Question

estimate the product by rounding the numbers so that the resulting arithmetic can easily be performed by hand or in your head. then use a calculator to perform the computation. how reasonable is your estimate when compared to the actual answer?
20.91×1.5
an estimate of the product is
(type an integer)

Explanation:

Step1: Round 20.91

Round 20.91 to a whole number. 20.91 is close to 21, so we round it to 21.

Step2: Estimate the product

Now, multiply the rounded number (21) by 1.5. So, $21\times1.5 = 31.5$. But we can also round 1.5 as is, or maybe round 20.91 to 20 for a different estimate, but a common rounding for 20.91 is to 21. Alternatively, if we round 20.91 to 20, then $20\times1.5 = 30$. Let's check the actual product: $20.91\times1.5 = 31.365$. If we rounded 20.91 to 21, the estimate is 31.5, which is very close to the actual 31.365. If we rounded to 20, estimate is 30, which is less close. So a reasonable estimate by rounding 20.91 to 21: $21\times1.5 = 31.5$. But maybe the problem expects rounding 20.91 to 20 (as a whole number approximation), so $20\times1.5 = 30$. Wait, 20.91 is closer to 21, but let's see the problem says "rounding the numbers so that the resulting arithmetic can easily be performed by hand or in your head". So 20.91 can be rounded to 21, 1.5 is already easy. So $21\times1.5 = 31.5$, and the actual is 31.365. So the estimate (if we use 21) is 31.5, which is very close. Alternatively, maybe round 20.91 to 20, then 20*1.5=30. But let's do the actual calculation first: 20.91*1.5. 20*1.5=30, 0.91*1.5=1.365, so total is 30+1.365=31.365. So if we round 20.91 to 21, estimate is 31.5, which is 0.135 more than actual. If we round to 20, estimate is 30, which is 1.365 less. So a better estimate is rounding 20.91 to 21. So the estimated product (using rounding 20.91 to 21) is 31.5, and the actual is 31.365. The question also asks "How reasonable is your estimate when compared to the actual answer?" The estimate (31.5) is very close to the actual (31.365), so it's a reasonable estimate. But the first part is to estimate the product by rounding, then find the actual product. Wait, the problem has two parts? Wait the image shows "Estimate the product by rounding the numbers... Then use a calculator to perform the computation. How reasonable is your estimate...? In estimate of the product is [ ] (Type an integer)". Wait maybe they want to round 20.91 to 20 (as an integer), so 20*1.5=30. Let's check: 20.91 rounded to the nearest integer is 21, but maybe the problem wants rounding to the nearest ten? No, 20.91 is between 20 and 21. Wait, maybe the problem has a typo, and the number is 20.91 or 2091? Wait the image shows "20.91x1.5"? Wait the user's image: "20.91x1.5"? Let me check again. The image text: "Estimate the product by rounding the numbers so that the resulting arithmetic can easily be performed by hand or in your head. Then use a calculator to perform the computation. How reasonable is your estimate when compared to the actual answer? 20.91x1.5 In estimate of the product is [ ] (Type an integer)". So first, estimate 20.91*1.5 by rounding. Let's round 20.91 to 21 (nearest whole number), then 21*1.5=31.5, but they want an integer, so maybe round 31.5 to 32? No, wait, maybe round 20.91 to 20 (as a whole number), then 20*1.5=30. So the estimate as an integer could be 30 or 31. Let's calculate the actual: 20.91*1.5. 20*1.5=30, 0.91*1.5=1.365, so total 31.365. So the actual is 31.365, which is close to 31. So if we round 20.91 to 21, 21*1.5=31.5, which is 31.5, and as an integer, maybe 31 or 32. But the problem says "Type an integer", so maybe the estimate is 30 (if we round 20.91 to 20) or 31 (if we round to 21). Let's do the estimation step by step:

Step1: Round 20.91

Round 20.91 to the nearest whole number: 21 (since 0.91 > 0.5).

Step2: Multiply by 1.5

Now, multiply 21 by 1.5. $21\times1.5 = 31.5$. Since we need an integer, we can round 31.5 to 32, but the actual product is 31.365, which is closer to 31. Alternatively, maybe the problem expects rounding 20.91 to 20 (as a simpler number), so 20*1.5=30. Let's check both:

- Estimate with 21: 31.5 (close to actual 31.365)
- Estimate with 20: 30 (less close)

So a more reasonable estimate is 31 (by rounding 31.5 to 31) or 32, but the actual is 31.365, so 31 is very close. Wait, maybe the problem is to round 20.91 to 20, so 20*1.5=30. Let's see the problem says "rounding the numbers", so maybe round 20.91 to 20 (a whole number) and 1.5 stays, so 20*1.5=30. Then the actual is 31.365, so the estimate (30) is a bit low, but easy to calculate. Alternatively, round 20.91 to 21, 21*1.5=31.5, which is 31.5, and as an integer, 31 or 32. But the problem says "Type an integer", so maybe 30 or 31. Let's go with the rounding to 21: 21*1.5=31.5, which is 31.5, and the actual is 31.365, so the estimate (as an integer) could be 31.

Wait, maybe I made a mistake. Let's recalculate the actual product: 20.91 * 1.5. Let's do 20.91 * 1.5:

20.91 * 1 = 20.91

20.91 * 0.5 = 10.455

Add them together: 20.91 + 10.455 = 31.365. So actual is 31.365.

Now, estimate by rounding 20.91 to 21: 21 * 1.5 = 31.5. So 31.5 is very close to 31.365. So the estimate (as an integer) would be 31 (since 31.5 is halfway, but 31.365 is closer to 31). So the estimate is 31.

Step3: Check the actual product

Actual product is 31.365, which is very close to the estimate of 31 (or 31.5). So the estimate (31) is reasonable because it's very close to the actual 31.365.

But maybe the problem expects rounding 20.91 to 20, so 20*1.5=30. Then the estimate is 30, and the actual is 31.365, so the estimate is a bit low, but easy to calculate.

Wait, the problem says "rounding the numbers so that the resulting arithmetic can easily be performed by hand or in your head". 20*1.5 is easy (30), 21*1.5 is also easy (31.5). So either is easy. But the actual is 31.365, so 31.5 (from 21*1.5) is closer. So the estimate as an integer could be 31 (rounding 31.5 to 31) or 32, but 31 is closer to the actual 31.365.

So, to summarize:

- Round 20.91 to 21 (easy to multiply by 1.5)
- 21 * 1.5 = 31.5
- The actual product is 31.365, so the estimate (31.5) is very close, and as an integer, 31 (or 32, but 31 is closer)

But the problem says "Type an integer", so maybe 31.

Wait, maybe the problem is to round 20.91 to 20, so 20*1.5=30. Let's check the problem again. The user's image: "Estimate the product by rounding the numbers so that the resulting arithmetic can easily be performed by hand or in your head. Then use a calculator to perform the computation. How reasonable is your estimate when compared to the actual answer? 20.91x1.5 In estimate of the product is [ ] (Type an integer)".

So, step by step:

1. Round 20.91: Let's round to the nearest whole number, which is 21.
2. Multiply by 1.5: 21 * 1.5 = 31.5.
3. Since we need an integer, we can take 31 (because 31.5 is closer to 31 than 32? Wait, 31.5 is exactly halfway, but the actual is 31.365, which is closer to 31. So 31 is a reasonable integer estimate.

Alternatively, if we round 20.91 to 20, 20*1.5=30, which is also an integer.

But which is more reasonable? The actual is 31.365, so 31 is closer to the actual than 30. So the estimate (as an integer) is 31.

Answer:

31 (or 30, but 31 is more reasonable. If we round 20.91 to 21, 21*1.5=31.5, which is 31.5, and as an integer, 31 is very close to the actual 31.365. So the answer is 31.)