Proportions and relationships

1. Dependent and independent variables

In the picture, we see 8 people waiting in line to buy movie tickets. If more people come to the movies, the line will be longer. If there were few people, the line would be smaller.
The line length and the number of people are dependent on each other, that is, they are related.
When a thing depends on another, we say that they are related.
But the number of people in the line and the number of cars on the street are not related. In this case, the quantities are independent.

Do these exercises:

 1. The hours that I work and the daily wage that I earn are… 2. The number of wheels of a truck and its speed are... 3. The weight and the amount of water are... 4. The height of a man and his intelligence are... 5. The color of a car and its maximum speed are... 6. The speed of the train and the time it takes to go from one city to another are...

2. Related and proportional amounts

If an egg cost 5 pesetas, 2 eggs will cost 10 pesetas and 3 eggs will cost 15 pesetas.
In the Table 2, we see that the above series of numbers were multiplied by 5 to calculate the bottom series.
Two series of numbers are proportional when exists a multiplication or division operator that allows passing from one series to another.
But there are related amounts that are not proportional. For example: the age and weight of a boy. If a 6 year old boy weighs 30 kilos, another 12 year old boy will not necessarily weighs 60 kilos: (6 x 2 and 30 x 2). At 24 he will not necessarily weigh 120 kilos: (6x 4 and 30 x 4).

Answer these questions and indicate if they are related or proportional amounts.

 1. The kilos of rice that I sell and the money that I receive from them are... 2. The age and weight of a girl are... 3. The fertilizer that we put to the land and the harvest results are... 4. The number of carpenters and the number of seats that they manufacture are...

3. Direct and inverse relationships

The relationship that we studied were directs, that is, when one variable increases, so does the other or vice versa.
Now we see inverse relationships in which one variable decreases as other increases or vice versa.
Example: The age of an old man and his strength. When the age increases, the strength decreases. This is an inverse relationship.

Indicate if these statements are direct or inverse relationships:

 1. The speed of a train and the time it takes to travel between two cities is a... 2. The number of workers and the number of furniture that they make is... 3. The power of a motor vehicle and its speed is... 4. The time to build a road and the number of workers is... 5. The number of couples getting married and the number of children that born is... 6. The time to empty a barrel and the size of the hole is...

4. Directly proportional and inversely proportional quantities

When the relationship between two quantities is equal, it is called proportion. As one amount increases, another amount increases at the same quantity. It is called directly proportional.
Example: A pack of cigarettes cost 2 Euros, 3 three packs will cost 6 Euros (2 x 3).

As one amount decreases, another amount increases at the same quantity. It is called inversely proportional.
Example: If a reaper takes to mow a field 21 hours, 7 Reapers will take 3 hours. (21:7).
Indicate if these proportions are direct or inverse:
 1. The number of woodcutter and the number of trees that they can cut is a… 2. The speed of an airplane and the time it takes to make a trip is… 3. The number of cigarettes that I smoke and the money that I spend smoking is... 4. The number of books that I buy and what I have to pay is... 5. If I have 12 Euros to buy books, the number of books that I can buy and the price is... 6. The number of painters and the time it takes to paint a house is...

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®Arturo Ramo García.-Record of intellectual property of Teruel (Spain) No 141, of 29-IX-1999
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