Problem 42 - Prealgebra

Question

Find each quotient. Write in simplest form. $$\frac{10}{3 x} \div \frac{5}{2 x}$$

Step-by-Step Solution

Answer

The quotient is \( \frac{4}{3} \).

Step 1: Rewrite the Division as Multiplication

When you divide by a fraction, it's equivalent to multiplying by its reciprocal. Given the expression \( \frac{10}{3x} \div \frac{5}{2x} \), we can rewrite this as a multiplication by flipping the second fraction: \( \frac{10}{3x} \times \frac{2x}{5} \).

Step 2: Simplify the Expression

Next, multiply the numerators together and the denominators together: \( \frac{10 \times 2x}{3x \times 5} \). This results in \( \frac{20x}{15x} \).

Step 3: Cancel Common Factors

Both the numerator and the denominator contain \( x \), which can be cancelled out. This simplifies to \( \frac{20}{15} \).

Step 4: Simplify the Fraction

Simplify \( \frac{20}{15} \) by dividing the numerator and the denominator by their greatest common divisor, which is 5. This gives \( \frac{4}{3} \).

Key Concepts

Simplifying Fractions

Simplifying a fraction means reducing it to its simplest form, where the numerator and denominator no longer share any common factors other than one. It helps in making fractions easier to work with and understand. Every step of simplification brings the fraction to its most basic form.
To simplify a fraction, follow these steps:

• Identify the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.
• Repeat the process until the fraction cannot be simplified further.

For example, to simplify \( \frac{20}{15} \):

• Find the GCD of 20 and 15. In this case, it is 5.
• Divide both 20 and 15 by 5.
• The simplified form is \( \frac{4}{3} \).

By simplifying fractions, you ensure that calculations are more efficient and results are presented as clearly as possible.

Multiplying Fractions

Multiplying fractions is an essential arithmetic skill that can be more intuitive than it seems at first. When you multiply fractions, you simply multiply their numerators and denominators.
Here's a step-by-step guide to multiply fractions:

• List both fractions you want to multiply.
• Multiply the numerators (the top numbers) together to get the new numerator.
• Multiply the denominators (the bottom numbers) together to get the new denominator.
• Simplify the resulting fraction if necessary.

In the expression \( \frac{10}{3x} \times \frac{2x}{5} \):

• Multiply the numerators: 10 and 2x, resulting in 20x.
• Multiply the denominators: 3x and 5, resulting in 15x.
• The result before simplification is \( \frac{20x}{15x} \).

This brings us eventually to simplifying the result and getting a clean answer, emphasizing the practical application in various mathematical problems.

Reciprocal of a Fraction

Understanding the reciprocal of a fraction is crucial because it is key to dividing fractions. A reciprocal flips the numerator and the denominator of a fraction.

• If you have a fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \).
• The product of a fraction and its reciprocal is always 1, i.e., \( \frac{a}{b} \times \frac{b}{a} = 1 \).

Using reciprocals in division:

• To divide by a fraction, multiply by its reciprocal instead. This is because division by a certain number is equivalent to multiplying by its reciprocal.
• For example, in \( \frac{10}{3x} \div \frac{5}{2x} \), you would multiply by the reciprocal of \( \frac{5}{2x} \), which is \( \frac{2x}{5} \).

This method simplifies the process of solving division problems involving fractions, making it more straightforward and efficient.