Simplify the Nested Fraction: Calculating 32m Divided by (8t Divided by 3m)

Q: Why do I need to find the reciprocal of the inner fraction?

When you divide by a fraction, it's the same as multiplying by its reciprocal. So $ 32m ÷ \frac{8t}{3m} $ becomes $ 32m × \frac{3m}{8t} $!

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

The colon (:) is another way to write division. So $ 8t:3m $ means exactly the same thing as $ 8t ÷ 3m $ or $ \frac{8t}{3m} $.

Q: Why is my answer different from the simple division 32÷8?

Because this isn't simple division! The nested structure means you're dividing 32m by the result of (8t÷3m), not just dividing numbers. The variables and order of operations matter.

Q: How do I simplify the final fraction correctly?

Look for common factors in numerator and denominator. Here, both 96 and 8 are divisible by 8: $ \frac{96m^2}{8t} = \frac{96÷8 × m^2}{8÷8 × t} = \frac{12m^2}{t} $

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:04 Let's write division as a fraction

00:13 Division is also multiplication by the reciprocal

00:28 Make sure to multiply numerator by numerator and denominator by denominator

00:33 Let's factor 32 into 8 and 4

00:46 Let's reduce what we can

00:57 Let's solve the multiplication

01:01 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

$32m:(8t:3m)=\text{?}$

2

Step-by-step solution

To solve this problem, follow these steps:

Step 1: Simplify the inner division $8t : 3m$ . This can be rewritten as $\frac{8t}{3m}$ .
Step 2: Find the reciprocal of $\frac{8t}{3m}$ . The reciprocal is $\frac{3m}{8t}$ .
Step 3: Multiply $32m$ by the reciprocal of the inner division result, which is $\frac{3m}{8t}$ .

Carrying out the multiplication:

$32m \times \frac{3m}{8t} = \frac{32m \times 3m}{8t} = \frac{96m^2}{8t}$

Simplify the expression $\frac{96m^2}{8t}$ by dividing the numerator and the denominator by 8:

$= \frac{12m^2}{t}$

Thus, the final simplified solution is $12\frac{m^2}{t}$ .

The correct choice from the given options is $12\frac{m^2}{t}$ .

3

Final Answer

$12\frac{m^2}{t}$

Key Points to Remember

Essential concepts to master this topic

Rule: Division by a fraction means multiply by its reciprocal
Technique: Convert $8t:3m$ to $\frac{8t}{3m}$ , then find reciprocal $\frac{3m}{8t}$
Check: Multiply variables carefully: $32m \times \frac{3m}{8t} = \frac{96m^2}{8t} = 12\frac{m^2}{t}$ ✓

Common Mistakes

Avoid these frequent errors

Treating nested division as simple multiplication
Don't just multiply 32m × 8t × 3m = wrong answer with wrong variables! This ignores the division structure completely. Always convert the inner division to a fraction first, then use the reciprocal rule for division.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

Why do I need to find the reciprocal of the inner fraction?

+

When you divide by a fraction, it's the same as multiplying by its reciprocal. So $32m ÷ \frac{8t}{3m}$ becomes $32m × \frac{3m}{8t}$ !

How do I handle the variables m and t in the calculation?

+

Treat variables like numbers when multiplying and dividing. Multiply the m's together: $m × m = m^2$ , and keep t in the denominator throughout the calculation.

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

+

The colon (:) is another way to write division. So $8t:3m$ means exactly the same thing as $8t ÷ 3m$ or $\frac{8t}{3m}$ .

Why is my answer different from the simple division 32÷8?

+

Because this isn't simple division! The nested structure means you're dividing 32m by the result of (8t÷3m), not just dividing numbers. The variables and order of operations matter.

How do I simplify the final fraction correctly?

+

Look for common factors in numerator and denominator. Here, both 96 and 8 are divisible by 8: $\frac{96m^2}{8t} = \frac{96÷8 × m^2}{8÷8 × t} = \frac{12m^2}{t}$

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Simplify the Nested Fraction: Calculating 32m Divided by (8t Divided by 3m)

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