Problem 74 - Prealgebra

Question

In a survey of 1,000 cell phone subscribers it was determined that 160 subscribers owned more than one cell phone and used different carriers for each phone. Represent the number of cell phone subscribers with more than one carrier as a fraction.

Step-by-Step Solution

Answer

The fraction is \(\frac{4}{25}\).

Step 1: Understand the Problem

The problem involves finding the fraction of cell phone subscribers who own more than one phone and use different carriers for each.

Step 2: Identify Given Information

From the survey, there are 1,000 total subscribers, and 160 own more than one cell phone with different carriers.

Step 3: Set Up the Fraction

We want to express 160, the number of subscribers with more than one carrier, as a fraction of the total number of subscribers, 1,000.

Step 4: Calculate the Fraction

Divide the number of subscribers with multiple carriers by the total number of subscribers:\[\frac{160}{1000}\]

Step 5: Simplify the Fraction

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, 40:\[\frac{160}{1000} = \frac{160 \div 40}{1000 \div 40} = \frac{4}{25}\]

Key Concepts

Simplifying Fractions

Fractions represent a part of a whole by having a numerator above a line and a denominator below. Simplifying fractions means reducing them to their simplest form. This occurs when the greatest common divisor (GCD) of the numerator and denominator is found and both numbers are divided by it.
For instance, in the exercise, 160 subscribers over a total of 1,000 give the initial fraction \(\frac{160}{1000}\).
To simplify, you divide both by their GCD, which is 40. This step by step division gives the simplified form \(\frac{4}{25}\). This fraction now represents the same quantity but in the simplest way possible.

• Start by identifying the GCD of both numbers.
• Divide the numerator and the denominator by this GCD.
• The result is a simplified fraction that is easier to understand and use.

Simplifying fractions is crucial for making comparisons and calculations more straightforward. It reduces the numbers, but the value remains the same, preserving the proportion.

Numerator

The numerator of a fraction is the number on the top. It signifies how many parts of the whole you are considering. In the fraction \(\frac{4}{25}\), 4 is the numerator, denoting the specific portions of the total being counted.
For example, in our scenario, 160 subscribers represent the "pieces" out of the total 1,000, making 160 the original numerator of the fraction \(\frac{160}{1000}\).
After simplification, our focus is downscaled to 4, which is the direct result of simplifying, showing the reduced yet equivalent portion.
Always remember:

• The numerator deals with how many parts of the whole we have.
• It directly impacts the value of the fraction.
• In simplification, it is the adjusted numerator that echoes the fraction’s same value.

Guiding through fraction operations, starting with the numerator helps in maintaining clarity.

Denominator

The denominator is the number found at the bottom of the fraction. It outlines the total number of equal parts the whole is divided into.
In the fraction \(\frac{4}{25}\), 25 is the denominator, implying the entire group into which the portions (numerator) are divided.
For an initial fraction like \(\frac{160}{1000}\), 1,000 represents the full set of subscribers surveyed, showing the context of what the numerator is part of.
When reducing, our denominator changes to 25, by dividing it through its GCD 40, producing a friendlier number but still accurately reflecting the entire surveyed group.
Some key aspects about denominators include:

• Determining the scale of total parts in consideration.
• A changing denominator via simplification still equals the original whole in scaled form.
• Maintaining it correctly ensures the fraction’s representation is unaltered.

Understanding denominators is essential for grasping fractions overall, as they define the scope and setting of the parts being measured.