Problem 65 - Prealgebra

Question

Many of us focus our attention on things other than driving when we are behind the wheel of our car. In a survey of 150 drivers, it was noted that 48 drivers spend time reading or writing while they are driving. Represent the number of drivers who spend time reading or writing while driving as a fraction in lowest terms.

Step-by-Step Solution

Answer

The fraction in lowest terms is \( \frac{8}{25} \).

Step 1: Identify the total and specific numbers

We know from the problem that the total number of drivers is 150 and that 48 of them spend time reading or writing while driving.

Step 2: Understand the fraction

The fraction representing the drivers who spend time reading or writing is calculated by taking the specific number (48 drivers) and dividing it by the total number of drivers (150). So, the fraction will be \( \frac{48}{150} \).

Step 3: Simplify the fraction

To simplify \( \frac{48}{150} \), we need to find the greatest common divisor (GCD) of 48 and 150, which is 6. Divide both the numerator and denominator by their GCD: \( \frac{48 \div 6}{150 \div 6} = \frac{8}{25} \).

Step 4: Verify that the fraction is in lowest terms

Check to ensure \( \frac{8}{25} \) is in its simplest form. Since 8 and 25 have no common factors other than 1, \( \frac{8}{25} \) is indeed the simplest form.

Key Concepts

Greatest Common Divisor

Finding the greatest common divisor (GCD) of two numbers is an essential step when simplifying fractions. The GCD is the largest number that evenly divides both of the numbers in question. To simplify a fraction like \( \frac{48}{150} \), we need to determine the GCD of 48 and 150.

One of the methods to find the GCD is listing out the factors of each number:

• For 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
• For 150: 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150

The largest common factor here is 6. Therefore, 6 is the GCD of 48 and 150. This GCD allows us to divide both the numerator and the denominator of our fraction, simplifying it.Another method to find the GCD is by using the Euclidean algorithm, which continually takes the remainder of the division until a remainder of zero is reached. The last non-zero remainder is the GCD.

Dividing Fractions

Dividing fractions actually involves an inverse operation compared to multiplying fractions. When we divide by a fraction, we multiply by its reciprocal.

The reciprocal of a fraction is created by swapping its numerator and denominator. For example, if we need to divide \( \frac{1}{2} \) by \( \frac{3}{4} \), we instead multiply \( \frac{1}{2} \) by \( \frac{4}{3} \). This can be expressed as:\[\frac{1}{2} \div \frac{3}{4} = \frac{1}{2} \times \frac{4}{3} = \frac{1 \times 4}{2 \times 3} = \frac{4}{6}\]This result can be further simplified by finding its GCD, which in this case is 2, to make it \( \frac{2}{3} \). Understanding this process is crucial when working with divisions involving fractions.

Mathematical Fractions

Mathematical fractions represent a part of a whole. They consist of two numbers: the numerator and the denominator. The numerator is the top number that indicates how many parts of the whole are being considered, while the denominator is the bottom number that shows how many parts the whole is divided into.

In the exercise mentioned, the fraction \( \frac{48}{150} \) represents the part of drivers in the survey who were reading or writing while driving. The numerator is 48, representing those specific drivers, and the denominator is 150, representing the total number of drivers surveyed.When working with fractions, it is always a good practice to simplify them to ensure they are in their lowest terms, as this makes them easier to understand and work with. Simplifying fractions involves finding the GCD of the numerator and the denominator and dividing both by this number, ensuring no common factors remain other than one.