Solve Compound Division: 42÷(7÷2½) Step-by-Step Solution

Q: Why can't I just work from left to right?

In math, parentheses always come first according to the order of operations (PEMDAS/BODMAS). The colon : means division, so you must solve $ 7 \div 2\frac{1}{2} $ before dividing 42 by that result.

Q: Why do I multiply by the reciprocal when dividing fractions?

Division by a fraction is the same as multiplication by its reciprocal. So $ 7 \div \frac{5}{2} = 7 \times \frac{2}{5} $. This is a fundamental rule that makes fraction division possible.

Q: Can I use decimals instead of fractions?

Yes! $ 2\frac{1}{2} = 2.5 $, so the problem becomes $ 42 \div (7 \div 2.5) $. Just remember: parentheses first, then the outer division.

Q: How can I check if my answer of 15 is correct?

Work backwards: if the answer is 15, then $ 42 \div 15 $ should equal $ 7 \div 2\frac{1}{2} $. Since $ 42 \div 15 = 2.8 $ and $ 7 \div 2.5 = 2.8 $, our answer is correct!

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:05 Let's solve this problem, step by step.

00:13 We'll start by using the formula. Ready?

00:21 We'll tackle one operation at a time, nice and easy.

00:29 Convert the whole number and fraction into a single fraction.

00:33 Great job! That's the solution to our question.

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

$42:(7:2\frac{1}{2})=$

2

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Convert the mixed number to an improper fraction.
Step 2: Simplify the inner division using fraction rules.
Step 3: Simplify the outer division with the result obtained from Step 2.

Now, let's work through each step:
Step 1: Convert the mixed number $2\frac{1}{2}$ to an improper fraction. We have: $\large 2\frac{1}{2} = \frac{5}{2}$
Step 2: Simplify the inner division $7 \div 2\frac{1}{2}$ . This becomes: $\large 7 \div \frac{5}{2} = 7 \times \frac{2}{5} = \frac{14}{5}$
Step 3: Now, simplify the overall expression $42 \div$ $\frac{14}{5}$ . This converts to multiplication: $\large 42 \times \frac{5}{14}$
Calculating this gives: $42 \times \frac{5}{14} = \frac{42 \times 5}{14} = \frac{210}{14} = 15$

Therefore, the solution to the problem is $15$ .

3

Final Answer

15

Key Points to Remember

Essential concepts to master this topic

Order of Operations: Always solve inner parentheses first, then outer operations
Mixed to Improper: Convert $2\frac{1}{2}$ to $\frac{5}{2}$ before dividing
Verification: Check that $42 \div (7 \div 2.5) = 42 \div 2.8 = 15$ ✓

Common Mistakes

Avoid these frequent errors

Solving left to right without respecting parentheses
Don't calculate 42 ÷ 7 first = 6, then divide by 2½! This ignores order of operations and gives the wrong answer of 2.4. Always solve what's inside parentheses first: 7 ÷ 2½ = 14/5, then 42 ÷ 14/5 = 15.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

Why can't I just work from left to right?

+

In math, parentheses always come first according to the order of operations (PEMDAS/BODMAS). The colon : means division, so you must solve $7 \div 2\frac{1}{2}$ before dividing 42 by that result.

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

+

Multiply the whole number by the denominator, then add the numerator: $2\frac{1}{2} = \frac{(2 \times 2) + 1}{2} = \frac{5}{2}$ . This makes division much easier!

Why do I multiply by the reciprocal when dividing fractions?

+

Division by a fraction is the same as multiplication by its reciprocal. So $7 \div \frac{5}{2} = 7 \times \frac{2}{5}$ . This is a fundamental rule that makes fraction division possible.

Can I use decimals instead of fractions?

+

Yes! $2\frac{1}{2} = 2.5$ , so the problem becomes $42 \div (7 \div 2.5)$ . Just remember: parentheses first, then the outer division.

How can I check if my answer of 15 is correct?

+

Work backwards: if the answer is 15, then $42 \div 15$ should equal $7 \div 2\frac{1}{2}$ . Since $42 \div 15 = 2.8$ and $7 \div 2.5 = 2.8$ , our answer is correct!

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