Question

A roll of pennies holds 50 pennies. What fraction of a roll is 237 pennies? Give your answer as a mixed number and as a fraction.

Short answer

Mixed number: 4 \frac{37}{50} Fraction: \frac{237}{50}

Step-by-step solution

- Understanding the Problem

First, understand that we need to figure out how many rolls of pennies are there in 237 pennies, considering each roll holds 50 pennies.

- Divide the Total Number of Pennies by the Pennies per Roll

Divide 237 by 50 to find out how many rolls fit into 237 pennies. \[ 237 \, \text{divided by} \, 50 = \frac{237}{50} \]

- Perform the Division

Perform the division to get the fraction: \[ \frac{237}{50} \]

- Simplify the Fraction

Convert the improper fraction \( \frac{237}{50} \) into a mixed number. Divide 237 by 50, which equals 4 with a remainder of 37. The mixed number is \( 4 \frac{37}{50} \).

Key concepts

Improper Fractions

When we talk about improper fractions, we refer to fractions where the numerator is larger than the denominator. In our exercise, when we divided 237 by 50 and got \( \frac{237}{50} \), we handled an improper fraction. Improper fractions are essential in mathematics because they help represent quantities greater than one.

Learning to work with improper fractions is crucial. They often appear in mathematical calculations, and knowing how to convert them to mixed numbers or simplify them is key to mastering fraction problems.

Mixed Numbers

Mixed numbers are a combination of whole numbers and fractions. They provide a more intuitive way to handle numbers greater than one. For example, in our solution, we converted the improper fraction \( \frac{237}{50} \) to the mixed number \( 4 \frac{37}{50} \).

When converting an improper fraction to a mixed number, you divide the numerator by the denominator, noting down the quotient and the remainder. Here, 237 divided by 50 gives 4 with a remainder of 37. Therefore, we express it as:

• 4 (the whole number part)
• \frac{37}{50} (the fractional part)

Learning to express improper fractions as mixed numbers makes it easier to understand and visualize quantities.

Division in Mathematics

Division is a fundamental arithmetic operation that helps us determine how many times one number is contained within another. In our exercise, we used division to find out how many rolls of 50 pennies fit into 237 pennies. \( 237 \) divided by \( 50 \) translates to \( \frac{237}{50} \).

Division also plays a critical role in simplifying fractions and converting improper fractions to mixed numbers. Understanding division's basics and how it applies to fractions provides a solid foundation for more complex mathematical concepts.

Simplification of Fractions

Simplifying fractions means reducing them to their simplest form where the numerator and the denominator have no common factors other than 1. In our problem, the fraction \( \frac{237}{50} \) is already simplified because 237 and 50 have no common factors other than 1.

Simplifying can make fractions easier to work with. Here are steps to simplify a fraction:

• Find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.

With practice, simplifying fractions becomes second nature, making mathematical operations more straightforward.