Problem 14 - Fundamentals of Mathematics

Question

Perform the following divisions. $$ \frac{5}{6} \div \frac{2}{3} \cdot \frac{8}{25} $$

Step-by-Step Solution

Answer

The result is \( \frac{2}{5} \).

Step 1: Rewrite the Division as Multiplication

First, remember that dividing by a fraction is the same as multiplying by its reciprocal. Rewrite the division as \( \frac{5}{6} \times \frac{3}{2} \cdot \frac{8}{25} \).

Step 2: Multiply the Fractions

Multiply the fractions: \( \frac{5}{6} \times \frac{3}{2} \) first. This is equal to \( \frac{5 \times 3}{6 \times 2} = \frac{15}{12} \).

Step 3: Simplify the First Product

Simplify the fraction \( \frac{15}{12} \) by finding the greatest common divisor of 15 and 12, which is 3. Thus, \( \frac{15}{12} = \frac{5}{4} \).

Step 4: Multiply with the Last Fraction

Now, multiply the result \( \frac{5}{4} \) by \( \frac{8}{25} \): \( \frac{5}{4} \times \frac{8}{25} = \frac{5 \times 8}{4 \times 25} = \frac{40}{100} \).

Step 5: Simplify the Final Result

Simplify \( \frac{40}{100} \) by finding the greatest common divisor of 40 and 100, which is 20. Thus, \( \frac{40}{100} = \frac{2}{5} \).

Key Concepts

Understanding the Reciprocal

The reciprocal of a number is essentially what you multiply that number by to get 1. This is a key concept in fraction division. When dealing with fractions, we flip the numerator and the denominator to find the reciprocal. For example, the reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \). It’s important to remember that division by a fraction is the same as multiplying by its reciprocal. This trick simplifies problems greatly.

• To find the reciprocal of a fraction, interchange the numerator and the denominator.
• The product of a number and its reciprocal always equals 1.

This method allows us to rewrite division problems as multiplication problems, making calculations easier.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. This happens when the numerator and the denominator are divided by their greatest common divisor (GCD). For example, consider the fraction \( \frac{15}{12} \).

• First, determine the GCD of the numerator and the denominator. In this case, the GCD of 15 and 12 is 3.
• Then, divide both the numerator and the denominator by this GCD to obtain the simplest form: \( \frac{15 \div 3}{12 \div 3} = \frac{5}{4} \).

Simplification helps make answers neater and easier to understand, and it’s an essential step in most fraction problems.

Multiplying Fractions

Once fractions are simplified, multiplying them becomes straightforward. Just multiply the numerators together and the denominators together. For instance, consider multiplying \( \frac{5}{4} \) by \( \frac{8}{25} \). Multiply the numerators: 5 and 8, and the denominators: 4 and 25.

• Calculate \( 5 \times 8 = 40 \).
• Calculate \( 4 \times 25 = 100 \).
• The product is \( \frac{40}{100} \).

After obtaining the product, simplify the resulting fraction if possible. By simplifying \( \frac{40}{100} \), we find it reduces to \( \frac{2}{5} \).
This process is repetitive and thus becomes easier with practice. Remember always to simplify your answers!