Question

Suppose 10 friends share 4 medium-sized pizzas so that each friend gets the same amount. What fraction of a pizza does each friend receive?

Short answer

Each friend receives \( \frac{2}{5} \) of a pizza.

Step-by-step solution

Understand the Problem

First, identify the total number of pizzas and the total number of friends sharing the pizzas. In this case, 10 friends are sharing 4 medium-sized pizzas.

Determine the Fraction of Pizza Each Friend Gets

To find out how much pizza each friend receives, divide the total number of pizzas by the total number of friends. This can be expressed as a fraction: \( \frac{4}{10} \).

Simplify the Fraction

Simplify the fraction \( \frac{4}{10} \) by finding the greatest common divisor (GCD) of 4 and 10, which is 2. Divide both the numerator and the denominator by the GCD: \( \frac{4 \div 2}{10 \div 2} = \frac{2}{5} \).

Key concepts

Simplifying Fractions

When you have a fraction, you sometimes need to simplify it to make it easier to understand and work with. Simplifying a fraction means making the numerator (top number) and the denominator (bottom number) as small as possible while keeping the same value. For example, in the exercise where 10 friends share 4 medium-sized pizzas, we got the fraction \( \frac{4}{10} \). This fraction can be simplified by finding a common factor that both the numerator and the denominator share. Here, the numbers 4 and 10 both can be divided by the same number: 2. So, \( \frac{4}{10} \) can be simplified to \( \frac{2}{5} \). Simplifying fractions often makes it easier to work with them in other math problems.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that can divide both the numerator and denominator of a fraction without leaving a remainder. To simplify a fraction, we use the GCD to make the numerator and the denominator as small as possible. In the given problem, we simplified the fraction \( \frac{4}{10} \) by using the GCD.

Here’s how it works:

• Find all the factors of the numerator (4): 1, 2, 4.
• Find all the factors of the denominator (10): 1, 2, 5, 10.
• Identify the largest number that appears on both lists. Here, that number is 2.

Divide both the numerator and the denominator by the GCD. So, divide 4 by 2 and divide 10 by 2, giving you the simplified fraction: \( \frac{2}{5} \). This tells you how much pizza each friend gets in its simplest form.

Division of Whole Numbers

Division of whole numbers involves splitting a total into equal parts. When you divide one whole number by another, you create a fraction. This helps to represent how much each part gets from the total. In the given exercise, we divided 4 medium-sized pizzas among 10 friends.

To do this, we set up our division as a fraction: \( \frac{4}{10} \). Each fraction represents one whole number (4 pizzas) being divided by another (10 friends). This means every friend gets an equal share of the pizzas.

• First, identify the total quantity to be divided (4 pizzas).
• Then, identify how many parts you are dividing this quantity into (10 friends).
• Write this as a fraction: \( \frac{4}{10} \).
• Simplify if possible, as we did in earlier steps. The fraction \( \frac{4}{10} \) simplifies to \( \frac{2}{5} \), showing each friend gets \( \frac{2}{5} \) of a pizza.

Division helps determine fair shares and understand proportions better.