Solve Complex Expression: 315÷(3/x)-(12x-4x÷(y/x))

Q: Why do I flip the fraction when dividing?

When you divide by a fraction, you're asking "how many groups of that size fit?" Flipping and multiplying gives the same result but is much easier to calculate. Think: $ 6 \div \frac{1}{2} = 12 $ because there are 12 half-pieces in 6 whole pieces!

Q: How do I handle the colon symbol (:) in division?

The colon (:) is just another way to write division (÷). So $ 315:\frac{3}{x} $ means exactly the same as $ 315 \div \frac{3}{x} $. Use whichever symbol you're more comfortable with!

Q: What's the order of operations with these complex fractions?

Follow PEMDAS/BODMAS: Handle what's inside parentheses first, then division/multiplication from left to right. For $ 4x:\frac{y}{x} $, treat the whole fraction $ \frac{y}{x} $ as one unit before dividing.

Q: Why does the final answer have two terms that can't be combined?

The terms $ 93x $ and $ \frac{4x^2}{y} $ are not like terms! One has degree 1 in x, the other has degree 2 in x and involves y. You can only combine terms with identical variable parts.

Q: How can I check if my algebra is correct?

Pick simple values like x = 2, y = 4 and substitute into both the original expression and your answer. If you get the same number from both, your algebra is correct! This is much faster than re-doing all the steps.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:03 Division is also multiplication by the reciprocal

00:28 Let's break down 315 into factors 105 and 3

00:40 Negative times positive always equals negative

00:45 Negative times negative always equals positive

00:55 Let's reduce what we can

01:11 Let's group the factors

01:20 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

$315:\frac{3}{x}-(12x-4x:\frac{y}{x})=\text{?}$

2

Step-by-step solution

To solve this problem, we need to address each term in the expression separately before combining them:

Simplify $315:\frac{3}{x}$ :
This is equivalent to $315 \div \left(\frac{3}{x}\right)$ , which can be rewritten using the property $a \div \frac{b}{c} = a \times \frac{c}{b}$ . Thus, $315 \cdot \frac{x}{3} = 105x.$
Simplify $12x-4x:\frac{y}{x}$ :
First, simplify $4x:\frac{y}{x}$ .
By the same property as above, we have: $4x \div \left(\frac{y}{x}\right) = 4x \cdot \frac{x}{y} = \frac{4x^2}{y}.$ Now, substitute back into the expression: $12x - \frac{4x^2}{y} = 12x - \frac{4x^2}{y}.$
Subtract the second simplified expression from the first:
$105x - (12x - \frac{4x^2}{y})$ Distribute the negative sign: $105x - 12x + \frac{4x^2}{y}.$ Combine like terms: $93x + \frac{4x^2}{y}.$

Therefore, the solution to the given expression is $93x + \frac{4x^2}{y}$ .

3

Final Answer

$93x+\frac{4x^2}{y}$

Key Points to Remember

Essential concepts to master this topic

Division Rule: Dividing by a fraction means multiply by its reciprocal
Technique: $315 \div \frac{3}{x} = 315 \times \frac{x}{3} = 105x$
Check: Substitute values to verify $93x + \frac{4x^2}{y}$ equals original expression ✓

Common Mistakes

Avoid these frequent errors

Incorrectly handling division by fractions
Don't leave division by fractions as $\frac{315}{\frac{3}{x}}$ = confusing mess! This makes the problem impossible to solve correctly. Always convert division by a fraction to multiplication by its reciprocal immediately.

Practice Quiz

Test your knowledge with interactive questions

$$ 100-(5+55)= $$

FAQ

Everything you need to know about this question

Why do I flip the fraction when dividing?

+

When you divide by a fraction, you're asking "how many groups of that size fit?" Flipping and multiplying gives the same result but is much easier to calculate. Think: $6 \div \frac{1}{2} = 12$ because there are 12 half-pieces in 6 whole pieces!

How do I handle the colon symbol (:) in division?

+

The colon (:) is just another way to write division (÷). So $315:\frac{3}{x}$ means exactly the same as $315 \div \frac{3}{x}$ . Use whichever symbol you're more comfortable with!

What's the order of operations with these complex fractions?

+

Follow PEMDAS/BODMAS: Handle what's inside parentheses first, then division/multiplication from left to right. For $4x:\frac{y}{x}$ , treat the whole fraction $\frac{y}{x}$ as one unit before dividing.

Why does the final answer have two terms that can't be combined?

+

The terms $93x$ and $\frac{4x^2}{y}$ are not like terms! One has degree 1 in x, the other has degree 2 in x and involves y. You can only combine terms with identical variable parts.

How can I check if my algebra is correct?

+

Pick simple values like x = 2, y = 4 and substitute into both the original expression and your answer. If you get the same number from both, your algebra is correct! This is much faster than re-doing all the steps.

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Solve Complex Expression: 315÷(3/x)-(12x-4x÷(y/x))

Division by Fractions with Variable Expressions

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