Question

1. open ended two 3-digit numbers are added. give an example where you would not want to round both 3-digit numbers to the nearest hundred to get the best estimate.

1. marin evaluated $-0.238 \cdot \frac{2}{10} - (1,003 \cdot 0.0062)$ and got $-6.65$ rounded to the nearest hundredth.

a. estimate the value of the expression.

b. construct arguments is the estimate an overestimate or underestimate? explain.

Explanation:

Question 9

Step1: Understand the Problem

We need two 3 - digit numbers where rounding both to the nearest hundred gives a poor estimate of their sum. Let's consider numbers where the tens and units digits have a significant impact on the sum. For example, take 149 and 151.

Step2: Analyze Rounding and Sum

• Rounding 149 to the nearest hundred: 149 is closer to 100 (since 49 < 50).
• Rounding 151 to the nearest hundred: 151 is closer to 200 (since 51>50).
• The sum of the rounded numbers: 100 + 200 = 300.
• The actual sum: 149+151 = 300. Wait, that's not a good example. Let's try 140 and 160.
• Rounding 140 to nearest hundred: 100, rounding 160 to nearest hundred: 200. Sum of rounded: 300. Actual sum: 140 + 160 = 300. Hmm. Let's take 125 and 174.
• Rounding 125 to nearest hundred: 100 (because 25 < 50? Wait, 125 is exactly halfway between 100 and 200, but the rule is usually to round up for 50 or more. Wait, 125: the tens digit is 2, so 125 rounded to nearest hundred is 100. 174: tens digit is 7, so rounded to 200. Sum of rounded: 100 + 200 = 300. Actual sum: 125+174 = 299. The difference between the estimated sum (300) and actual sum (299) is 1, which is small. Let's try 149 and 249.
• Rounding 149 to nearest hundred: 100, rounding 249 to nearest hundred: 200. Sum of rounded: 300. Actual sum: 149 + 249 = 398. The difference is 98, which is large. Wait, no, 149 + 249: 149 is ~100, 249 is ~200, sum ~300, actual 398. But maybe a better example: 199 and 201.
• Rounding 199 to nearest hundred: 200, rounding 201 to nearest hundred: 200. Sum of rounded: 400. Actual sum: 199+201 = 400. Not good. Let's take 130 and 170.
• Rounding 130: 100, 170: 200. Sum of rounded: 300. Actual sum: 130 + 170 = 300. Still the same. Wait, maybe 110 and 190.
• Rounding 110: 100, 190: 200. Sum of rounded: 300. Actual sum: 110+190 = 300. Hmm. Let's think about numbers where the sum of the non - rounded numbers is much closer to the sum of the numbers rounded to the nearest ten. For example, 145 and 155.
• Rounding to nearest hundred: 100 and 200, sum 300. Rounding to nearest ten: 150 and 160, sum 310. Actual sum: 145 + 155 = 300. Wait, actual sum is 300. Rounding to nearest ten gives 310, which is a worse estimate. Wait, maybe my approach is wrong. Let's recall that when the two numbers are close to a number that is halfway between two hundreds, but in opposite directions. For example, 149 and 151: as before, but their sum is 300, and rounded to hundreds is 100 and 200, sum 300. Wait, maybe the key is that when the two numbers are such that one is just below a hundred mark and the other is just above, but their sum is close to a multiple of 100. Wait, perhaps a better example: 139 and 161.
• Rounding 139 to nearest hundred: 100, 161 to nearest hundred: 200. Sum of rounded: 300.
• Actual sum: 139+161 = 300. No. Wait, let's take 249 and 349.
• Rounding 249 to nearest hundred: 200, 349 to nearest hundred: 300. Sum of rounded: 500.
• Actual sum: 249+349 = 598. The difference between 500 and 598 is 98, which is significant. So in this case, rounding to the nearest hundred (200 and 300) gives a sum of 500, but the actual sum is 598, so rounding to the nearest hundred is a poor estimate.

Step1: Simplify the Expression for Estimation

We have the expression \(-0.238\cdot\frac{2}{10}-(1003\cdot0.0062)\).

• First, simplify \(\frac{2}{10}=0.2\).
• For the first term: \(-0.238\cdot0.2\). We can estimate \(-0.238\approx - 0.24\), so \(-0.24\cdot0.2=-0.048\approx - 0.05\).
• For the second term: \(1003\cdot0.0062\). We can estimate \(1003\approx1000\), so \(1000\cdot0.0062 = 6.2\).
• Now, the expression becomes approximately \(-0.05-6.2=-6.25\).

Step2: Refine the Estimation (Optional)

• A more accurate estimate for the first term: \(-0.238\cdot0.2=-0.0476\approx - 0.05\) (since we are estimating).
• A more accurate estimate for the second term: \(1003\cdot0.0062=(1000 + 3)\cdot0.0062=1000\cdot0.0062+3\cdot0.0062 = 6.2+0.0186 = 6.2186\approx6.22\).
• Then the expression is approximately \(-0.0476-6.2186=-6.2662\approx - 6.27\). But a simpler estimate is using \(1000\) for \(1003\) and \(-0.2\) for \(-0.238\): \(-0.2\cdot0.2-(1000\cdot0.0062)=-0.04 - 6.2=-6.24\).

Step1: Recall the Original Expression and Estimation

The original expression is \(E=-0.238\cdot\frac{2}{10}-(1003\cdot0.0062)\) and our estimate (from part A) was done by approximating \(-0.238\) as a smaller magnitude (more positive) number (e.g., \(-0.2\) or \(-0.24\) instead of \(-0.238\)) and \(1003\) as \(1000\) (a smaller number).

Step2: Analyze Overestimation/Underestimation

• Let's analyze the first term: \(-0.238\cdot\frac{2}{10}=-0.0476\). If we estimate the first term as \(-0.04\) (by using \(-0.2\) instead of \(-0.238\)), we are making the first term larger (less negative) because \(-0.04>-0.0476\).
• For the second term: \(1003\cdot0.0062 = 6.2186\). If we estimate it as \(6.2\) (by using \(1000\) instead of \(1003\)), we are making the second term smaller.
• Now, the expression is of the form \(a - b\) where \(a=-0.238\cdot\frac{2}{10}\) and \(b = 1003\cdot0.0062\). Our estimate was \(\text{Estimate}=-0.04 - 6.2=-6.24\) (using \(-0.2\) and \(1000\)). The actual value of \(a=-0.0476\) (more negative than \(-0.04\)) and \(b = 6.2186\) (larger than \(6.2\)). So the actual value is \(a - b=-0.0476-6.2186=-6.2662\). Our estimate \(-6.24\) is larger (less negative) than the actual value \(-6.2662\). So the estimate is an overestimate.
• Another way: When we round \(-0.238\) to \(-0.2\) (a less negative number), the first term \(-0.2\times0.2=-0.04\) is greater than \(-0.238\times0.2=-0.0476\). When we round \(1003\) to \(1000\), the second term \(1000\times0.0062 = 6.2\) is less than \(1003\times0.0062 = 6.2186\). So in the expression \(-0.238\times0.2-(1003\times0.0062)\), we have replaced the first part with a greater number (less negative) and the second part with a smaller number. So the overall estimate \((\text{greater first part}-\text{smaller second part})\) is greater (less negative) than the actual value. Hence, the estimate is an overestimate.

Answer:

An example is 249 and 349. When we round 249 to the nearest hundred, we get 200, and when we round 349 to the nearest hundred, we get 300. The sum of the rounded numbers is \(200 + 300=300\), but the actual sum is \(249 + 349 = 598\), and the difference between the estimated sum (300) and the actual sum (598) is large, so rounding both to the nearest hundred does not give a good estimate of their sum. (Other valid examples are also possible)

Question 10A