Equations with Decimals
Learn to solve linear equations that contain decimal numbers using multiplication strategies.
Definition
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Worked Examples
Solve:
Identify the operation
is being multiplied by → Division will isolate
Divide both sides by
→
Check the solution
\checkmark → Correct!
Answer:
Common Mistakes
Forgetting to multiply ALL terms when clearing decimals
Why it's wrong: If you multiply by , you must multiply every term: , not .
Correct: Write out each multiplication: to avoid missing terms.
Using the wrong power of 10
Why it's wrong: For , using gives (still has decimals).
Correct: Count the maximum decimal places (2 in ), then use : .
Decimal arithmetic errors
Why it's wrong: Mistakes like happen when dividing with decimals.
Correct: Double-check division: . Converting to whole numbers helps!
Not checking the solution in the original equation
Why it's wrong: Errors in decimal arithmetic can go unnoticed without verification.
Correct: Always substitute your answer back into the original (decimal) equation to verify.
Interactive Visual
Equation Solver
Ready to solve some equations?
Follow along as we solve the equation step by step.
Place Value Chart
| Ones 1 | Tenths 0.1 |
|---|---|
| 2 | 5 |
Expanded Form:
2 × 1 + 5 × 0.1
Click on any place value to highlight it and see its value.
Balance Scale
Solution: x = 4
Interactive Sandbox
Expression Calculator
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Practice Problems
16 problemsSolve:
Why It Matters
- Shopping: If an item costs 2.50 dollars per unit and you spend 17.50 dollars total, how many did you buy?
- Science: Calculating concentrations, measurements, and experimental data
- Finance: Interest rates, tax calculations, and budgeting often involve decimals
- Engineering: Precise measurements and tolerances are expressed as decimals
Real World Applications
Shopping and Unit Prices
Stores often price items with decimals. Finding how many items you can buy requires solving decimal equations.
Example:
Apples cost 1.25 dollars each. If you spend 8.75 dollars, how many apples did you buy? Solve:
You buy notebooks at 2.50 dollars each and a pen for 1.75 dollars. Your total is 14.25 dollars.
How many notebooks did you buy?
Step 1: Write the mathematical expression
Set up the equation:
Temperature Conversion
Converting between Celsius and Fahrenheit uses decimals.
Example:
To convert Celsius to Fahrenheit: . If , solve for :
The temperature is 95 degrees Fahrenheit. Use to find Celsius.
What is 95 degrees F in Celsius?
Step 1: Write the mathematical expression
Set up:
Gas Mileage and Travel
Fuel efficiency calculations involve decimal equations.
Example:
If gas costs 1.85 euros per liter and you spend 55.50 euros, how many liters did you buy? Solve:
Your car uses 0.08 liters per kilometer. You want to travel until you have used 12.8 liters of fuel.
How far can you travel?
Step 1: Write the mathematical expression
Set up:
Key Takeaways
- 1Decimal equations can be solved directly or by multiplying to clear decimals first
- 2To clear decimals, multiply every term by the appropriate power of 10
- 3Count decimal places: 1 place = multiply by 10, 2 places = multiply by 100, etc.
- 4Always check your solution by substituting back into the original equation
- 5The same equation-solving rules apply: what you do to one side, do to the other
Frequently Asked Questions
Glossary
- Coefficient
- The number multiplied by a variable (e.g., in )
- Constant
- A number without a variable (e.g., in )
- Clear decimals
- Multiply all terms by a power of 10 to convert decimals to whole numbers
- Power of 10
- Numbers like , , (, , )
Formula Card
Clearing Decimals Strategy
Where $n$ = maximum number of decimal places in the equation
Decimal Place Rule
Use 10 for 1 decimal place, 100 for 2 places, 1000 for 3 places