Problem 63 - Prealgebra

Question

There are 60 minutes in 1 hour. What fraction of an hour is 20 minutes? Write your answer in lowest terms.

Step-by-Step Solution

Answer

20 minutes is \( \frac{1}{3} \) of an hour.

Step 1: Understanding the Problem

We need to find out what fraction of an hour 20 minutes represents, given that 1 hour is equal to 60 minutes. This is essentially asking us to express 20 minutes as a fraction of 60 minutes.

Step 2: Setting Up the Fraction

Since 20 minutes is part of a total of 60 minutes, we can express this as a fraction: \( \frac{20}{60} \). This represents the portion of the hour that 20 minutes occupies.

Step 3: Simplifying the Fraction

To express the fraction in its simplest form, we need to divide both the numerator and the denominator of \( \frac{20}{60} \) by their greatest common divisor (GCD). The GCD of 20 and 60 is 20.

Step 4: Calculating the Simplified Fraction

Divide the numerator and the denominator by the GCD, which is 20: \( \frac{20 \div 20}{60 \div 20} = \frac{1}{3} \). This fraction is in simplest form.

Key Concepts

Simplifying Fractions

When we talk about fractions, simplifying them means finding an equivalent fraction that has the smallest possible numerator and denominator. This makes it easier to work with and understand.
To simplify a fraction, we must divide both the top number (numerator) and the bottom number (denominator) by the same number, which is called their greatest common divisor (GCD). Here's a simple way to go about it:

• Identify the numerator and denominator of your fraction, like 20 and 60 in our exercise example.
• Find the GCD of the two numbers. We'll discuss how in the next section.
• Divide both the numerator and the denominator by the GCD.

After simplifying, the fraction takes up less space and makes calculations or comparisons much easier. In our case, simplifying \( \frac{20}{60} \) gives \( \frac{1}{3} \). This means 20 minutes is a third of an hour when simplified.

Greatest Common Divisor

Finding the greatest common divisor (GCD) is crucial for simplifying fractions. It is the largest number that divides both the numerator and denominator without any remainder. This process streamlines fractions effectively. Here's how you find the GCD:

• List the factors of each number. For example, for 20: 1, 2, 4, 5, 10, 20, and for 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.
• Identify the largest number that appears in both lists of factors. In our case, it is 20.

Once you have the GCD, use it to divide both the numerator and the denominator of the fraction as earlier illustrated. Knowing how to find the GCD helps not just in simplifying fractions but also in solving a variety of math problems efficiently.

Unit Conversion in Time

Understanding how to convert units of time is an essential skill, whether you're calculating everyday durations or working on math problems. In time conversion, the key is knowing how many smaller units make up a larger unit. Here's an easy guide:

• Recall that 1 hour equals 60 minutes. This is fundamental.
• To find what fraction of an hour another number of minutes represents, place those minutes over 60 in a fraction. For the exercise, 20 minutes becomes \( \frac{20}{60} \).
• Then simplify this fraction to get your final answer, which can be easily interpreted as a portion of an hour.

Converting time units and expressing them as simplified fractions will make your computations more intuitive and manageable. In this exercise, converting 20 minutes into \( \frac{1}{3} \) of an hour helps us quickly understand how much of the hour we've used.