- Calculate the exact value: \(326 - 278 = 48\).
- Estimate by rounding to the nearest ten: \(330 - 280 = 50\), resulting in an error of \(|48 - 50| = 2\).
- Estimate by rounding to the nearest hundred: \(300 - 300 = 0\), resulting in an error of \(|48 - 0| = 48\).
- Rounding to the nearest ten is clearly more accurate, as \(\boxed{\text{nearest ten}}\) provides an estimate significantly closer to the actual difference.
Explanation
- Calculation of the Exact Value
To estimate the value of the subtraction \(326 - 278\), we first calculate the exact difference. Subtracting \(278\) from \(326\) gives \(326 - 278 = 48\). Our goal is to determine whether rounding both numbers to the nearest ten or to the nearest hundred provides a more accurate estimate of this true value.
- Rounding to the Nearest Ten
First, we round to the nearest ten. The number \(326\) rounds to \(330\) because the digit in the units place is \(6\), which is \(5\) or greater. The number \(278\) rounds to \(280\) because the digit in the units place is \(8\), which is also \(5\) or greater. The estimated difference is \(330 - 280 = 50\). The absolute error of this estimate is \(|48 - 50| = 2\).
- Rounding to the Nearest Hundred
Next, we round to the nearest hundred. The number \(326\) rounds to \(300\) because the digit in the tens place is \(2\), which is less than \(5\). The number \(278\) rounds to \(300\) because the digit in the tens place is \(7\), which is \(5\) or greater. The estimated difference is \(300 - 300 = 0\). The absolute error of this estimate is \(|48 - 0| = 48\).
- Conclusion and Evaluation
Comparing the two methods, rounding to the nearest ten yields an estimate of \(50\), which is very close to the actual value of \(48\). Rounding to the nearest hundred yields an estimate of \(0\), which is quite far from the actual value. Therefore, rounding to the nearest ten provides a much more accurate estimate for the subtraction.
Examples
Imagine you are shopping at a store and have a budget. You want to estimate the total cost of two items priced at \(\$326\) and \(\$278\) to ensure you have enough money. If you round the prices to the nearest hundred, you might think you need \(\$0\) to cover the difference or simply miscalculate your total budget significantly. By rounding to the nearest ten, you get a much more reliable estimate of \(\$50\), helping you make better financial decisions while at the store.