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The teacher decides to select 5 students from the class of 30, using random numbers. She writes down the names of all 30 students and allocates to each name a two-digit number, such as 01, 02, 03, 04,..., 10, 12,..., 28, 29, 30. Then she generates 5 random numbers, using her calculator. The calculator can be preset to only include numbers from 01 to 30. If the calculator is not preset, it means that any numbers that turn up that are outside the range 01 to 30 are discarded. Suppose the random numbers are 07, 18, 02, 27, and 11. This means the students that have been allocated those numbers are the ones whose books are sent to the Principal.
References: Bias, Digit, Population, Random sample.
This is a term used in statistics. A random sample is part of a population and is chosen without bias from the whole population, and is hoped to have the same characteristics as the whole population. For the sample to be random, every member of the population must have the same chance of being selected. Since the random sample is expected to have the same characteristics as the population, it is used to make predictions about, the whole population. Jack is the manager of a jam factory of 1500 workers and wanted to know how they felt about overtime. He could ask all of them, which would take a lot of time and expense, or he could select a random sample of workers and question the sample. The information obtained from the sample would not be as reliable as from the whole population, but provided the sample included a good cross section of the work force, it would have, it is hoped, pretty much the same characteristics. To ensure a good cross section of ages Jack may wish randomly to select, say, 20 workers from five different age groups.
References: Bias, Population, Random Numbers, Sample.
References: Cartesian Coordinates, Correspondence.
Reference: Standard Deviation.
A rate compares one quantity with another quantity, which has different units, by dividing the quantities. Rate is not to be confused with ratio, which is the division of one quantity by another quantity which has the same units. When we divide the quantity of distance in kilometers by the quantity of time in hours, we obtain a rate of kilometers per hour, which is speed. We can say that speed is the rate at which distance is changing with respect to time. There are many examples of rates in everyday life, such as the running speed measured in meters per second of an athlete in a race, an individual’s heart beat measured in beats per second, and fuel consumption in kilometers per liter or miles per hour on a car trip.
Example 1. In a recent athletics meet David ran the 10,000 meters race in a time of 28 minutes, 55 seconds. Calculate his average speed for the race in meters per second.
Solution. The time must be expressed in seconds: 28 minutes, 55 seconds = 1735 seconds. Write
io ooo „ distance
Average speed = u. Speed =------------
i, m ^me
= 5.76 (to 2 dp)
David’s running rate is 5.76 meters per second.
Example 2. Which is the better buy, a 3-kg bag of potatoes for $1.55 or a 10-kg bag for $5.30?
Rate for the 3-kg bag = $0.52 per kg. Dividing $1.55 by 3 kg to 2 dp Rate for the 10-kg bag = $0.53 per kg. Dividing $5.30 by 10 kg
The 3-kg bag is the better buy, because it has a lower cost per kilogram.
Example 3. Darren uses 14 liters of gasoline for a trip of 189 km in his car. How far would he travel on 37 liters at the same rate?
Rate = ^ Dividing kilometers by liters
= 13.5 km per liter
For a trip of 37 liters he would travel
13.5 x 37 = 499.5 km Distance = rate x number of liters
Darren would travel 499.5 km on 37 liters of gasoline.
Example 4. Pat and her friends are doing a vacation job in an orchard picking apricots. Three girls can pick six baskets of fruit in 5 hours. How many baskets of fruit could four girls pick in 4 hours, working at the same average rate?
Solution. Using a table is a good way to keep track of the information in this difficult problem. Reducing the number in the Number of girls column to 1 while keeping the number in the Number of hours column fixed is a suitable way to begin solving the problem. Restrict calculations to two columns at a time.
Number of Girls Number of Baskets Number of Hours
3 6 5
3 = 3 = 1 6 = 3 = 2 5
1x4=4 2x4 = 8 5
4 8 = 5 = 1.6 5 = 5=1
4 1.6 x 4 = 6.4 1x4 = 4
Result: Four girls pick 6.4 baskets of apricots in 4 hours.
If a problem on rates involve a graph it should be remembered that the gradient of the graph is a rate.
Example 5. The graph in the figure is about Nathan’s bath time. It shows the volume of water in the bath after a certain time. V is the volume of water in the bath in liters and T is the time in minutes. The graph is made up of three straight lines. The first line shows the bath being filled. The second line represents Nathan being in the bath. The third line shows the bath emptying. Find the rate at which the bath is filling and the rate at which it is emptying.