Problem 35 - Fundamentals of Mathematics

Question

For the following problems, find each value. $$\frac{25}{49} \div \frac{4}{9}$$

Step-by-Step Solution

Answer

The value is \( \frac{225}{196} \).

Step 1: Rewrite the Division as Multiplication

To divide two fractions, convert the division problem into a multiplication problem by taking the reciprocal of the divisor (the second fraction). This gives us: \( \frac{25}{49} \div \frac{4}{9} = \frac{25}{49} \times \frac{9}{4} \).

Step 2: Multiply the Fractions

Now, perform the multiplication of the fractions by multiplying their numerators and their denominators: \( \frac{25}{49} \times \frac{9}{4} = \frac{25 \cdot 9}{49 \cdot 4} = \frac{225}{196} \).

Step 3: Simplify the Fraction

Check if the fraction can be simplified. In this case, \( \frac{225}{196} \) is already in its simplest form because there are no common factors between 225 and 196 other than 1. Therefore, \( \frac{225}{196} \) is the simplest form of the fraction.

Key Concepts

Fraction Multiplication

Fraction multiplication might sound intimidating, but it's quite straightforward once you break it down. To multiply fractions, follow these simple steps:

• Multiply the numerators (the numbers on top) of the fractions to get the new numerator.
• Multiply the denominators (the numbers on the bottom) of the fractions to get the new denominator.

This results in a new fraction. For instance, if you have \( \frac{25}{49} \times \frac{9}{4} \), you multiply 25 by 9 and 49 by 4, which gives \( \frac{225}{196} \).

It's important to remember that the fractions must remain as fractions when multiplying; you do not need to find a common denominator like you would in addition or subtraction. This makes multiplying fractions relatively straightforward!

Reciprocal of a Fraction

Understanding the reciprocal is crucial for fraction division. A reciprocal of a fraction is simply swapping its numerator and denominator. For example, the reciprocal of \( \frac{4}{9} \) is \( \frac{9}{4} \).

To divide by a fraction, you multiply by its reciprocal. This is one of the most important tricks in fraction division. Instead of dividing, you change the division into a multiplication problem by taking the reciprocal of the fraction you're dividing by. In the original exercise, this changed \( \frac{25}{49} \div \frac{4}{9} \) into \( \frac{25}{49} \times \frac{9}{4} \).

Remember: "Dividing by a fraction is the same as multiplying by its reciprocal." Recognizing and using reciprocals can simplify your calculations and help you solve fraction problems efficiently.

Simplifying Fractions

Once you've multiplied or performed any operation on fractions, you might end up with a large numerator and a denominator. In this situation, simplification becomes key. Simplifying a fraction means reducing it to its smallest form by canceling out any common factors from the numerator and denominator.

To simplify \( \frac{225}{196} \), first find any common factors. If both numbers have a common factor other than 1, you divide them by this common factor. In our case, since 225 and 196 share no common factors (except 1), the fraction is already in its simplest form.

Simplifying helps to make fractions more presentable and easier to work with in further calculations. Being able to identify when a fraction cannot be simplified is just as important.