4.3.5: Percent Error
Lesson
Let's use percentages to describe other situations that involve error.
Exercise \(\PageIndex{1}\): Number Talk: Estimating a Percentage of a Number
Estimate.
25% of 15.8
9% of 38
1.2% of 127
0.53% of 6
0.06% of 202
Exercise \(\PageIndex{2}\): Plants, Bicycles, and Crowds
- Instructions to care for a plant say to water it with \(\frac{3}{4}\) cup of water every day. The plant has been getting 25% too much water. How much water has the plant been getting?
- The pressure on a bicycle tire is 63 psi. This is 5% higher than what the manual says is the correct pressure. What is the correct pressure?
- The crowd at a sporting event is estimated to be 3,000 people. The exact attendance is 2,486 people. What is the percent error ?
Are you ready for more?
A micrometer is an instrument that can measure lengths to the nearest micron (a micron is a millionth of a meter). Would this instrument be useful for measuring any of the following things? If so, what would the largest percent error be?
- The thickness of an eyelash, which is typically about 0.1 millimeters.
- The diameter of a red blood cell, which is typically about 8 microns.
- The diameter of a hydrogen atom, which is about 100 picometers (a picometer is a trillionth of a meter).
Exercise \(\PageIndex{3}\): Measuring in the Heat
A metal measuring tape expands when the temperature goes above \(50^{\circ}F\). For every degree Fahrenheit above 50, its length increases by 0.00064%.
- The temperature is 100 degrees Fahrenheit. How much longer is a 30-foot measuring tape than its correct length?
- What is the percent error?
Summary
Percent error can be used to describe any situation where there is a correct value and an incorrect value, and we want to describe the relative difference between them. For example, if a milk carton is supposed to contain 16 fluid ounces and it only contains 15 fluid ounces:
- the measurement error is 1 oz, and
- the percent error is 6.25% because \(1\div 16=0.0625\).
We can also use percent error when talking about estimates. For example, a teacher estimates there are about 600 students at their school. If there are actually 625 students, then the percent error for this estimate was 4%, because \(625-600=25\) and \(25\div 625=0.04\).
Glossary Entries
Definition: Measurement Error
Measurement error is the positive difference between a measured amount and the actual amount.
For example, Diego measures a line segment and gets 5.3 cm. The actual length of the segment is really 5.32 cm. The measurement error is 0.02 cm, because \(5.32-5.3=0.02\).
Definition: Percent Error
Percent error is a way to describe error, expressed as a percentage of the actual amount.
For example, a box is supposed to have 150 folders in it. Clare counts only 147 folders in the box. This is an error of 3 folders. The percent error is 2%, because 3 is 2% of 150.
Practice
Exercise \(\PageIndex{4}\)
A student estimated that it would take 3 hours to write a book report, but it actually took her 5 hours. What is the percent error for her estimate?
Exercise \(\PageIndex{5}\)
A radar gun measured the speed of a baseball at 103 miles per hour. If the baseball was actually going 102.8 miles per hour, what was the percent error in this measurement?
Exercise \(\PageIndex{6}\)
It took 48 minutes to drive downtown. An app estimated it would be less than that. If the error was 20%, what was the app’s estimate?
Exercise \(\PageIndex{7}\)
A farmer estimated that there were 25 gallons of water left in a tank. If this is an underestimate by 16%, how much water was actually in the tank?
Exercise \(\PageIndex{8}\)
For each story, write an equation that describes the relationship between the two quantities.
- Diego collected \(x\) kg of recycling. Lin collected \(\frac{2}{5}\) more than that.
- Lin biked \(x\) km. Diego biked \(\frac{3}{10}\) less than that.
- Diego read for \(x\) minutes. Lin read \(\frac{4}{7}\) of that.
(From Unit 4.1.4)
Exercise \(\PageIndex{9}\)
For each diagram, decide if \(y\) is an increase or a decrease of \(x\). Then determine the percentage.
(From Unit 4.3.3)
Exercise \(\PageIndex{10}\)
Lin is making a window covering for a window that has the shape of a half circle on top of a square of side length 3 feet. How much fabric does she need?
(From Unit 3.3.1)