Solving the Fraction Equation: Uncover X in 2/5x = 40

Q: What exactly is a reciprocal and why do I need it?

A reciprocal is when you flip a fraction upside down. For $ \frac{2}{5} $, the reciprocal is $ \frac{5}{2} $. When you multiply a fraction by its reciprocal, you get 1, which perfectly isolates your variable!

Q: Can I just multiply both sides by 5 and then divide by 2?

Yes, that works too! Multiplying by 5 gives you $ 2x = 200 $, then dividing by 2 gives $ x = 100 $. Both methods are correct - use whichever feels easier!

Q: How do I multiply a whole number by a fraction?

To multiply 40 by $ \frac{5}{2} $, think: 40 × 5 = 200, then 200 ÷ 2 = 100. Or write it as $ \frac{40 \times 5}{2} = \frac{200}{2} = 100 $.

Q: What if the fraction coefficient is negative?

The process is exactly the same! If you have $ -\frac{2}{5}x = 40 $, multiply both sides by the reciprocal $ -\frac{5}{2} $. The negative signs will give you a negative answer.

Q: How can I check my work quickly?

Substitute your answer back: $ \frac{2}{5} \times 100 = \frac{200}{5} = 40 $ ✓. If you get the original right side (40), your answer is correct!

Linear Equations with Fractional Coefficients

Solve for X:

$\frac{2}{5}x=40$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:03 We want to isolate the unknown X

00:07 We'll multiply by the reciprocal fraction to eliminate the fraction

00:12 We'll reduce what we can

00:24 We'll factor 40 into 20 and 2

00:29 We'll reduce what we can

00:36 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve for X:

$\frac{2}{5}x=40$

Step-by-step solution

To solve the equation $\frac{2}{5}x = 40$ , follow these steps:

Step 1: Identify the reciprocal of the fraction $\frac{2}{5}$ , which is $\frac{5}{2}$ .
Step 2: Multiply both sides of the equation by $\frac{5}{2}$ to isolate $x$ .
Step 3: Simplify the equation to find the value of $x$ .

Let's perform the calculation:

Step 1: Multiply both sides by the reciprocal:
$\frac{5}{2} \times \frac{2}{5}x = \frac{5}{2} \times 40$ .

The left side simplifies to:
$x = \frac{5}{2} \times 40$ .

Step 2: Calculate the right side:
$x = 200 \div 2 = 100$ .

Therefore, the solution to the equation $\frac{2}{5}x = 40$ is $x = 100$ .

Final Answer

$100$

Key Points to Remember

Essential concepts to master this topic

Isolation Rule: Multiply both sides by reciprocal of fraction coefficient
Technique: Multiply $\frac{2}{5}x = 40$ by $\frac{5}{2}$ to get x
Check: Substitute x = 100: $\frac{2}{5} \times 100 = 40$ ✓

Common Mistakes

Avoid these frequent errors

Dividing by the fraction instead of multiplying by reciprocal
Don't divide both sides by $\frac{2}{5}$ = wrong calculation! Division by fractions is confusing and leads to errors. Always multiply both sides by the reciprocal $\frac{5}{2}$ to isolate x cleanly.

Practice Quiz

Test your knowledge with interactive questions

Solve for X:

$$ 5x=25 $$

FAQ

Everything you need to know about this question

What exactly is a reciprocal and why do I need it?

A reciprocal is when you flip a fraction upside down. For $\frac{2}{5}$ , the reciprocal is $\frac{5}{2}$ . When you multiply a fraction by its reciprocal, you get 1, which perfectly isolates your variable!

Can I just multiply both sides by 5 and then divide by 2?

Yes, that works too! Multiplying by 5 gives you $2x = 200$ , then dividing by 2 gives $x = 100$ . Both methods are correct - use whichever feels easier!

How do I multiply a whole number by a fraction?

To multiply 40 by $\frac{5}{2}$ , think: 40 × 5 = 200, then 200 ÷ 2 = 100. Or write it as $\frac{40 \times 5}{2} = \frac{200}{2} = 100$ .

What if the fraction coefficient is negative?

The process is exactly the same! If you have $-\frac{2}{5}x = 40$ , multiply both sides by the reciprocal $-\frac{5}{2}$ . The negative signs will give you a negative answer.

How can I check my work quickly?

Substitute your answer back: $\frac{2}{5} \times 100 = \frac{200}{5} = 40$ ✓. If you get the original right side (40), your answer is correct!

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