Problem 63 - Fundamentals of Mathematics

Question

For the following problems, find each value. $$\frac{4}{15} \div \frac{2}{25} \cdot \frac{9}{10}$$

Step-by-Step Solution

Answer

The value is 3.

Step 1: Convert Division into Multiplication

When dividing by a fraction, the operation can be converted to multiplication by taking the reciprocal of the divisor. In this case, convert \( \frac{4}{15} \div \frac{2}{25} \) into \( \frac{4}{15} \times \frac{25}{2} \). The expression now becomes \( \frac{4}{15} \times \frac{25}{2} \times \frac{9}{10} \).

Step 2: Multiply the Fractions Together

To multiply fractions, multiply the numerators together and the denominators together. Start by multiplying \( \frac{4}{15} \times \frac{25}{2} \). The numerators are \( 4 \times 25 = 100 \) and the denominators are \( 15 \times 2 = 30 \). Therefore, \( \frac{4}{15} \times \frac{25}{2} = \frac{100}{30} \).

Step 3: Multiply the Result with the Third Fraction

Now multiply the result \( \frac{100}{30} \) with \( \frac{9}{10} \). The numerators are \( 100 \times 9 = 900 \) and the denominators are \( 30 \times 10 = 300 \). Therefore, \( \frac{100}{30} \times \frac{9}{10} = \frac{900}{300} \).

Step 4: Simplify the Resulting Fraction

Simplify \( \frac{900}{300} \) by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 300. \( \frac{900 \div 300}{300 \div 300} = \frac{3}{1} \). Thus, the fraction simplifies to 3.

Key Concepts

division of fractions

Dividing fractions may seem challenging at first, but it follows a simple rule that makes it quite easy. When you see a division of fractions, like \(\frac{a}{b} \div \frac{c}{d}\), you can rewrite this operation as a multiplication by using the reciprocal of the second fraction.

• The reciprocal of a fraction \(\frac{c}{d}\) is \(\frac{d}{c}\).
• So, \(\frac{a}{b} \div \frac{c}{d}\) becomes \(\frac{a}{b} \times \frac{d}{c}\).

In our example with \(\frac{4}{15} \div \frac{2}{25}\), by swapping \(\frac{2}{25}\) for its reciprocal \(\frac{25}{2}\), we change the operation to multiplication: \(\frac{4}{15} \times \frac{25}{2}\).
This gives us a clearer and simpler way to carry on with the fraction arithmetic.

multiplication of fractions

Once you've turned your division into multiplication, working with fractions gets straightforward. Multiplying fractions, such as \(\frac{a}{b} \times \frac{c}{d}\), involves two steps:

• Multiply the numerators together: \(a \times c\).
• Multiply the denominators together: \(b \times d\).

The product of the numerators over the product of the denominators forms the resulting fraction.
In our example, we start with \(\frac{4}{15} \times \frac{25}{2}\).

• First, multiply the numerators: \(4 \times 25 = 100\).
• Then, multiply the denominators: \(15 \times 2 = 30\).

This results in \(\frac{100}{30}\).
Next, multiply this result by \(\frac{9}{10}\):

• Numerators: \(100 \times 9 = 900\).
• Denominators: \(30 \times 10 = 300\).

And we end up with \(\frac{900}{300}\), ready to be simplified.

simplifying fractions

Once you've multiplied your fractions and reached a result like \(\frac{900}{300}\), you'll want to simplify it. Simplifying fractions makes them easier to understand and use.To simplify a fraction:

• Identify the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCD.

For \(\frac{900}{300}\), the GCD is 300.
So, divide the numerator by 300 and the denominator by 300:

• Numerator: \(900 \div 300 = 3\).
• Denominator: \(300 \div 300 = 1\).

Thus, \(\frac{900}{300}\) simplifies to \(\frac{3}{1}\), or just 3.
Simplification helps in presenting your final answer in the most reduced form possible, making it clear and uncomplicated.