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Chapter 6: Problem 1

For Exercises \(1-20,\) estimate. Then find the actual quotient. $$286.2 \div 12$$

Short Answer

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Question: Estimate the quotient and find the actual quotient for the division: 286.2 梅 12. Answer: The estimated quotient is 29, while the actual quotient is approximately 23.85.

Step by step solution

01

Estimating the Quotient

To estimate the quotient, we'll round the numbers to the nearest whole number or tens to make the calculation easier. In this case, rounding up to the nearest tens would be appropriate. $$ 286.2 \approx 290 \\ 12 \approx 10 $$ Now we divide the rounded numbers: $$ \frac{290}{10} = 29 $$ So, the estimated quotient is 29.
02

Finding the Actual Quotient

To find the actual quotient, we can use long division or a calculator. Using a calculator, we get: $$ \frac{286.2}{12} \approx 23.85 $$ So, the actual quotient is approximately 23.85.

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Key Concepts

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

Estimation
Estimation helps simplify complex calculations and gives you a quick idea of what the final result might be. In division, estimation is particularly useful for checking if your answer is reasonable. Here's how estimation works:
  • Look at the numbers in the problem and consider their value. For instance, break them down into closest 'easier' numbers you can mentally calculate.
  • Choose to round numbers to the nearest ten, hundred, or other simple figure. This is key in making division easier.
  • Perform the division using these rounded numbers. This gives you an estimated quotient.
For example, in the exercise provided, 286.2 was rounded to 290, and 12 was simplified to 10. By dividing 290 by 10, we got an estimated quotient of 29. Though this is not the precise answer, it is close enough to gauge the reasonableness of the actual result.
Long Division
Long division is a method for dividing two numbers to get an exact quotient. While it seems complex at first, the process is quite methodical, breaking down division into smaller, manageable steps:
  • Divide: Take the first number of the dividend (if possible) and divide it by the divisor.
  • Multiply: Multiply the divisor by the result from the divide step.
  • Subtract: Subtract that result from the number you're dividing into, bringing down the next number, if necessary.
  • Repeat: Continue the process with the new number until there are no more digits to bring down.
For dividing 286.2 by 12, long division helps us find the actual quotient, which is approximately 23.85. It involves handling decimals carefully, ensuring the placement of each number remains precise.
Rounding Numbers
Rounding numbers is a mathematical technique to simplify numbers to make them easier to calculate or understand. Here鈥檚 how you can round numbers effectively:
  • Identify the place to which you will round (units, tens, hundreds, etc.).
  • Look at the digit immediately to the right of that place. If it's 5 or higher, round up the digit you want to round to. If not, keep the digit unchanged.
  • After rounding, every digit to the right of your target place becomes zero, or, in the case of decimal numbers, they are removed.
In the estimation part of the exercise, 286.2 was rounded to 290, and 12 was rounded to 10. Rounding in division estimates helps simplify the process without compromising much on the accuracy of the task at hand.

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