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Chapter 6: Problem 11

For Exercises \(1-20,\) estimate. Then find the actual quotient. $$1 \div 0.008$$

Short Answer

Expert verified
Answer: The quotient is 125.

Step by step solution

01

Multiply both the dividend and the divisor by 1000

We will multiply both 1 and 0.008 by \(1000\), in this way: $$(1 * 1000) \div (0.008 * 1000)$$
02

Perform the multiplication

Multiplying the numbers: $$(1000) \div (8)$$
03

Divide 1000 by 8

Use long division or simple division method to find the quotient: $$1000 \div 8 = 125$$
04

Write the final answer

The actual quotient of $$1 \div 0.008$$ is 125.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Decimal Division
Decimal division is a process where you divide one decimal number by another. Although the general concept is similar to whole number division, decimals add a layer of complexity. To simplify decimal division, it's often useful to eliminate the decimals.
  • Eliminate Decimals by Multiplying: Multiply both numbers by the same power of 10 to make the divisor a whole number. For example, in the problem \(1 \div 0.008\), multiply both numbers by 1000. This changes the expression to \(1000 \div 8\), making it easier to handle.
  • Retain the Decimal Point Properly: Ensure that the decimal point placement stays correct, especially during final steps, to maintain the integrity of the answer.
Decimal division becomes straightforward once you convert it to a simpler form using basic arithmetic operations.
Quotient
The quotient is the result of a division problem. It's the number that shows how many times the divisor fits into the dividend.
  • Understanding the Quotient: In a simple example like \(8 \div 2 = 4\), 4 is the quotient, indicating how many 2s fit into 8.
  • In Decimal Division: The quotient might initially seem confusing due to decimal places but converting to whole numbers, as shown in the exercise, clarifies how we get from \(1 \div 0.008\) to a quotient of 125.
In problems involving conversion from decimal to whole numbers, always revert back to check the quotient's meaning and correctness.
Long Division
Long division is a method to divide numbers using a step-by-step approach. This tried-and-true method is great for dividing larger figures or decimals by hand.
  • Prepare for Long Division: Align the numbers properly, and begin dividing from the largest place value. In the exercise, after converting \(1 \div 0.008\) to \(1000 \div 8\), long division becomes applicable.
  • Step-by-Step Process: Break the dividend into manageable parts. For instance, 8 goes into 1000 a total of 125 times, leaving no remainder and making operation simpler.
  • Remain Organized: Keep your work neat to avoid mistakes in calculations or misplacing digits, especially when dealing with possible remainders.
Long division turns what seems a complex problem into a manageable task via its structured step-by-step methodology.
Math Problem Solving
Math problem-solving involves understanding and applying mathematical concepts to find a solution. For division problems, clarity and step-by-step methods are keys to success.
  • Understand the Problem: Before jumping into calculation, read and dissect the question. Know what is being asked - such as finding a quotient or simplifying decimals.
  • Apply the Right Techniques: Use strategies like simplifying numbers or applying long division principles, as shown in the practice exercise. Converting \(1 \div 0.008\) to \(1000 \div 8\) shows a clever method to handle divisions with tricky decimals.
  • Review Your Solution: After solving, always double-check your steps and answers to identify any possible errors. Confirmation ensures correctness and improves understanding.
Solving math problems efficiently involves a blend of comprehension, technique application, and review, turning challenges into achievable tasks.

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