Direct And Inverse Problems
In mathematics, two quantities are said to be proportional if they vary in such a
way that one of the quantities is a constant multiple of the other, or equivalently if they have a constant ratio. Proportion also refers to the equality of two ratios. The mathematical symbol '?' is used to indicate that two values are proportional. For example, A ? B. In this article we shall discuss about direct proportion problems and indirect proportion problems. (Source: wikipedia)
Direct Proportions Problems:
Example 1:
Rose ran 200 meters in 50 minutes. How long did she take to run 1 meter?
Solution:
Step 1: Consider the given problem If 200 then 55. If 1 then how many?
Step 2: The proportional relationship
200=>50
1=> '1/200' x50= 0.25
Answer: Rose took 0.25 minutes
Example 2:
A bike travels 350 miles in 5 hours. How far would it travel in 8 hours?
Solution:
Step 1: consider the given problem as If 5 then 350. If 8 then how many?
Step 2: The proportional relationship:
4=>350
8=>'8/5' x350
=>560
Answer: The bike traveled 560 miles.
Inversely proportions problems:
Example 1:
It takes 10 men 12 hours to repair a machine. How long will it take 14 men to do the job if they work at the same rate?
Step 1: Consider the given problem as If 10 then 12. If 14 then how many?
Step 2: The inverse proportional relationship:
10=>12
14=>'10/14' x12
=>'120/14'
=>8.5 hours
Answer: They will take repair a machine 8.5 hours.
Example 2:
If 12 men 14 hours to build a machine. How long will it take 14 men to do the job if they work at the same rate?
Step 1: Consider the given problem as If 12 then 14. If 14 then how many?
Step 2: The inverse proportional relationship:
12=>14
14=>'12/14' x14
=>'168/14'
=>12 hours
Answer: They will take build a machine 12 hours.
Direct and Inverse Practice Problems
Problem 1:
Jessica ran 300 meters in 40 minutes. How long did she take to run 1 meter?
Answer: Jessica took 0.13 minutes
Problem 2:
If 5 men 8 hours to construct a house. How long will it take 10 men to do the job if they work at the same rate?
Answer: They will take construct a house 4 hours.
An inverse problem is a general framework that is used to convert observed measurements into information about a physical object or system that we are interested in. For example, if we have measurements of the Earth's gravity field, then we might ask the question: "given the data that we have available, what can we say about the density distribution of the Earth in that area?" The solution to this problem (i.e. the density distribution that best matches the data) is useful because it generally tells us something about a physical parameter that we cannot directly observe. Thus, inverse problems are some of the most important and well-studied mathematical problems in science and mathematics. Inverse problems arise in many branches of science and mathematics, including computer vision, natural language processing, machine learning, statistics, statistical inference, geophysics, medical imaging (such as computed axial tomography and EEG/ERP), remote sensing, ocean acoustic tomography, nondestructive testing, astronomy, physics and many other fields.
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