Direct & Inverse Proportional (Time & Work)
Category : 8th Class
Direct & Inverse Proportional (Time & Work)
FUNDAMENTALS
Let 3 pens cost Rs. 9, then 6 pens will cost Rs. is
Clearly. More pens will cost more.
Again, if 2 women can do a piece of work in 7 hours, then 1 woman alone can do it in 14 hours.
Thus, less people at work, more will be the time taken to finish it.
Thus, change in one quantity brings a change in the other.
Variation: If two quantities depend upon each other in a way such that the change in one results in a corresponding change in the other, then the two quantities are said to be in variation. This variation may be direct (i.e. increase in one quantity leads to increase in other quantity) as illustrated in the example of "cost of pens" above. Variation may also be indirect (i.e. increase in one quantity leads to decrease in other quantity) as illustrated in the example of "work done by women" above.
There are many situations in our daily life where the variation in one quantity brings a variation in the other.
ILLUSTRATIONS:
Direct Proportionality: Two quantities x and y are said to be in direct proportion if increase / decrease in the value of one variable x, leads to increase / decrease in the value of y, in such a way that the ratio \[\frac{x}{y}\] remains constant.
Hence, if x and y are directly proportional, then\[\frac{x}{y}=k\], where k is a constant. As x takes the values \[({{x}_{1}}={{x}_{2}}={{x}_{3}})\]and y. takes the values. \[({{y}_{1}},{{y}_{2}},{{y}_{3}})\]then,
\[\frac{{{x}_{1}}}{{{y}_{1}}}=\frac{{{x}_{2}}}{{{y}_{2}}}=\frac{{{x}_{3}}}{{{y}_{3}}}=....=K\]
Examples (1), (3) and (4) given above are the cases of direct proportion.
Consider a gas in a closed cylinder such that its volume (V) is kept constant. If you increase the temperature (T) of the gas, then its Pressure (P), which can be measured through a manometer, also increases.
If V= constant; \[\frac{P}{T}=k;\]
(in the example above, just try to understand \[\frac{{{x}_{1}}}{{{y}_{1}}}=\frac{{{x}_{2}}}{{{y}_{2}}}=\frac{{{x}_{3}}}{{{y}_{3}}}=....=k\] where the analogy for x is Pressure (P) and the analogy for y is temperature (T).
Temperature \[({}^\circ C)\] |
Temperature * (K) |
Pressure (kPa) |
-150 |
173 |
36.0 |
-100 |
223 |
46.4 |
-50 |
273 |
56.7 |
0 |
323 |
67.1 |
50 |
373 |
77.5 |
100 |
423 |
88.0 |
*Temperature in Absolute scale (Value in Column II = Value in Column I+273).
Inverse Proportional:
As we saw in examples (2) & (5) on previous page, more is the number of workers, less is the time taken to complete the work and faster is the speed of train, lesser will be the time taken to cover a given distance. (Both of these are examples of Indirect Variation).These are the cases wherein two variables are related in such a way that increasing one, deceases the other and vice versa.
Inverse proportional: Two quantities x and y are said to be in inverse proportion if xy = k, where k is a constant.
Thus, \[{{x}_{1}}{{y}_{1}}={{x}_{2}}{{y}_{2}}={{x}_{3}}{{y}_{3}}=.......=k.\]
Consider the gas example again. This time, its temperature (T) is kept constant. Here, analogy for x is Pressure (P) and the analogy for y is Volume (V),
Sample Data from Pressure Volume Measurement
Pressure (torr ) |
Volume (ml) |
760 |
29.0 |
960 |
23.0 |
1160 |
19.0 |
1360 |
16.2 |
1500 |
14.7 |
150 |
13.3 |
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