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2 8 - x -
The two fractions — and — are equivalent 24 x
References: Multiple, Prime Number.
In mathematics error does not necessarily mean that a mistake has been made, but that we may be talking about a quantity that is not expressed exactly. This kind of error may arise due to an estimation or because it is not possible to make measurements with 100% accuracy. Suppose we look at three types of quantities:
1. Quantities that are counted. Suppose the number of students in a classroom is known to be 30, and a student counted them and got the answer 32. The difference between the correct answer and the incorrect answer is the error. In this case the error is 2, and in this case is due to a mistake being made in the count.
Sometimes errors are made that are not due to mistakes, but due to errors in estimating and measuring, as identified below.
2. Quantities that are estimated. Estimate how many times the letter E appears on the next line, without counting them all:
Suppose a person estimated that the letter E appears 60 times, which is quite a good estimate, because by counting them you will find it appears 62 times. The error is 62 — 60 = 2. This is not a mistake, but an error due to estimating.
3. Quantities that are measured. For example, we may be attempting to find the length of a table in the dining room. It can be measured, perhaps to the nearest centimeter, or to the nearest millimeter, but the exact length of the table cannot be measured. This is because there is no instrument or device accurate enough to measure it exactly. The error here is the difference between the true length, if it is known, and the measured length. This is not a mistake, but an error in measuring due to the inaccuracies of the measuring instrument.
This is an opportune point to discuss limits of accuracy. Suppose we are asked to measure the length of a table to the nearest centimeter, and measured it correctly to be 124 cm. Figure a shows an enlarged view of the measuring tape that was used. The actual length of the table, measured to the nearest centimeter, lies somewhere in the interval of 123.5-124.5 cm. The greatest length of the table could not be
124.5 cm, because then it would be rounded up to 125 cm to the nearest centimeter. But any measurement just less than 124.5 cm is OK. The measurement of 124.5 cm is excluded with a hole at the upper end of the range, as shown in figure a. The least length of the table could be 123.5 cm, because that would be rounded up to 124 cm to the nearest centimeter. This measurement is included with a filled hole at the lower end of the interval, as in figure a.
123.0 123.5 124.0 124.5 125.0 cm
These two values 123.5 and 124.5 cm are called the limits of accuracy for the length of the table when it has been measured to be 124 cm, to the nearest centimeter.
This means we can assume that the actual length L of the table is between these two extremes. The limits of accuracy of the length L can be expressed as
123.5 < L < 124.5
Example 1. The number N of people in a crowd at a sports event is recorded as
23,000 to the nearest thousand. What are the limits of accuracy?
22,000 22,500 23,000 23,500 24,000
Solution. The size of the crowd is positioned on a number line as in figure b. The limits of accuracy are
22,500 < N < 23,500
Example 2. The length L of a room is measured as 5.67 meters to two decimal places. What are the limits of accuracy?
5.66 5.665 5.67 5.675 5.68
Solution. The length of the room is positioned on a number line as in figure c. The limits of accuracy are
5.665 <L< 5.675 References: Accuracy, Estimation, Measurement, Percentage Error, Rounding.
When we estimate we calculate an approximate value for the size of a certain quantity. There are various reasons why we estimate the size of a quantity instead of attempting to find its exact value. Some examples are given here.
Example 1. If we are writing a report on an athletics meet, we only need an approximate value of the size of the crowd, so it is quicker to estimate how many people are there rather than count everyone. Generally, people would not be interested in the exact size of the crowd. This is an example where estimating saves time.
A suitable method of estimating the number of spectators is to count 50 people, and then count roughly how many groups that size there are around the sports arena. If you count 20 such groups, then there are approximately 50 x 20 = 1000 spectators.
Example 2. Suppose you are in business manufacturing Chocko chocolates and wonder how many people would buy a new flavor, “crunchy” Chocko bar. Rather than ask everyone in the country, you decide to select a sample of 1000 people, ask them what they think, and then use this result to estimate how many people in the country would buy your new Chocko bar. This is an example where estimating can save the expense, and also the time, of asking everyone in the country.