Problem 40 - Fundamentals of Mathematics

Question

What part of \(\frac{8}{9}\) is \(\frac{3}{5}\) ?

Step-by-Step Solution

Answer

\( \frac{3}{5} \) is \( \frac{27}{40} \) of \( \frac{8}{9} \).

Step 1: Understand the Question

The question asks us to find what part of \( \frac{8}{9} \) is \( \frac{3}{5} \). This means we need to determine a fraction that represents how much \( \frac{3}{5} \) fits into \( \frac{8}{9} \).

Step 2: Set Up the Division

To find what part one fraction is of another, divide the first fraction by the second. Set up the division as follows: \[ \frac{3}{5} \div \frac{8}{9} \].

Step 3: Convert Division to Multiplication

Dividing by a fraction is the same as multiplying by its reciprocal. Therefore, convert the division into a multiplication: \[ \frac{3}{5} \times \frac{9}{8} \].

Step 4: Multiply the Fractions

Multiply the numerators and the denominators together: \[ \frac{3 \times 9}{5 \times 8} = \frac{27}{40} \].

Step 5: Simplify the Fraction (if necessary)

Check if the result \( \frac{27}{40} \) can be simplified. Since 27 and 40 have no common factors other than 1, this fraction is already in its simplest form.

Step 6: Conclusion

The fraction \( \frac{3}{5} \) is \( \frac{27}{40} \) of \( \frac{8}{9} \).

Key Concepts

Division of Fractions

When trying to determine what part one fraction is of another, we use division of fractions. This involves dividing the first fraction by the second. Think of it as trying to fit one fraction into another.
Let's say you want to find out what part of \( \frac{8}{9} \) is \( \frac{3}{5} \). The first step involves setting up a division equation where the dividend is the fraction that needs to be "fit" (\( \frac{3}{5} \)), and the divisor is the fraction you are comparing it to (\( \frac{8}{9} \)). This can be written as \( \frac{3}{5} \div \frac{8}{9} \).
Dividing fractions might sound tricky, but once you know the method, it becomes straightforward. You can transform the division into an easier multiplication task. This is where the concept of a reciprocal comes into play.

Fraction Multiplication

Multiplying fractions is a handy tool when dealing with divisions of fractions. Instead of dividing by a fraction, we multiply by its reciprocal. The reciprocal of a fraction simply means flipping its numerator and denominator.
In our exercise, to solve \( \frac{3}{5} \div \frac{8}{9} \), you'll change the operation from division to multiplication by using the reciprocal of \( \frac{8}{9} \), which is \( \frac{9}{8} \). So this becomes \( \frac{3}{5} \times \frac{9}{8} \).
When multiplying fractions, remember these steps:

• Multiply the numerators together. Here, it is \( 3 \times 9 = 27 \).
• Multiply the denominators together. Here, it is \( 5 \times 8 = 40 \).

The result is \( \frac{27}{40} \). This multiplication process is straightforward as long as you remember to multiply across numerators and denominators.

Simplifying Fractions

Simplifying fractions is an essential step to ensure your result is as neat and simple as possible. A fraction is simplified or reduced to its simplest form when its numerator and denominator have no common factors other than 1.
In our solved problem, the result of the multiplication gives us \( \frac{27}{40} \). To simplify a fraction:

• Check if both the numerator and the denominator have common factors. You can often do this by testing smaller factors.
• If there are no common factors (other than 1), the fraction is already in its simplest form.

For the fraction \( \frac{27}{40} \), since 27 and 40 share no common factors, they cannot be divided by any common number to simplify further. That means \( \frac{27}{40} \) is the simplest form of the fraction resulting from our operation.