Problem 30 - Prealgebra

Question

Approximately three-fourths of the students at Davis Middle School have a telephone in their room. Two-fifths of those students have call waiting on their phones. About what fraction of students have call waiting?

Step-by-Step Solution

Answer

About \( \frac{3}{10} \) of the students have call waiting.

Step 1: Represent Fraction of Students with Phones

According to the problem, three-fourths of the students at Davis Middle School have a telephone in their room. We can represent this as the fraction \( \frac{3}{4} \).

Step 2: Find Fraction of Students with Call Waiting

Two-fifths of the students who have a telephone also have call waiting. This can be written as \( \frac{2}{5} \) of those students.

Step 3: Calculate Combined Fraction

To find the fraction of students with call waiting, multiply the two fractions: \( \frac{3}{4} \times \frac{2}{5} = \frac{6}{20} \).

Step 4: Simplify the Fraction

Simplify \( \frac{6}{20} \) by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 2. This gives us \( \frac{3}{10} \).

Key Concepts

Mathematical Problem Solving

Mathematical problem solving is like a fun puzzle game. It involves reading and understanding a problem, deciding what you need to find out, and figuring out how to solve it. In our exercise, you start by getting your information from the problem, like knowing that three-fourths of students have phones and two-fifths of those have call waiting.
By carefully breaking down each part of the problem, you know what numbers and operations to use. This clear understanding makes calculating much easier.
Remember to:

• Read the problem carefully.
• Identify the fractions or numbers involved.
• Decide on the operations needed, like multiplication or addition.
• Break down the steps and work systematically.

Following these tips ensures you don't miss important details and helps you reach the correct answer.

Simplifying Fractions

Simplifying fractions makes math easier and your answers cleaner. When you simplify a fraction, you're making the numbers in it smaller and friendlier, without changing its value.
In our exercise, after multiplying the two fractions, we got \( \frac{6}{20} \). While technically correct, that fraction can be simplified.

• Find the greatest common divisor (GCD) for both the numerator and the denominator.
• In this case, 2 is the GCD of 6 and 20.
• Divide both numbers by 2 to get a simplified fraction, \( \frac{3}{10} \).

This new fraction \( \frac{3}{10} \) is much cleaner and easier to understand. Simplifying fractions is always a useful step to take in order to make your calculations clearer and more accurate.

Multiplying Fractions

Multiplying fractions might seem tricky, but it's straightforward with a little practice. When you're faced with multiplying fractions, such as in our exercise, follow these simple guidelines:

• Multiply the numerators across each other. For \( \frac{3}{4} \) and \( \frac{2}{5} \), you multiply 3 (numerator) by 2, resulting in 6.
• Next, multiply the denominators. Here, it's 4 times 5, giving you 20 as your new denominator.
• This results in the fraction \( \frac{6}{20} \).

Note that after multiplying, you often should simplify the fraction. This process of multiplying directly and then simplifying ensures your fractions are easy to interpret and handle.
Remember, practice makes perfect when it comes to fractions. Try multiplying different ones to get the hang of it!