Question

nori measures the length of her belt. the tag says that the belt is 62 centimeters long, but she measures that the length is actually 61.07 centimeters. what is the percent of error in the belt length according to the tag, to the nearest hundredth? %

Explanation:

Step1: Find the absolute error

The absolute error is the absolute difference between the measured value and the actual value. The tag value (measured) is 62 cm, and the actual value is 61.07 cm. So, absolute error = |62 - 61.07| = 0.93 cm.

Step2: Calculate percent error

The formula for percent error is $\frac{\text{Absolute Error}}{\text{Measured Value}} \times 100\%$. Substituting the values, we get $\frac{0.93}{62} \times 100\%$. First, calculate $\frac{0.93}{62} \approx 0.015$, then multiply by 100% to get $0.015 \times 100\% = 1.5\%$? Wait, no, wait: 0.93 divided by 62: 0.93 ÷ 62 = 0.015, then 0.015 × 100 = 1.5? Wait, no, let's do it more accurately. 0.93 / 62 = 0.015, so 1.5%? Wait, no, wait, 62 - 61.07 is 0.93. Then (0.93 / 62) * 100. Let's compute 0.93 ÷ 62: 62 × 0.01 = 0.62, 0.93 - 0.62 = 0.31, 0.31 / 62 = 0.005, so total is 0.01 + 0.005 = 0.015, so 1.5%? Wait, but let's check with a calculator: 0.93 ÷ 62 = 0.015, so 1.5%? Wait, but the problem says "to the nearest hundredth of a percent". Wait, maybe I made a mistake. Wait, 0.93 divided by 62: 0.93 ÷ 62 = 0.015, which is 1.5%, but as a decimal, 0.015 is 1.50% when rounded to the nearest hundredth? Wait, no, 0.015 is 1.50%? Wait, no, 0.015 × 100 = 1.5, so 1.50%? Wait, maybe I messed up the measured and actual. Wait, percent error is usually (|measured - actual| / actual) × 100? Wait, no, different definitions: sometimes it's (|error| / measured) or (|error| / actual). Wait, the problem says "percent of error in the belt length according to the tag", so the tag is the measured value, so we use measured as the reference. Wait, let's check the formula. The formula for percent error is:

Percent Error = $\frac{|\text{Measured Value} - \text{Actual Value}|}{\text{Measured Value}} \times 100\%$

So Measured = 62, Actual = 61.07.

So |62 - 61.07| = 0.93.

Then (0.93 / 62) × 100 = (0.93 × 100) / 62 = 93 / 62 ≈ 1.50 (rounded to the nearest hundredth). Wait, 93 ÷ 62: 62 × 1 = 62, 93 - 62 = 31, 31/62 = 0.5, so 1.5. So 1.50%? Wait, 1.5 is 1.50 when rounded to the nearest hundredth. Wait, but let's do the division more accurately: 0.93 ÷ 62 = 0.015, so 0.015 × 100 = 1.5, so 1.50% (since hundredth place is two decimal places). Wait, 1.5 is 1.50 when rounded to the nearest hundredth. So the percent error is approximately 1.50%.

Wait, but let's check again. Maybe the formula is (|actual - measured| / actual) × 100? Let's try that. |61.07 - 62| = 0.93. Then (0.93 / 61.07) × 100 ≈ (0.93 / 61.07) ≈ 0.01523, ×100 ≈ 1.523%, which is approximately 1.52%. Wait, now I'm confused. Which formula is correct?

The percent error formula can be (|experimental - accepted| / accepted) × 100, where accepted is the actual value. Wait, in this case, the tag is the experimental (measured) value, and the actual is 61.07. So if we take accepted as actual, then percent error is (|62 - 61.07| / 61.07) × 100. Let's calculate that: 0.93 / 61.07 ≈ 0.01523, ×100 ≈ 1.523%, which is approximately 1.52% when rounded to the nearest hundredth.

Wait, the problem says "percent of error in the belt length according to the tag". So "according to the tag" means the tag's value is the one we're comparing against? Wait, maybe the formula is (|tag - actual| / tag) × 100. So tag is 62, actual is 61.07. So (0.93 / 62) × 100 ≈ 1.50%. But which is correct?

Wait, let's check the definition. Percent error is typically calculated as (|measured - true| / true) × 100, where true is the actual value. So in this case, true is 61.07, measured is 62. So (|62 - 61.07| / 61.07) × 100 = (0.93 / 61.07) × 100 ≈ 1.523%, which is 1.52% when rounded to the nearest hundredth.

Ah, that makes more sense. Because the percent error is the error relative to the actual (true) value. So let's recalculate:

0.93 ÷ 61.07 ≈ 0.01523, multiply by 100: 1.523%, which rounds to 1.52% (nearest hundredth).

Wait, so where did I go wrong earlier? The formula is (|measured - actual| / actual) × 100, not divided by measured. Because the error is in the actual length, so we compare the error to the actual length. Wait, no, sometimes it's divided by the measured value. It depends on the context. Let's check the problem statement: "percent of error in the belt length according to the tag". So "according to the tag" means the tag's length is the stated length, and we're finding the error in that stated length. So the tag says 62, actual is 61.07. So the error is (62 - 61.07) = 0.93, and we want the percent of that error relative to the tag's length? Or relative to the actual length?

The problem says "percent of error in the belt length according to the tag". So "according to the tag" implies that the tag's length is the reference. So percent error = (error / tag length) × 100. So error is |62 - 61.07| = 0.93, tag length is 62. So (0.93 / 62) × 100 ≈ 1.50%. But when I calculate 0.93 / 62, that's 0.015, times 100 is 1.5, so 1.50% (to the nearest hundredth).

But now I'm confused because different sources define percent error as (|measured - actual| / actual) × 100. Let's check with an example. Suppose a ruler says 10 cm, but actual is 9 cm. Percent error: (10 - 9)/9 × 100 ≈ 11.11%, or (10 - 9)/10 × 100 = 10%. Which is correct?

The definition of percent error is:

Percent Error = $\frac{|\text{Experimental Value} - \text{Accepted Value}|}{\text{Accepted Value}} \times 100\%$

Where the accepted value is the actual (true) value. So in this case, experimental value is 62 (tag), accepted is 61.07 (actual). So:

Percent Error = $\frac{|62 - 61.07|}{61.07} \times 100\% = \frac{0.93}{61.07} \times 100\% \approx 1.52\%$

Yes, that's the correct formula. So the percent error is approximately 1.52% when rounded to the nearest hundredth.

Let's compute 0.93 ÷ 61.07:

61.07 × 0.015 = 0.91605

0.93 - 0.91605 = 0.01395

0.01395 / 61.07 ≈ 0.000228

So total is 0.015 + 0.000228 ≈ 0.015228, ×100 ≈ 1.5228%, which rounds to 1.52% (nearest hundredth).

So the correct percent error is approximately 1.52%.

Answer:

1.52