Direct Method: One-Step Decimal Equation

Solve: $0.4x = 2.8$

Identify the operation: $x$ is being multiplied by $0.4$ = Division will isolate $x$

Divide both sides by $0.4$: $\frac{0.4x}{0.4} = \frac{2.8}{0.4}$ = $x = 7$

Check the solution: $0.4 \times 7 = 2.8$ \checkmark = Correct!

Answer: $x = 7$

Multiply-to-Clear: Two-Step Equation

Solve: $0.3x + 0.5 = 2.0$

Find the power of 10 to clear decimals: All decimals have 1 decimal place, so multiply by $10$ = Multiply everything by $10$

Multiply each term by $10$: $10(0.3x) + 10(0.5) = 10(2.0)$ = $3x + 5 = 20$

Subtract $5$ from both sides: $3x + 5 - 5 = 20 - 5$ = $3x = 15$

Divide both sides by $3$: $\frac{3x}{3} = \frac{15}{3}$ = $x = 5$

Check in original equation: $0.3(5) + 0.5 = 1.5 + 0.5 = 2.0$ \checkmark = Correct!

Answer: $x = 5$

Mixed Decimal Places

Solve: $0.25x - 1.5 = 3.75$

Find the largest number of decimal places: $0.25$ has 2 places, $1.5$ has 1 place, $3.75$ has 2 places = Multiply by $100$ (for 2 decimal places)

Multiply each term by $100$: $100(0.25x) - 100(1.5) = 100(3.75)$ = $25x - 150 = 375$

Add $150$ to both sides: $25x - 150 + 150 = 375 + 150$ = $25x = 525$

Divide both sides by $25$: $\frac{25x}{25} = \frac{525}{25}$ = $x = 21$

Check in original equation: $0.25(21) - 1.5 = 5.25 - 1.5 = 3.75$ \checkmark = Correct!

Answer: $x = 21$

Decimals on Both Sides

Solve: $1.2x + 0.8 = 0.6x + 2.6$

Multiply by $10$ to clear decimals: $10(1.2x) + 10(0.8) = 10(0.6x) + 10(2.6)$ = $12x + 8 = 6x + 26$

Subtract $6x$ from both sides: $12x - 6x + 8 = 6x - 6x + 26$ = $6x + 8 = 26$

Subtract $8$ from both sides: $6x + 8 - 8 = 26 - 8$ = $6x = 18$

Divide both sides by $6$: $\frac{6x}{6} = \frac{18}{6}$ = $x = 3$

Check in original equation: $1.2(3) + 0.8 = 3.6 + 0.8 = 4.4$ $0.6(3) + 2.6 = 1.8 + 2.6 = 4.4$ \checkmark = Correct!

Answer: $x = 3$

Mistake: Forgetting to multiply ALL terms when clearing decimals

Why: If you multiply $0.5x + 2 = 3.5$ by $10$, you must multiply every term: $5x + 20 = 35$, not $5x + 2 = 35$.

Correct: Write out each multiplication: $10(0.5x) + 10(2) = 10(3.5)$ to avoid missing terms.

Mistake: Using the wrong power of 10

Why: For $0.25x = 1.5$, using $\times 10$ gives $2.5x = 15$ (still has decimals).

Correct: Count the maximum decimal places (2 in $0.25$), then use $10^2 = 100$: $25x = 150$.

Mistake: Decimal arithmetic errors

Why: Mistakes like $2.8 \div 0.4 = 0.7$ happen when dividing with decimals.

Correct: Double-check division: $2.8 \div 0.4 = 28 \div 4 = 7$. Converting to whole numbers helps!

Mistake: Not checking the solution in the original equation

Why: Errors in decimal arithmetic can go unnoticed without verification.

Correct: Always substitute your answer back into the original (decimal) equation to verify.

Shopping and Unit Prices

Stores often price items with decimals. Finding how many items you can buy requires solving decimal equations.

Apples cost 1.25 dollars each. If you spend 8.75 dollars, how many apples did you buy? Solve: $1.25x = 8.75$

Temperature Conversion

Converting between Celsius and Fahrenheit uses decimals.

To convert Celsius to Fahrenheit: $F = 1.8C + 32$. If $F = 77$, solve for $C$: $77 = 1.8C + 32$

Gas Mileage and Travel

Fuel efficiency calculations involve decimal equations.

If gas costs 1.85 euros per liter and you spend 55.50 euros, how many liters did you buy? Solve: $1.85x = 55.50$

Q: When should I clear decimals versus work with them directly?

A: For simple one-step equations like $0.5x = 4$, working directly is often faster. For multi-step equations or those with many decimals, clearing first usually reduces errors.

Q: What if my answer is a decimal?

A: That is perfectly fine! If clearing decimals gives you $5x = 12$, then $x = 2.4$. Not all solutions are whole numbers.

Q: Can I multiply by any number to clear decimals?

A: Technically yes, but powers of 10 work best because they simply shift the decimal point. Multiplying $0.5$ by $10$ gives exactly $5$.