Problem 55 - Algebra 1

Question

$$ \frac{6}{5} \div \frac{3}{10} $$

Step-by-Step Solution

Answer

The result of the division \(\frac{6}{5} \div \frac{3}{10}\) is \(4\).

Step 1: Identify the two fractions and the operation

The first fraction is \(\frac{6}{5}\) and the second fraction is \(\frac{3}{10}\). The operation to be completed is division.

Step 2: Find the reciprocal of the second fraction

To find the reciprocal of a fraction, simply interchange the numerator and denominator. The reciprocal of \(\frac{3}{10}\) is \(\frac{10}{3}\).

Step 3: Multiply the first fraction by the reciprocal of the second fraction

Now, instead of dividing, one is going to multiply the first fraction \(\frac{6}{5}\) by the reciprocal of the second fraction \(\frac{10}{3}\). Write it down as: \(\frac{6}{5} \times \frac{10}{3}\).

Step 4: Solve the multiplication problem

To solve the multiplication problem, multiply the numerators together to get the numerator of the result, and then multiply the denominators together to get the denominator of the result. This yields: \(\frac{6 \times 10}{5 \times 3} = \frac{60}{15}\).

Step 5: Simplify the resulting fraction

The fraction \(\frac{60}{15}\) simplifies to \(4\). This is done by dividing both the numerator and the denominator of the fraction by their greatest common divisor (GCD), which in this case is \(15\).

Key Concepts

Reciprocal of a Fraction

Understanding the reciprocal of a fraction is a vital part of dividing fractions. When you see a fraction, like \( \frac{3}{10} \), and you need to find its reciprocal, think of flipping the fraction upside down. This means you swap the numerator and the denominator. So, the reciprocal of \( \frac{3}{10} \) becomes \( \frac{10}{3} \).

• Take the original fraction \( \frac{a}{b} \)
• Flip to get \( \frac{b}{a} \)

The purpose of finding a reciprocal is to transform the division of fractions into multiplication. This makes the calculation easier because multiplying fractions is more straightforward than dividing them directly. Just remember: flip the second fraction, multiply, and simplify.

Multiplying Fractions

After finding the reciprocal, the division problem changes into a multiplication task. Let's take our example with the fractions: we initially had \( \frac{6}{5} \div \frac{3}{10} \). Applying the reciprocal makes us multiply \( \frac{6}{5} \) by \( \frac{10}{3} \).
When multiplying fractions:

• Multiply the numerators: 6 times 10, which equals 60.
• Multiply the denominators: 5 times 3, which equals 15.

Once multiplied, the result is a new fraction, \( \frac{60}{15} \). This process of multiplying helps in dealing with fractional division, turning it into a simpler and less error-prone operation.

Simplifying Fractions

Simplifying fractions is an essential step that comes after you've multiplied them. This means reducing the fraction to its smallest equivalent form. For the fraction \( \frac{60}{15} \), simplifying involves using the greatest common divisor (GCD).

• Identify the GCD of the numerator and denominator. For 60 and 15, the GCD is 15.
• Divide both the numerator and denominator by this number: \( \frac{60}{15} \div \frac{15}{15} \).
• You're left with the simplest form of the fraction: 4.

Simplification makes the resulting fraction easy to understand and use. It also ensures that your answer is neat and precise, especially in contexts where further calculations might be needed.