Problem 64 - Fundamentals of Mathematics

Question

State the numerator and denominator and write in words each of the fractions appearing in the statements for the following 10 problems. The mean (average) of the numbers \(21,25,43,\) and 36 is \(\frac{125}{4}\).

Step-by-Step Solution

Answer

Numerator: 125; Denominator: 4; Fraction: "one hundred twenty-five divided by four."

Step 1: Identify the Mean Formula

The mean (average) of a set of numbers is calculated by adding up all the numbers and dividing by the count of numbers in the set.

Step 2: Write Numerator

The numerator is the sum of the numbers. In this case, add: \(21 + 25 + 43 + 36 = 125\). Thus, the numerator of the fraction for the mean is 125.

Step 3: Identify Denominator

The denominator represents the number of values in the set. Here, there are four numbers: 21, 25, 43, and 36. Therefore, the denominator is 4.

Step 4: Express Fraction in Words

The fraction \(\frac{125}{4}\) can be written in words as 'one hundred twenty-five divided by four.'

Key Concepts

Numerator and Denominator Identification

To understand fractions, it's essential to know what numerators and denominators are. They are the two numbers that compose a fraction, with the numerator on the top and the denominator on the bottom.
- The **numerator** represents the part of the whole or the total you have. In simple terms, it's the number above the fraction bar.
- The **denominator** tells you the total number of equal parts the whole is divided into. It's the number below the fraction bar.
In the fraction \(\frac{125}{4}\), based on the exercise, 125 is the numerator and 4 is the denominator. This tells us that 125 is being divided by 4, reflecting the total sum of the numbers divided by the total count of these numbers.

Writing Fractions in Words

Fractions can represent numbers in a different format, and writing them in words helps in comprehending their value and meaning. To convert a fraction into words, follow these simple steps:
1. State the numerator in words.
2. Use the word 'divided by.'
3. State the denominator in words.
For instance, the fraction \(\frac{125}{4}\) can be expressed in words as 'one hundred twenty-five divided by four.' This is an easy process but crucial when you need to clearly and effectively communicate mathematical information in a verbal format.

Average of Numbers

Finding the mean, or average, of a set of numbers helps us understand the central tendency of the data. This is helpful in summarizing a large dataset into a single representative value. Here's how to calculate it:
- **Add up all the numbers.**
For the numbers given - 21, 25, 43, and 36 - you calculate their total: \(21 + 25 + 43 + 36 = 125\).
- **Count the number of numbers.** In this case, 4 numbers.
- **Divide the sum by their count.** This gives us the average, so: \(\frac{125}{4}\).
Therefore, the mean or average is 31.25. This value provides a simple summary of the overall number set.