Understanding Dependencies

There are many things in the real world whose values are dependent on one or more other values.

For example, the amount of food that is consumed at a party is dependent on the number of people eating at the party. The more the number of people the more food is consumed. Of course, this can also depend on many other things like the amount of food consumed by each person, the taste of the food. The value which you are calculating are call Dependent Entity (entity means object or thing), and the value or values which you are changing are called Independent Entities.

We will understand this more with the examples given in this topic.

The price that you pay for buying milk is dependent on the quantity of milk you buy. The more quantity you buy the more you pay.

These quantities are called Directly Proportional.

Then there is the other type of dependencies, where increasing one quantity decreases the other.

The time taken to build a given house, depends on the number of people building the house. The more the number of people, the less is the time required to build the house.

When travelling from one place to another, the time taken to reach is dependent on the speed of the vehicle. The more the speed is, lesser is the time taken to reach.

These quantities are called Inversely Proportional. (Remember, Inverse means opposite).

Although it is not important to remember these names at this stage, for you to be able to solve these problems, you should be able to identify the dependency between different quantities an the type of dependencies, that is, if you increase one whether the other one will increase or decrease. Then we calculate the dependent quantity for a value of 1 of the quantity on the left.

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Now let us look at the steps we need to follow to solve a problem using the unitary method.

Step 1:

1. Identify the dependent entity, that is the quantity you are calculating.

2. Identify the independent entity or entities, that is the quantities that you are going to change.

3. Identify the relationship, that is if you change the independent quantity, will the dependent quantity increase or decrease.

4. Write your statement, such that the independent quantity is on the left and the dependent quantity is on the right. This will make forming the expression easy.

5. Once you have written the complete expression for the independent quantity.

6. Although not required, but it is a good practice to do the calculations at the last step. Since many problems are such where numbers cancel out and your calculations become much easier.

Example Problem - 1:

If $10$ workers produce $200$ chocolates in a day, how much will $25$ workers produce in a day?

Solution:

Here the independent entity, that you are calculating is the number of chocolates, and the dependent entity is the number of workers.

We also know that more the number of people the more chocolate they will produce in the same time period. Therefore they are directly proportional.

Now let us form the expression:

$10$ workers produce $200$ chocolates in a day.

$1$ worker will produce $\dfrac{200}{10} = 20$ chocolates in a day.

$25$ workers will produce $25 \times 20 = 500$ chocolates in a day.

Answer: $200$ chocolate.

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Example Problem - 2:

It takes $30$ days for $5$ farmers to cultivate a farm. How many days will $15$ farmer take?

Independent entity: number of days.

Dependent entity: number of farmers.

We know that increasing the number of farmers will reduce the time taken to cultivate the farm. Hence they are inversely proportional.

Now let us write the steps:

$5$ farmers take $30$ days to cultivate a farm.

$1$ farmer will take $30 \times 5$ days to cultivate the farm.

$15$ farmers will take $\dfrac{30 \times 5}{15} = 10$ days to cultivate the farm.

Answer: $10$ days.