Solve Complex Division: 27½ ÷ (2⅕ ÷ ⅘) Step-by-Step

Q: Why do I have to solve the parentheses first?

The order of operations (PEMDAS) requires you to work inside parentheses first! If you ignore this, you'll get a completely different answer. Think of parentheses as a "do this calculation first" instruction.

Q: How do I convert mixed numbers to improper fractions?

Use the formula: multiply whole number by denominator, then add numerator. For $ 2\frac{1}{5} $: $ (2 \times 5) + 1 = 11 $, so it becomes $ \frac{11}{5} $.

Q: When I divide fractions, which one gets flipped?

Always flip the second fraction (the divisor) and change division to multiplication. So $ \frac{11}{5} \div \frac{4}{5} $ becomes $ \frac{11}{5} \times \frac{5}{4} $.

Q: How do I simplify the final multiplication?

Look for common factors to cancel. In $ \frac{55 \times 4}{2 \times 11} $, notice that 55 ÷ 11 = 5, so you get $ \frac{5 \times 4}{2} = \frac{20}{2} = 10 $.

Q: Can I work with decimals instead of fractions?

Technically yes, but fractions are more accurate here! Converting $ 2\frac{1}{5} = 2.2 $ and $ \frac{4}{5} = 0.8 $ might lead to rounding errors in complex calculations.

Q: What if I get a different answer than the choices?

Double-check your order of operations and fraction conversions! Most errors happen from solving left-to-right instead of parentheses-first, or making arithmetic mistakes when converting mixed numbers.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:03 Division is also multiplication by reciprocal

00:07 Therefore when opening the parentheses, at the end it will be multiplied by C

00:12 We will use this formula in our exercise

00:22 Let's solve one operation at a time

00:29 Convert from number to fraction

00:36 Division is also multiplication by reciprocal

00:42 Divide 55 by 11

00:48 Multiply 5 by 5

00:52 Let's write down the solution we found in the exercise and continue solving

00:57 Divide 25 by 5

01:00 Multiply 5 by 4

01:03 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

$27\frac{1}{2}:(2\frac{1}{5}:\frac{4}{5})=$

2

Step-by-step solution

Let's solve the problem step-by-step.

Step 1: Convert the mixed numbers to improper fractions.

Convert $27\frac{1}{2}$ to an improper fraction:
$27\frac{1}{2} = \frac{2 \times 27 + 1}{2} = \frac{55}{2}$ .

Convert $2\frac{1}{5}$ to an improper fraction:
$2\frac{1}{5} = \frac{5 \times 2 + 1}{5} = \frac{11}{5}$ .

Step 2: Simplify $2\frac{1}{5} : \frac{4}{5}$ using reciprocal multiplication.

$\frac{11}{5} : \frac{4}{5} = \frac{11}{5} \times \frac{5}{4} = \frac{11 \times 5}{5 \times 4} = \frac{11}{4}$ .

Step 3: Divide $27\frac{1}{2}$ by the result from Step 2.

$\frac{55}{2} : \frac{11}{4} = \frac{55}{2} \times \frac{4}{11} = \frac{55 \times 4}{2 \times 11}$ .

Simplify the expression:
The $55$ in the numerator and $11$ in the denominator simplify as follows: $55 \div 11 = 5$ , leaving us with:
$\frac{5 \times 4}{2} = \frac{20}{2} = 10$ .

Therefore, the solution to the problem is $10$ , matching the correct choice.

3

Final Answer

10

Key Points to Remember

Essential concepts to master this topic

Order of Operations: Work from inner parentheses outward first
Technique: Convert mixed numbers: $27\frac{1}{2} = \frac{55}{2}$
Check: Division by reciprocal: $\frac{55}{2} \div \frac{11}{4} = \frac{55}{2} \times \frac{4}{11} = 10$ ✓

Common Mistakes

Avoid these frequent errors

Working left to right instead of following order of operations
Don't solve $27\frac{1}{2} \div 2\frac{1}{5}$ first = wrong result! This ignores the parentheses which must be solved first. Always solve the expression in parentheses $2\frac{1}{5} \div \frac{4}{5}$ before dividing by $27\frac{1}{2}$ .

Practice Quiz

Test your knowledge with interactive questions

$70:(14\times5)=$

FAQ

Everything you need to know about this question

Why do I have to solve the parentheses first?

+

The order of operations (PEMDAS) requires you to work inside parentheses first! If you ignore this, you'll get a completely different answer. Think of parentheses as a "do this calculation first" instruction.

How do I convert mixed numbers to improper fractions?

+

Use the formula: multiply whole number by denominator, then add numerator. For $2\frac{1}{5}$ : $(2 \times 5) + 1 = 11$ , so it becomes $\frac{11}{5}$ .

When I divide fractions, which one gets flipped?

+

Always flip the second fraction (the divisor) and change division to multiplication. So $\frac{11}{5} \div \frac{4}{5}$ becomes $\frac{11}{5} \times \frac{5}{4}$ .

How do I simplify the final multiplication?

+

Look for common factors to cancel. In $\frac{55 \times 4}{2 \times 11}$ , notice that 55 ÷ 11 = 5, so you get $\frac{5 \times 4}{2} = \frac{20}{2} = 10$ .

Can I work with decimals instead of fractions?

+

Technically yes, but fractions are more accurate here! Converting $2\frac{1}{5} = 2.2$ and $\frac{4}{5} = 0.8$ might lead to rounding errors in complex calculations.

What if I get a different answer than the choices?

+

Double-check your order of operations and fraction conversions! Most errors happen from solving left-to-right instead of parentheses-first, or making arithmetic mistakes when converting mixed numbers.

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Solve Complex Division: 27½ ÷ (2⅕ ÷ ⅘) Step-by-Step

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