Decoding Ratios: Solve the Equation for 2x : (7y : 4x)

Q: What does the colon symbol (:) mean in math?

The colon symbol means division or "to". So $ 2x : 7y $ is the same as $ \frac{2x}{7y} $. It's commonly used in ratios and proportions.

Q: Why do I need to flip the fraction when dealing with nested ratios?

When you have $ a : (b : c) $, you're dividing by a division! Dividing by a fraction means multiplying by its reciprocal. So $ b : c $ becomes $ \frac{c}{b} $.

Q: How do I convert the improper fraction to a mixed number?

$ \frac{8x^2}{7y} = \frac{7x^2 + x^2}{7y} = \frac{7x^2}{7y} + \frac{x^2}{7y} = \frac{x^2}{y} + \frac{x^2}{7y} = 1\frac{1}{7}\frac{x^2}{y} $Divide the numerator by the coefficient of the denominator!

Q: Can I solve this problem without using the reciprocal rule?

Not easily! The reciprocal rule $ a : (b : c) = a \times \frac{c}{b} $ is specifically designed for nested ratios. Other methods will make the problem much more complicated.

Q: What if there are more than two levels of nesting?

Work from the innermost parentheses outward, applying the reciprocal rule at each level. For example: $ a : (b : (c : d)) $ would become $ a : (b \times \frac{d}{c}) $ first.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:03 Let's write division as a fraction

00:09 Division is also multiplication by the reciprocal

00:14 Make sure to multiply numerator by numerator and denominator by denominator

00:19 Convert from fraction to whole number and remainder

00:23 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

$2x:(7y:4x)=\text{?}$

2

Step-by-step solution

To solve this problem, we'll simplify the expression $2x : (7y : 4x)$ using properties of division:

Step 1: Recognize $a : (b : c) = a \times \frac{c}{b}$ . This changes the nested division into a singular operation.
Step 2: Substitute into our expression: $2x : (7y : 4x) = 2x \times \frac{4x}{7y}$ .
Step 3: Simplify the expression $= 2 \times \frac{4x^2}{7y}$ .
Step 4: Calculate the product: $= \frac{8x^2}{7y}$ .
Step 5: Convert to a mixed number if necessary: $= 1\frac{1}{7}\frac{x^2}{y}$ .

Therefore, the solution to the problem is $\frac{8x^2}{7y}$ , which can be expressed as a mixed number $1\frac{1}{7}\frac{x^2}{y}$ .

3

Final Answer

$1\frac{1}{7}\frac{x^2}{y}$

Key Points to Remember

Essential concepts to master this topic

Rule: Transform nested ratios using $a : (b : c) = a \times \frac{c}{b}$
Technique: Convert $2x : (7y : 4x)$ to $2x \times \frac{4x}{7y} = \frac{8x^2}{7y}$
Check: Verify by substituting test values like x=2, y=1 into original expression ✓

Common Mistakes

Avoid these frequent errors

Treating nested ratios like regular division
Don't just divide left to right like $2x ÷ 7y ÷ 4x$ = wrong result! This ignores the grouping and gives $\frac{1}{14xy}$ instead of $\frac{8x^2}{7y}$ . Always recognize that parentheses create nested division, requiring the reciprocal rule.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

What does the colon symbol (:) mean in math?

+

The colon symbol means division or "to". So $2x : 7y$ is the same as $\frac{2x}{7y}$ . It's commonly used in ratios and proportions.

Why do I need to flip the fraction when dealing with nested ratios?

+

When you have $a : (b : c)$ , you're dividing by a division! Dividing by a fraction means multiplying by its reciprocal. So $b : c$ becomes $\frac{c}{b}$ .

How do I convert the improper fraction to a mixed number?

+

$\frac{8x^2}{7y} = \frac{7x^2 + x^2}{7y} = \frac{7x^2}{7y} + \frac{x^2}{7y} = \frac{x^2}{y} + \frac{x^2}{7y} = 1\frac{1}{7}\frac{x^2}{y}$

Divide the numerator by the coefficient of the denominator!

Can I solve this problem without using the reciprocal rule?

+

Not easily! The reciprocal rule $a : (b : c) = a \times \frac{c}{b}$ is specifically designed for nested ratios. Other methods will make the problem much more complicated.

What if there are more than two levels of nesting?

+

Work from the innermost parentheses outward, applying the reciprocal rule at each level. For example: $a : (b : (c : d))$ would become $a : (b \times \frac{d}{c})$ first.

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Decoding Ratios: Solve the Equation for 2x : (7y : 4x)

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