Equations with Fractions
Learn to solve equations that contain fractions by clearing denominators and using inverse operations.
Definition
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Worked Examples
Solve:
Identify what operation is applied to x
is divided by 4 → Division by 4
Apply the inverse operation
Multiply both sides by 4 →
Simplify
→
Check
✓ → Solution verified
Answer:
Common Mistakes
Multiplying only some terms by the LCD
Why it's wrong: When clearing denominators, you must multiply EVERY term (including whole numbers) by the LCD to keep the equation balanced.
Correct: In , multiply ALL terms: gives
Forgetting to distribute after clearing fractions
Why it's wrong: When you have expressions like and multiply by 4, you get , not .
Correct: , not
Using wrong LCD
Why it's wrong: An incorrect LCD leads to non-integer coefficients and more complex calculations.
Correct: LCD of 4 and 6 is 12, not 24. Find the LEAST common multiple, not just any common multiple.
Not checking the answer
Why it's wrong: Fraction equations can sometimes produce extraneous solutions, especially with variables in denominators.
Correct: Always substitute your answer back into the original equation to verify it works.
Interactive Visual
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Balance Scale
Solution: x = 4
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Practice Problems
17 problemsWhat is the first step to solve ?
Why It Matters
- Cooking: If a recipe serves 6 but you need to serve 4, you solve cups of flour needed
- Finance: Calculating how long to save when putting away of your income monthly
- Science: Rate problems like (distance, time, rate)
- Construction: Scaling blueprints where scale means solving
Real World Applications
Recipe Scaling
Chefs often need to adjust recipes when cooking for different numbers of people.
Example:
A recipe uses cup of flour to make 12 cookies. How much flour for 20 cookies? Solve .
A smoothie recipe calls for cup of berries for 2 servings. You want to make 5 servings.
How many cups of berries do you need?
Step 1: Write the mathematical expression
Set up the proportion:
Shared Work Problems
When two people work together, their combined work rate determines how fast they finish.
Example:
If Maria can paint a room in 4 hours and John in 6 hours, working together they complete of the room per hour.
Pipe A fills a tank in 3 hours. Pipe B fills it in 6 hours. With both pipes open, what fraction of the tank fills in 1 hour?
What fraction of the tank is filled per hour?
Step 1: Write the mathematical expression
Add the rates:
Speed and Distance
The formula $d = rt$ (distance = rate times time) often involves fractions when times are not whole numbers.
Example:
If you travel of an hour at 60 mph, you cover miles.
A cyclist travels 15 miles in of an hour.
What is the cyclist's average speed in miles per hour?
Step 1: Write the mathematical expression
Use :
Key Takeaways
- 1To solve equations with fractions, find the LCD and multiply every term to clear denominators
- 2The reciprocal method works well when a fraction multiplies the variable: multiply both sides by the reciprocal
- 3Always distribute carefully when clearing fractions with expressions in the numerator
- 4Check your answer by substituting back into the original equation
Frequently Asked Questions
Glossary
- LCD (Least Common Denominator)
- The smallest number that is a multiple of all denominators in an equation
- Reciprocal
- The flip of a fraction; the reciprocal of is
- Clear denominators
- Multiply every term by the LCD to eliminate all fractions
- Extraneous solution
- A solution that emerges from solving but does not satisfy the original equation
Formula Card
LCD Method
Clears all fractions at once
Reciprocal Method
Multiply by the reciprocal of the coefficient
Cross-Multiplication
Only for proportions