Question

Which answer expresses the rate 24 miles per 36 minutes as a fraction in lowest terms? A. \(\frac{24}{36}\) B. \(\frac{2}{3}\) C. \(\frac{3 \text { minutes }}{2 \text { miles }}\) D. \(\frac{2 \text { miles }}{3 \text { minutes }}\)

Short answer

D. \[\frac{2 \text{ miles}}{3 \text{ minutes}}\].

Step-by-step solution

Identify the given rate

The rate given is 24 miles per 36 minutes. It can be expressed as the fraction \[\frac{24 \text{ miles}}{36 \text{ minutes}}\].

Simplify the fraction

To simplify \[\frac{24}{36}\], find the greatest common divisor (GCD) of 24 and 36. The GCD of 24 and 36 is 12. Divide both the numerator and the denominator by 12: \[\frac{24 \text{ miles}}{36 \text{ minutes}} = \frac{24 \text{ miles} \div 12}{36 \text{ minutes} \div 12} = \frac{2 \text{ miles}}{3 \text{ minutes}}\].

Compare with the given options

The simplified fraction \[\frac{2 \text{ miles}}{3 \text{ minutes}}\] matches option D. Therefore, the correct answer is D.

Key concepts

Rate Problems

Rate problems involve comparing two different quantities. For example, if you drive 24 miles in 36 minutes, you're comparing distance and time. To express rates, we often use fractions. In this instance, the rate can be written as \ \(\frac{24 \text{ miles}}{36 \text{ minutes}}\).

Understanding rate problems is crucial in everyday life. Rates tell us how fast or slow something happens. You see rates in speed (miles per hour), pricing (cost per item), and even in cooking (teaspoons per ingredient).

When solving rate problems:

• First, identify the two quantities being compared.
• Write them down as a fraction showing the relationship.
• Simplify the fraction to its lowest terms for clarity.

Greatest Common Divisor (GCD)

Simplifying fractions often requires finding factors that both the numerator and denominator share. This is where the greatest common divisor (GCD) comes in. The GCD is the largest number that can evenly divide both parts of a fraction.

Let's look at the fraction from the exercise: \ \(\frac{24}{36}\).

To find the GCD of 24 and 36, start by listing their factors:

• Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
• Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

The largest common factor is 12, so the GCD of 24 and 36 is 12. Knowing this, we can simplify \ \(\frac{24}{36}\) by dividing both numbers by their GCD.

GCD helps in reducing fractions, making them easier to understand and interpret.

Fraction Simplification

Simplifying fractions means reducing them to their simplest form. This involves dividing both the numerator (top part) and the denominator (bottom part) by their greatest common divisor (GCD).

For the exercise's fraction \ \(\frac{24}{36}\), the steps to simplify are:

• Find the GCD, which we know is 12.
• Divide both the numerator and the denominator by 12.
• \frac{24}{36} = \frac{24 \text{ miles} / 12}{36 \text{ minutes} / 12} = \frac{2 \text{ miles}}{3 \text{ minutes}}

Hence, \frac{24 miles}{36 minutes} simplifies to \frac{2 miles}{3 minutes}, which is a cleaner, easily interpretable rate.

Simplification makes fractions easier to work with, understand, and communicate. That's why it's a crucial skill in both mathematics and real-life applications.