Problem 110 - Algebra 1: Concepts and Skills

Question

Divide. Write the answer as a fraction or as a mixed number in simplest form. $$ \frac{4}{5} \div 10 $$

Step-by-Step Solution

Answer

The division of \( \frac{4}{5} \) by 10 gives the fraction \( \frac{2}{25} \).

Step 1: Find the Reciprocal of the Whole Number

To divide by a number, you multiply by its reciprocal. The reciprocal of a number is found by dividing 1 by the number. Therefore, the reciprocal of 10 is \( \frac{1}{10} \).

Step 2: Multiply the Fraction by Reciprocal of the Whole Number

The fraction \( \frac{4}{5} \) should now be multiplied by the reciprocal of 10, which is \( \frac{1}{10} \). So, \( \frac{4}{5} \times \frac{1}{10} \) equals \( \frac{4}{50} \) after multiplication.

Step 3: Simplify the Fraction

The resulting fraction \( \frac{4}{50} \) can be simplified by finding the highest number that can divide both the numerator and the denominator. In this case, that's 2. So \( \frac{4}{50} \) simplifies to \( \frac{2}{25} \)

Key Concepts

Reciprocal in Division

When dividing fractions, you often need to use the concept of a reciprocal. A reciprocal of a number is simply what you multiply with that number to get 1. For instance, the reciprocal of a whole number like 10 can be found by flipping it upside down, which means you turn it into a fraction by placing it under 1, giving you \( \frac{1}{10} \).

• This process is essential because dividing by a number is the same as multiplying by its reciprocal.
• The use of reciprocals turns the division of fractions into a multiplication problem, which is usually simpler to solve.

Let's look at our problem: \( \frac{4}{5} \div 10 \). Here, the reciprocal of 10 is \( \frac{1}{10} \), which transforms the problem into \( \frac{4}{5} \times \frac{1}{10} \), making it easier to manage.

Simplifying Fractions

Simplifying fractions is a key step in working with fractions and ensuring the result is in its simplest form. A fraction is simplified when the greatest common divisor (GCD) of both the numerator and the denominator is 1. This means there are no common factors other than 1.

• To simplify a fraction, divide both the numerator and the denominator by their GCD.
• Often, this step makes both the calculation and final answer easier to interpret.

For example, after multiplying \( \frac{4}{5} \times \frac{1}{10} \), we got \( \frac{4}{50} \). The numbers 4 and 50 have a GCD of 2.
Dividing both by 2, we find that \( \frac{4}{50} \) simplifies to \( \frac{2}{25} \), making our fraction much cleaner and simpler in its final form.

Multiplying Fractions

Multiplying fractions is one of the more straightforward operations in fraction arithmetic. To multiply two fractions, you multiply the numerators together and the denominators together. Unlike adding or subtracting fractions, you do not need a common denominator.

• This process usually results in a product that might need simplification, as seen in the previous section.
• Multiplying fractions directly addresses operations involving reciprocals, as shown in the division example with \( \frac{4}{5} \div 10 \).

In our problem, converting the division to a multiplication of fractions gave us \( \frac{4}{5} \times \frac{1}{10} \), resulting in \( \frac{4 \times 1}{5 \times 10} = \frac{4}{50} \).
By learning how to multiply fractions, you can handle division problems more efficiently, using their reciprocal trick!