Using rates to make predictions :
When there are two quantities, unit rate describes how many units of the first type of quantity corresponds to one unit of the second type of quantity.
In many real world problems, we can use unit rates to make predictions.
Example 1 :
Look at the picture below.
Janet drives from Clarkson to Humbolt in 2 hours. Suppose Janet drives for 10 hours. If she maintains the same driving rate, can she drive more than 600 miles? Justify your answer.
A double number line is useful, because the regular intervals represent equivalent rates that compare different quantities. The one shown compares the number of miles driven to the time driven for different amounts of time.
She covers 112 miles of distance in 2 hours. Since she maintains the same driving rate, every 2 hours, the distance covered will be increased by 112 miles.
In 8 hours she would have covered 448 miles of distance. After 2 hours, that is in 10 hours, the distance covered will be increased by 112 miles.
So, the distance covered in 10 hours is 560 miles.
It is shown in the picture given below.
From the above working, it is clear that she can not cover more than 600 miles in 10 hours, if she maintains the same driving rate.
Example 2 :
In 15 minutes, Lena can finish 2 math problems. At that rate, how many math problems can she finish in 75 minutes ? Use a double number line to find the answer.
The one shown compares the number of minutes to the number of math problems.
She finishes 2 math problems in 15 minutes. At this rate, every 15 minutes, the number of math problems finished will be increased by 2.
Let us mark this situation in the above double number line.
Than, we have
From the double number line, it is clear that she can finish 10 math problems in 75 minutes.
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