MAGNITUDES INVERSAMENTE PROPORCIONALES-EJERCICIOS RESUELTOS

Inversely Proportional Magnitudes Explained

Definition and Examples of Inversely Proportional Magnitudes

• Two magnitudes are inversely proportional when an increase in one results in a decrease in the other, and vice versa. An example is given with two painters taking 12 hours to paint a house; if four painters work together, they will take less time.

• To analyze this relationship, creating a table helps organize information. The problem illustrates that more painters lead to fewer hours needed for the task.

• The relationship confirms that as the number of painters increases, the time taken decreases, establishing that these magnitudes are indeed inversely proportional.

• If one magnitude is halved (e.g., number of days), the other must be multiplied by two (e.g., time taken). This leads to calculations showing that one painter would take 24 hours alone.

• For four painters, dividing the original time by two gives 6 hours. This demonstrates how inverse relationships can yield exact numerical answers through consistent operations.

Finding Constant of Proportionality

• The constant of proportionality can be found by multiplying corresponding values from both magnitudes. For instance, multiplying pairs like 1 and 24 or 2 and 12 consistently yields a constant value of 24.

• A practical application shows that three painters would take approximately eight hours to complete the same job based on this constant.

Application Example: Travel Time Calculation

• A new scenario involves traveling from Ibagué to Bogotá at a speed of 60 km/h over six hours. A table is used again to illustrate this relationship clearly.

• It’s established that higher speeds result in shorter travel times—confirming another case of inverse proportionality where increased speed reduces duration.

• By doubling the speed from 60 km/h to 120 km/h, travel time halves; conversely, reducing speed results in longer travel times (e.g., down to half).

Further Exploration with Tables

• The process continues with finding constants for different scenarios involving varying speeds and their respective travel times using multiplication for verification.

• Another example introduces filling a pool with open faucets: more faucets mean less filling time—again confirming inverse proportions between faucet count and fill duration.

Solving for Unknown Variables

• When given tables with known values, calculating unknown variables involves determining constants through multiplication (e.g., verifying if they equal each other).

• Once constants are established (like equating products), solving for unknown quantities requires rearranging equations appropriately based on inverse relationships demonstrated earlier.

Understanding Inverse Proportionality in Time and Speed

Filling the Table with Data

• The speaker explains how to fill a table based on the relationship between the number of open taps and time taken to fill a pool, noting that 9 taps take 44 hours while more taps reduce this time significantly.

Graphing Time vs. Speed

• Exercise number 3 involves graphically representing the inverse relationship between time and speed, emphasizing that increased time results in decreased speed, illustrating their inversely proportional nature.

Finding the Constant of Proportionality

• The speaker calculates the constant of proportionality using various values (e.g., 6 by 50, etc.), leading to an equation where dividing gives insights into how these variables interact.

Representing Values on Cartesian Plane

• The discussion transitions to plotting points on a Cartesian plane, highlighting that inverse relationships create hyperbolic curves; as one variable increases, the other decreases.

Final Thoughts and Engagement