Solve: 1/(mn) - (mn-(m:1/n+n/m:(mn))) Complex Fraction Problem

Q: What does the colon notation mean in this expression?

The colon : represents division. So $ m:\frac{1}{n} $ means m divided by $ \frac{1}{n} $, which equals $ m \times n = mn $.

Q: Why do we work from the inside out?

Following the order of operations (PEMDAS/BODMAS) requires us to solve expressions in parentheses first. Start with the innermost parentheses and work outward to avoid errors.

Q: How do I multiply fractions like n/m × 1/(mn)?

Multiply straight across: $ \frac{n}{m} \times \frac{1}{mn} = \frac{n \times 1}{m \times mn} = \frac{n}{m^2n} $. Then cancel the common factor n to get $ \frac{1}{m^2} $.

Q: What happens when I subtract a negative number?

Subtracting a negative is the same as adding a positive! So $ \frac{1}{mn} - (-\frac{1}{m^2}) = \frac{1}{mn} + \frac{1}{m^2} $.

Q: Can I combine the final fractions into one?

You could find a common denominator, but the answer $ \frac{1}{mn} + \frac{1}{m^2} $ is already in simplest form. Sometimes keeping fractions separate is cleaner than combining them.

Complex Fraction Simplification with Order of Operations

$\frac{1}{mn}-(mn-(m:\frac{1}{n}+\frac{n}{m}:(mn))=?$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:07 Negative times positive is always negative

00:11 Division is also multiplication by the reciprocal

00:25 Let's move the division to the denominator

00:35 Let's simplify what we can

00:46 Negative times negative is always positive

00:50 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

$\frac{1}{mn}-(mn-(m:\frac{1}{n}+\frac{n}{m}:(mn))=?$

Step-by-step solution

To solve this problem, follow these steps:

Simplify the innermost portion of the expression: $(m:\frac{1}{n}+\frac{n}{m}:(mn))$ .
We interpret $m:\frac{1}{n}$ as $m \cdot n$ , translating division by a fraction into multiplication.
The entire sub-expression becomes $(m \cdot n + \frac{n}{m} \cdot \frac{1}{mn})$ .
Recognize that $\frac{n}{m} \cdot \frac{1}{mn} = \frac{n}{m^2n} = \frac{1}{m^2}$ .
This simplifies to $mn + \frac{1}{m^2}$ .

Now simplify the outer expression:

Substitute back into the original expression:
The expression becomes $\frac{1}{mn} - (mn - (mn + \frac{1}{m^2}))$ .
Calculate inside the bracket: $mn - mn - \frac{1}{m^2} = -\frac{1}{m^2}$ .
Therefore, the expression becomes $\frac{1}{mn} - (-\frac{1}{m^2}) = \frac{1}{mn} + \frac{1}{m^2}$ .

Therefore, the simplified form of the problem is $\frac{1}{mn} + \frac{1}{m^2}$ .

Final Answer

$\frac{1}{mn}+\frac{1}{m^2}$

Key Points to Remember

Essential concepts to master this topic

Division Rule: Converting division by fraction to multiplication by reciprocal
Technique: Simplify innermost expressions first: $m:\frac{1}{n} = m \cdot n$
Check: Substitute back to verify: $\frac{1}{mn} + \frac{1}{m^2}$ matches final answer ✓

Common Mistakes

Avoid these frequent errors

Incorrectly interpreting the colon notation as simple division
Don't treat $m:\frac{1}{n}$ as $\frac{m}{\frac{1}{n}}$ = confusion with complex fractions! This leads to wrong simplification and incorrect final answers. Always convert division by a fraction to multiplication by its reciprocal first.

Practice Quiz

Test your knowledge with interactive questions

$$ 100-(30-21)= $$

FAQ

Everything you need to know about this question

What does the colon notation mean in this expression?

The colon : represents division. So $m:\frac{1}{n}$ means m divided by $\frac{1}{n}$ , which equals $m \times n = mn$ .

Why do we work from the inside out?

Following the order of operations (PEMDAS/BODMAS) requires us to solve expressions in parentheses first. Start with the innermost parentheses and work outward to avoid errors.

How do I multiply fractions like n/m × 1/(mn)?

Multiply straight across: $\frac{n}{m} \times \frac{1}{mn} = \frac{n \times 1}{m \times mn} = \frac{n}{m^2n}$ . Then cancel the common factor n to get $\frac{1}{m^2}$ .

What happens when I subtract a negative number?

Subtracting a negative is the same as adding a positive! So $\frac{1}{mn} - (-\frac{1}{m^2}) = \frac{1}{mn} + \frac{1}{m^2}$ .

Can I combine the final fractions into one?

You could find a common denominator, but the answer $\frac{1}{mn} + \frac{1}{m^2}$ is already in simplest form. Sometimes keeping fractions separate is cleaner than combining them.

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