Problem 18 - Fundamentals of Mathematics

Question

\(\frac{1}{100}\) of \(\frac{1}{10}\) is what number?

Step-by-Step Solution

Answer

The number is \(\frac{1}{1000}\).

Step 1: Understand the Problem

The question asks us to find what number is equivalent to \( \frac{1}{100} \) of \( \frac{1}{10} \). This means we need to multiply \( \frac{1}{100} \) by \( \frac{1}{10} \).

Step 2: Set Up the Multiplication

We set up the multiplication of the two fractions: \( \frac{1}{100} \times \frac{1}{10} \).

Step 3: Multiply the Fractions

The multiplication of fractions involves multiplying the numerators and the denominators separately. So, we multiply the numerators: \(1 \times 1 = 1\) and the denominators: \(100 \times 10 = 1000\). This gives us the fraction \( \frac{1}{1000} \).

Step 4: Simplify If Possible

The fraction \( \frac{1}{1000} \) is already in its simplest form as the numerator and the denominator have no common factors other than 1.

Key Concepts

Fractions

A fraction represents a part of a whole. It's like slicing a pie into equal pieces and taking some of them. Each fraction consists of two important parts: the numerator and the denominator.

• Numerator: The top number which indicates how many parts you have.
• Denominator: The bottom number which shows the total number of equal parts the whole is divided into.

Fractions are a lot like division problems and they tell us how many parts we have out of a total number. In the example given, we are dealing with fractions of a very small pie, as fractions like \( \frac{1}{100} \) and \( \frac{1}{10} \) both represent very small parts of their respective wholes.

Numerators

Numerators are like the players in a game, telling us how many parts of the whole we're talking about. When multiplying fractions, we multiply the numerators together. Like in our problem, multiplying the top numbers:

• \( 1 \times 1 = 1 \)

The numerator of the resulting fraction shows how many parts we have after multiplying. Even if we start with a very small numerator like 1, it stays small after multiplication, giving clear insight into how multiplication works on the top numbers in fractions.

Denominators

Denominators are the referees ensuring we're all on the same playing field. They tell us into how many equal parts the whole is split. For example:

• \( \frac{1}{100} \) has a denominator of 100, meaning the whole is split into 100 equal parts.
• \( \frac{1}{10} \) has a denominator of 10.

When multiplying fractions, like \( \frac{1}{100} \times \frac{1}{10} \), we multiply the denominators:

• \( 100 \times 10 = 1000 \)

The resulting denominator tells us the new total number of equal parts. It shows us the larger "whole" we're now dealing with, making it crucial in understanding fractions.

Simplifying Fractions

Simplifying fractions makes them easier to understand by reducing them to their simplest form. This involves finding any common factors between the numerator and the denominator and dividing them out.
When a fraction can't be broken down any further, it's in its simplest form. However, in our problem \( \frac{1}{1000} \), the numerator is 1. A fraction with 1 as a numerator is always in its simplest form unless the denominator is also 1, which would make it just the whole number 1.

Simplifying is all about finding the greatest common factor to make calculations more straightforward and fractions easier to visualize.