Question

Suppose 6 people share 8 sticks of gum so that each person gets the same amount. What fraction of a stick does each person receive? Express your answer as a fraction and as a mixed number.

Short answer

\(\frac{4}{3}\) or \(1\frac{1}{3}\)

Step-by-step solution

Understand the Problem

We need to divide 8 sticks of gum equally among 6 people. This can be represented as the division problem: 8 divided by 6.

Set Up the Division

Write the division problem as a fraction: \(\frac{8}{6}\). This fraction represents the amount of gum each person will get.

Simplify the Fraction

Simplify \(\frac{8}{6}\) by finding the greatest common divisor (GCD) of 8 and 6, which is 2. Divide both the numerator and the denominator by 2: \(\frac{8 ÷ 2}{6 ÷ 2} = \frac{4}{3}\). So, each person gets \(\frac{4}{3}\) sticks of gum.

Convert to a Mixed Number

Convert \(\frac{4}{3}\) to a mixed number. Divide 4 by 3 to get 1 with a remainder of 1. Hence, \(\frac{4}{3} = 1\frac{1}{3}\). Each person gets \(1\frac{1}{3}\) sticks of gum.

Key concepts

simplifying fractions

Simplifying fractions is an essential step when it comes to fraction division problems. When we simplify a fraction, we make it easier to understand and work with. To simplify a fraction, we reduce it to its lowest terms. For example, in the problem given, we had the fraction \(\frac{8}{6}\).
To simplify \(\frac{8}{6}\), we need to find the greatest common divisor (GCD) of the numerator (8) and the denominator (6). Once we find the GCD, we divide both the numerator and the denominator by this number. Here, the GCD of 8 and 6 is 2.
So, dividing both the numerator and the denominator by 2, we get \(\frac{8 ÷ 2}{6 ÷ 2} = \frac{4}{3}\). Now, \(\frac{4}{3}\) is in its simplest form. Always make sure to simplify your fractions to make the calculations easier and answers clearer.
Some handy points to remember:

• The GCD is the largest number that can divide both the numerator and the denominator without leaving a remainder.
• Always check if your fraction can be simplified further.
• Simplified fractions are easier to convert to other forms, like mixed numbers or decimals.

mixed numbers

A mixed number is another way to represent an improper fraction. It consists of a whole number and a proper fraction. In our problem, once we simplified \(\frac{8}{6}\) to \(\frac{4}{3}\), the next step was to convert \(\frac{4}{3}\) to a mixed number.
To do this, we see how many whole numbers fit into the improper fraction. We divide the numerator by the denominator. For \(\frac{4}{3}\), dividing 4 by 3 gives us 1 with a remainder of 1. So, \(\frac{4}{3}\) equals 1 whole number and a remainder of 1, which we write as \(\frac{1}{3}\).
Thus, \(\frac{4}{3}\) can be expressed as \1\frac{1}{3}\.
Why convert to mixed numbers?

• Mixed numbers are often easier to understand and visualize.
• They can be practical in real-world contexts, like in our example of gum sticks.
• They make adding and subtracting more intuitive, especially when dealing with other mixed numbers or whole numbers.

Remember:

• Improper fractions (where the numerator is greater than the denominator) can always be converted to mixed numbers.
• The process involves simple division and writing the remainder as a fraction.

greatest common divisor

The greatest common divisor (GCD) is a key concept in simplifying fractions. It is the largest number that divides both the numerator and the denominator of a fraction evenly. In our exercise, to simplify \(\frac{8}{6}\), we needed to find the GCD of 8 and 6.
To find the GCD, you can use the following methods:

• Prime Factorization: Write down the prime factors of each number and find the common factors. The common factor's highest power is the GCD.
• Euclidean Algorithm: This method involves repeated division. For 8 and 6, we divide 8 by 6 to get a remainder of 2. Then, we divide 6 by 2 to get a remainder of 0. Hence, the last non-zero remainder (2) is the GCD.
• Listing Factors: List all factors of both numbers and find the highest common one.

The GCD helps in reducing fractions to their simplest form. It ensures calculations are simplified and accurate. Recognizing the GCD is especially helpful in various mathematical problems beyond simplifying fractions, such as solving equations and finding LCM (Least Common Multiple).
Key points:

• GCD is vital in fraction simplification.
• Multiple methods can determine the GCD.
• Understanding GCD strengthens mathematical problem-solving skills.