Question

Area codes are used to distinguish phone numbers for which the last seven digits are the same. If you have 35,000,000 phone numbers in a state and an area code can distinguish approximately 900,000 phone numbers, how many area codes are needed to distinguish the phone numbers of this state?

Short answer

39 area codes are needed.

Step-by-step solution

Understand the Problem

The problem requires us to determine how many area codes are necessary to distinguish a large number of phone numbers given that one area code can handle only a certain quantity of numbers.

Identify Known Values

We are given that there are 35,000,000 phone numbers in total and one area code can accommodate approximately 900,000 phone numbers.

Write Down the Formula for Number of Area Codes Needed

To find the number of area codes, divide the total number of phone numbers by the number each area code can handle: \( \frac{\text{Total phone numbers}}{\text{Phone numbers per area code}} \).

Perform the Calculation

Using the values provided, calculate the necessary number of area codes: \( \frac{35,000,000}{900,000} = 38.888... \).

Round Up the Result

Since we cannot have a fraction of an area code, round up to the next whole number. Hence, 39 area codes are needed.

Key concepts

Phone Number Distribution

In the context of telephony, phone number distribution is about how phone numbers are organized using area codes. Area codes are crucial for ensuring there is a manageable, organized system to handle a large number of telephone numbers within a geographical region. Each area code has a specific capacity to distinguish a certain number of unique phone numbers.
In our exercise, we have a large state with 35,000,000 phone numbers. These numbers must be organized efficiently using area codes. An individual area code can manage up to approximately 900,000 numbers. Understanding phone number distribution helps in planning how many area codes will be required. This process ensures every phone number is unique and can be easily categorized, promoting communication efficiency.
Phone number distribution allows us to accommodate millions of lines without confusion. It's like creating bins to categorize similar items - assigning every item a specific bin to avoid overlap and ensure organization.

Mathematical Problem-Solving

Mathematical problem-solving involves taking a complex issue and utilizing mathematical principles to find a solution. In this task, our objective is to ascertain how many area codes are necessary for a given number of phone numbers. To solve this, we apply logical reasoning and a step-by-step method.
First, understand what is required: finding out how many area codes are needed to cover all phone numbers. Break the problem down by using known values, such as the total number of phone numbers and the capacity of each area code. Next, identify the relationship between these numbers and choose an appropriate mathematical operation.
In this case, division is suitable because we need to see how many full groups (area codes) can fit into the larger total (total phone numbers). Problem-solving in maths often involves breaking down complex problems into manageable steps, making the situation easier and more intuitive.

Division and Rounding

Division and rounding are essential mathematical operations used in many problem-solving scenarios, including our exercise on area codes. To distribute the total number of phone numbers among the available area codes, we use division. This tells us how many groups or segments (area codes) can be formed.
Using the given numbers, execute the division of total phone numbers by the capacity per area code: \[ \frac{35,000,000}{900,000} = 38.888\ldots \]. The result, 38.888..., suggests we need slightly more than 38 area codes. However, because area codes must be whole numbers—since you can't have a fraction of an area code—we use the rounding operation.
In mathematics, when dealing with division results in a non-integer, rounding up ensures that all items (or in this case, phone numbers) are properly accounted for. Hence, rounding 38.888... up to 39 ensures no phone number is left without an area code, ensuring completeness and accuracy in our solution.