numerator is 5.
4
Spot the Trap
heads-up

Wrong: Only multiplying one term by the LCD. In $\\frac{x}{3} + 2 = 7$, multiplying only $\\frac{x}{3}$ by 3 gives $x + 2 = 7$.

Right: Multiply every term on both sides by the LCD: $x + 6 = 21$.

Wrong: Using cross-multiplication when there are multiple terms on one side, such as $\\frac{x}{2} + 3 = \\frac{x}{4}$.

Right: Cross-multiplication only works for $\\frac{a}{b} = \\frac{c}{d}$ (one fraction each side). For multiple terms, use the LCD method.

5
Clearing Fractions with the LCD
+5 XP

To solve an equation with fractions, find the lowest common denominator of all fractions, then multiply every term on both sides by that number. This clears all denominators in one step and leaves a simple linear equation.

$\\frac{x}{3} + \\frac{x}{6} = 5$: LCD is 6. Multiply every term by 6. $2x + x = 30$.

x/3 + x/6 = 5 2x + x = 30 x = 10
multiply every term
Include every term
Multiply constants too: 2 becomes 2 times the LCD. Do not skip any term.
Find the LCD first
List multiples or use prime factors. The LCD is the smallest number all denominators share.
Simplify after clearing
Collect like terms before trying to isolate the variable.
6
Cross-Multiplication
+5 XP

When there is exactly one fraction on each side, cross-multiplication is a fast shortcut. Multiply the numerator of the left fraction by the denominator of the right, and set it equal to the denominator of the left times the numerator of the right.

$\\frac{3}{x} = \\frac{2}{5}$: cross-multiply to get $3 \\times 5 = 2 \\times x$.

3/x = 2/5 15 = 2x x = 7.5
one fraction each side
Check the form
Cross-multiplication only works for a/b = c/d. If there are extra terms, use the LCD method.
It is just the LCD shortcut
Cross-multiplication is what happens when you multiply both sides by b and d (the LCD).
Watch for restrictions
x cannot be 0 in 3/x = 2/5. State this before solving.
7
Algebraic Fractions and Restrictions
+5 XP

When the variable appears in the denominator, some values are forbidden. A denominator can never equal zero. Identify these restrictions before you solve, and check your final answer does not violate them.

$\\frac{5}{x-2} = 3$: restriction is $x \\neq 2$. Then solve.

5/(x-2) = 3 x != 2, 5 = 3(x-2) x = 11/3
denominator != 0
State restrictions first
For 5/(x-2), write x != 2 before doing any algebra. This earns marks.
Multiply by the denominator
For 5/(x-2) = 3, multiply both sides by (x-2) to eliminate the fraction.
Check against restrictions
If your answer equals a restricted value, there is no solution.
8
Choosing the Right Method
+5 XP

Not every fraction equation needs the same treatment. Look at the structure first, then choose the most efficient method. Using the wrong method wastes time and invites errors.

One fraction each side → cross-multiply. Multiple terms → use LCD. Variable in denominator → state restriction first.

a/b = c/d ? Cross-multiply Use LCD Always check for restrictions
structure decides method
Scan before solving
Take two seconds to look at the equation structure. It saves minutes of wrong working.
LCD always works
When in doubt, use the LCD method. It is universal and reliable for any fraction equation.
Verify your choice
After clearing fractions, check that no denominators remain before solving.

Watch Me Solve It · 3 examples

Watch Me Solve It · Clearing fractions with LCD
+15 XP per step
Q1
PROBLEM
Solve $\\frac{x}{3} + \\frac{x}{6} = 5$.
  1. 1
    Identify the LCD
    Denominators are 3 and 6. LCD = 6.
    6 is the smallest number that both 3 and 6 divide into evenly.
  2. 2
    Multiply every term by 6
    $6 \\times \\frac{x}{3} + 6 \\times \\frac{x}{6} = 6 \\times 5$
    Every term on both sides must be multiplied, including the 5.
  3. 3
    Simplify each term
    $2x + x = 30$
    6 divided by 3 is 2, so the first term becomes 2x. 6 divided by 6 is 1.
  4. 4
    Collect and solve
    $3x = 30$    so    $x = 10$
    Check: $\\frac{10}{3} + \\frac{10}{6} = \\frac{20}{6} + \\frac{10}{6} = \\frac{30}{6} = 5$. Correct.
Answer$x = 10$
Watch Me Solve It · Cross-multiplication
+15 XP per step
Q2
PROBLEM
Solve $\\frac{3}{x} = \\frac{2}{5}$.
  1. 1
    Check the form and state restrictions
    One fraction each side. Restriction: $x \\neq 0$.
    x cannot be 0 because it would make the left denominator zero.
  2. 2
    Cross-multiply
    $3 \\times 5 = 2 \\times x$
    Multiply numerator left by denominator right, and denominator left by numerator right.
  3. 3
    Solve and check
    $15 = 2x$    so    $x = 7.5$
    Check: $\\frac{3}{7.5} = 0.4$ and $\\frac{2}{5} = 0.4$. Both match. x = 7.5 is not restricted.
Answer$x = 7.5$
Watch Me Solve It · Algebraic fraction with restriction
+15 XP per step
Q3
PROBLEM
Solve $\\frac{5}{x-2} = 3$. Identify any restrictions.
  1. 1
    State the restriction
    $x - 2 \\neq 0$    so    $x \\neq 2$
    The denominator cannot equal zero, so x cannot be 2.
  2. 2
    Multiply both sides by (x - 2)
    $5 = 3(x - 2)$
    This eliminates the fraction and leaves a simple linear equation.
  3. 3
    Expand and solve
    $5 = 3x - 6$  →  $11 = 3x$  →  $x = \\frac{11}{3}$
    Add 6 to both sides, then divide by 3.
  4. 4
    Check against restriction and verify
    $x = \\frac{11}{3} \\approx 3.67$, which is not 2. Valid.
    Check: $\\frac{5}{11/3 - 2} = \\frac{5}{5/3} = 5 \\times \\frac{3}{5} = 3$. Correct.
Answer$x = \\frac{11}{3}$

Brain Trainer · 4 problems

D
Brain Trainer · Fraction Equations
4 problems

Four equations covering LCD, cross-multiplication and algebraic fractions. Show each step, then reveal the answer to check.

  1. 1 Solve $\\frac{x}{4} + 2 = 7$.

    Subtract 2: $\frac{x}{4} = 5$. Multiply by 4: $x = 20$. Check: $\frac{20}{4} + 2 = 7$.

  2. 2 Solve $\\frac{2x}{5} = 8$.

    Multiply by 5: $2x = 40$. Divide by 2: $x = 20$. Check: $\frac{2(20)}{5} = \frac{40}{5} = 8$.

  3. 3 Solve $\\frac{x}{2} + \\frac{x}{3} = 10$.

    LCD = 6. Multiply all terms by 6: $3x + 2x = 60$. $5x = 60$. $x = 12$. Check: $\frac{12}{2} + \frac{12}{3} = 6 + 4 = 10$.

  4. 4 Solve $\\frac{3}{x+1} = 1$.

    Restriction: $x \neq -1$. Multiply by (x+1): $3 = x + 1$. $x = 2$. Check: $\frac{3}{2+1} = 1$. Valid.
Complete in your workbook.

Quick Check · 5 questions

1
What is the LCD of $\\frac{x}{4}$ and $\\frac{x}{6}$?
+10 XP
2
Solve $\\frac{x}{5} + 3 = 7$.
+10 XP
3
Solve $\\frac{3}{x} = \\frac{2}{5}$.
+10 XP
4
Which method is most appropriate for solving $\\frac{x}{2} + 4 = \\frac{x}{3} + 6$?
+10 XP
5
For the equation $\\frac{5}{x-3} = 2$, which value is not allowed?
+10 XP

Show Your Working · 3 questions

Show Your Working
9 marks total
Apply Medium 3 MARKS

Q6. Solve $\\frac{2x}{3} - \\frac{x}{4} = 5$. Show all steps, including finding the LCD and checking your answer.

Answer in your workbook.
Apply Medium 3 MARKS

Q7. Solve $\\frac{5}{2x-1} = 3$. Identify any restrictions on $x$, then show all working and verify your solution.

Answer in your workbook.
Evaluate Medium 3 MARKS

Q8. A recipe requires $\frac{2}{3}$ of a cup of sugar per batch. Jordan has $5\frac{1}{2}$ cups of sugar and wants to make as many full batches as possible.

(a) Write an equation to find the number of batches $n$. (1 mark)

(b) Solve the equation and state how many full batches Jordan can make. (1 mark)

(c) Explain why the answer must be a whole number and calculate how much sugar will be left over. (1 mark)

Answer in your workbook.
Comprehensive Answers

Quick Check

1. B -- Multiples of 4: 4, 8, 12, 16... Multiples of 6: 6, 12, 18... LCD = 12.

2. C -- Subtract 3: $\frac{x}{5} = 4$. Multiply by 5: $x = 20$.

3. B -- Cross-multiply: $3 \times 5 = 2x$, so $15 = 2x$, giving $x = 7.5$.

4. A -- Multiple terms on each side means LCD method. LCD of 2 and 3 is 6.

5. B -- Denominator is (x - 3). Setting x - 3 = 0 gives x = 3, which is not allowed.

Show Your Working Model Answers

Q6 (3 marks): LCD of 3 and 4 is 12 [0.5]. Multiply every term by 12: $12 \\times \\frac{2x}{3} - 12 \\times \\frac{x}{4} = 12 \\times 5$ [0.5]. Simplify: $8x - 3x = 60$ [0.5]. $5x = 60$ [0.5]. $x = 12$ [0.5]. Check: $\\frac{24}{3} - \\frac{12}{4} = 8 - 3 = 5$ [0.5].

Q7 (3 marks): Restriction: $2x - 1 \\neq 0$, so $x \\neq \\frac{1}{2}$ [0.5]. Multiply by $(2x - 1)$: $5 = 3(2x - 1)$ [0.5]. Expand: $5 = 6x - 3$ [0.5]. Add 3: $8 = 6x$ [0.5]. Divide by 6: $x = \\frac{4}{3}$ [0.5]. Check: $\\frac{5}{8/3 - 1} = \\frac{5}{5/3} = 3$ [0.5].

Q8 (3 marks): (a) $\\frac{2}{3}n = \\frac{11}{2}$ (since $5\\frac{1}{2} = \\frac{11}{2}$) [1]. (b) Multiply by 6: $4n = 33$, so $n = 8.25$. Jordan can make 8 full batches [1]. (c) The answer must be a whole number because you cannot make a fraction of a batch. Leftover sugar: $8 \\times \\frac{2}{3} = \\frac{16}{3}$ cups used. $\\frac{11}{2} - \\frac{16}{3} = \\frac{33}{6} - \\frac{32}{6} = \\frac{1}{6}$ cup left over [1].

Stretch Challenge · +25 XP, +10 coins

The Triple Fraction

Solve $\\frac{x+1}{2} - \\frac{x-1}{3} = \\frac{x+2}{4}$. Hint: find the LCD of all three denominators, multiply every term by it, then solve the resulting linear equation. Show every step.

Reveal solution

LCD of 2, 3 and 4 is 12.

Multiply every term by 12: $12 \\times \\frac{x+1}{2} - 12 \\times \\frac{x-1}{3} = 12 \\times \\frac{x+2}{4}$.

Simplify: $6(x+1) - 4(x-1) = 3(x+2)$.

Expand: $6x + 6 - 4x + 4 = 3x + 6$.

Collect: $2x + 10 = 3x + 6$.

Subtract 2x: $10 = x + 6$.

Subtract 6: $x = 4$.

Check: LHS = $\\frac{5}{2} - \\frac{3}{3} = 2.5 - 1 = 1.5$. RHS = $\\frac{6}{4} = 1.5$. Both match.

R
Quick Review

Find the LCD

The smallest number all denominators divide into

Multiply Every Term

Do not skip constants when clearing fractions

Cross-Multiply

Use only when there is one fraction on each side

State Restrictions

Denominator cannot equal zero

One Step Per Line

Clear working earns method marks

Always Check

Substitute back into the original equation

Interactive: Fraction Equation Solver

Practise clearing fractions, cross-multiplying and checking restrictions with instant feedback.

Your Badges

0 of 6
First Steps
3-Day Streak
3 in a Row
Lesson Ace
Stretch Seeker
Daily Warrior
DAILY CHALLENGE · TODAY
The Fraction Master
Solve $\\frac{2x}{3} + \\frac{x}{4} = \\frac{11}{2}$ in under 90 seconds for double coins!
NEXT UP · 0% UNLOCKED
Inequalities and Formulae
0/5
Finish this lesson's Practice MCs to unlock!

Printable Worksheets

Print or save as PDF — or build a custom worksheet from any module's questions.

Build Apply Master Build custom