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Seen on a wall is the following graffiti: “8 out of 5 Americans have trouble with fractions.” Comment: There cannot be more Americans having trouble with fractions than there are Americans! The statement is impossible.
In the following examples we will leam how to find a % of a quantity, and how to increase or decrease a quantity by a %.
326 PERCENTAGE ERROR
Example 6. Bill buys a motorbike for $350. He makes a few improvements to it, and sells it for a profit of 60%. How much profit does he make?
Solution. The profit is 60% of $350. Write
60% of 350 = x 350
The profit he makes is $210.
Example 7. After the Olympics were over David decided to reduce training by 45%. If he had been running 120 km per week, what would this reduce to?
Solution. To reduce his distance by 45% we multiply his distance by 55% to obtain the new distance he runs (100% — 45%):
120 x 55% = 120 x —
David now runs 66 km per week.
Note. When a quantity is increased by a certain %, and then decreased by the same %, we do not obtain the original quantity. The following example demonstrates this fact:
Example 8. Pat weighs 60 kg. Over Christmas her weight increases by 2%. After a diet her weight then decreases by 2%. Does her weight return to 60 kg?
Solution. After Christmas her weight is
60 x 1.02 = 61.2 kg 102% = 1.02 After the diet her weight is
61.2 x 0.98 = 59.976 kg 98% = 0.98 Her weight does not return to 60 kg.
This refers to an error written as a % of the correct amount. The formula is
% error =-----------------x 100
Error is the difference between the correct and wrong amounts. The following example brings out the meaning of percentage error.
Example. Tom is having new carpets in his lounge. He asks Pat to measure the length of the lounge, which she measures to be 16.34 meters. On the plans of the house the room is marked as shorter than her measurement, so Tom checks the tape and finds that the first meter has been cut off. The actual length of the lounge is therefore 15.34 meters. Find the % error in Pat’s original measurement.
Error = 16.34- 15.34 = 1 meter
= —-— x 100 Using the formula for % error
= 6.52 (to 2 dp) Using a calculator
The % error in Pat’s original measurement is 6.52%.
References: Canceling, Conversion Factor, Cost Price, Decimal, Error, Fractions, Percentage.
Reference: Cost Price.
Percentiles divide a set of data into 100 equal parts, and are often used to give information on how well a person has done in relation to other people in a competition or in an examination. Each of the 100 parts into which the data is divided is called a percentile.
Suppose Luke takes an examination and there are 200 students who take the same examination. Their results are recorded in a cumulative frequency graph, as shown in the figure. Suppose we wish to find out the examination mark of the 90th percentile. There are 200 students in the group, and 90% of 200 students = 180. We can read from the graph that the 180th student from the bottom of the ranking scored 75 in the examination. This mark of 75% is the 90th percentile. It means that 90% of the students, which is 180 students, scored less than 75% in the examination. In addition, it means that 10% of the students scored above a mark of 75%.
Example. In Luke’s examination, what examination mark is the 40th percentile? Solution. Write
40% of 200 students = 0.4 x 200 40% = 0.4
We count 80 students from the bottom of the list and obtain the mark that corresponds to the 80th person. This mark in the examination is 50%.
328 PERFECT NUMBER
The 40th percentile is an examination mark of 50%. The 40th percentile means that 40% of the students scored below that examination mark and 60% of them scored above it.
The median mark in the examination is the 50 percentile, which in this example is the mark of the 100th student. From the graph the median examination mark is about 53%. In this example 100 students scored below the median and 100 scored above it. The upper quartile is the 75th percentile, and 75% of the students, which is 150, scored below this mark. The lower quartile is the 25th percentile, and 25% of the students, which is 50, scored below this mark. From the graph, the upper quartile is about 63% and the lower quartile is about 42%.
A decile is a multiple of 10 percentiles. For example, the 20th percentile is the 2nd decile, and the 90th percentile is the 9th decile.
References: Box and Whisker Graph, Cumulative Frequency Graph, Interquartile Range.
A perfect number is a whole number that is the sum of all its factors, not including itself. For example, the first perfect number is 6. All the factors of 6 are 1,2, 3, and 6:
6 = 1 + 2 + 3.
So 6 is a perfect number, because it is the sum of all its factors except itself.
Example. Is 8 a perfect number?