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Chapter 1: Problem 122

Divide. Then check by estimating the quotient. $$2 1 9 \div { 3 3 2 , 0 0 4 }$$

Short Answer

Expert verified
The division of 219 by 332,004 yields an approximate quotient of 0.000660.

Step by step solution

01

Perform Division

Dividing 219 by 332,004, using a calculator or by hand, yields a quotient of approximately \(0.00065990\). This result is much less than 1, which makes sense considering that dividing a smaller number by a larger one should result in a fractional number.
02

Estimate the Quotient

As the second part of the question requires you to estimate the quotient, it's not necessary to use the precise decimal you got. To make estimation easier, you can round the quotient to six decimal places, resulting in \(0.000660\).
03

Check The Estimation

To check the estimation, multiply your estimate back by the divisor (332,004). So, \(0.000660 * 332,004 = 219.12264\) which is reasonably close to 219. Small differences are acceptable as this is merely an estimation.

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Key Concepts

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

Estimating Quotients
When faced with division tasks, estimating the quotient can be a helpful strategy. Estimation involves finding an approximate result that is close enough to the actual answer. This is especially useful when dealing with large or awkward numbers. By rounding the numbers involved, the calculation becomes more manageable. For example, in our exercise, the exact quotient of dividing 219 by 332,004 is approximately \(0.00065990\).
  • If you round these numbers, perhaps to the nearest whole number or significant figure, you might estimate the quotient as \(0.000660\).
  • This quick estimation helps verify the actual division result and ensures that it makes sense.
By employing estimation, you can easily double-check your division work, saving time and minimizing errors.
Division Verification
Verifying your division work can be a reassuring step to ensure accuracy. To verify a division, you can multiply the quotient by the divisor鈥攊f the result is close to the original dividend, your division is likely correct. In our example, we performed the division and found the quotient to be around \(0.000660\).
  • By multiplying \(0.000660\) by \(332,004\), we get a product of approximately \(219.12264\).
  • This product is very close to our original dividend of \(219\), confirming that the division was precise enough for practical purposes.
Even though there might be small differences, they are generally acceptable, as perfect precision isn't always necessary in estimation exercises.
Fractional Numbers
When division results in a quotient that is less than one, the quotient is described as a fractional number. Fractional numbers are often encountered in mathematics, especially when dividing smaller numbers by larger ones. In our exercise, dividing 219 by 332,004 resulted in a quotient of approximately \(0.00065990\), indicating a fractional number as the result.
  • Such quotients signify that the divisor is much larger than the dividend, thus the result is a tiny portion of a whole number.
  • Understanding fractional numbers is essential as they appear frequently in real-world applications, such as measuring small quantities or financial transactions.
Grasping the concept of fractional numbers allows students to interpret division outcomes more effectively, especially when the results aren't whole numbers.

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