Question

Divide the following fractions and mixed numbers. Reduce to lowest terms. \(\frac{3}{10} \div \frac{5}{25}=\) ______

Short answer

\(\frac{3}{2}\)

Step-by-step solution

Understand Division of Fractions

When dividing two fractions, you multiply the first fraction by the reciprocal of the second fraction. The reciprocal of a fraction is obtained by swapping its numerator and denominator. Therefore, \(\frac{3}{10} \div \frac{5}{25}\) becomes \(\frac{3}{10} \times \frac{25}{5}\).

Multiply the Fractions

Multiply the numerators and then multiply the denominators. Here, the numerators are \(3\) and \(25\), and the denominators are \(10\) and \(5\). Thus, \(\frac{3}{10} \times \frac{25}{5} = \frac{3 \times 25}{10 \times 5}\).

Calculate the New Numerator and Denominator

Calculating the new numerator, \(3 \times 25 = 75\). Calculating the new denominator, \(10 \times 5 = 50\). This gives us \(\frac{75}{50}\).

Simplify the Fraction

To simplify the fraction \(\frac{75}{50}\), find the greatest common divisor (GCD) of 75 and 50, which is 25. Divide both the numerator and the denominator by 25. \(\frac{75 \div 25}{50 \div 25} = \frac{3}{2}\).

Write in Final Form

The fraction is already simplified. Thus, \(\frac{3}{10} \div \frac{5}{25} = \frac{3}{2}\) in its lowest terms.

Key concepts

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form so they are easier to work with. To simplify a fraction, determine its greatest common divisor (GCD). The GCD is the largest number that can divide both the numerator and the denominator without leaving a remainder.

For instance, to simplify the fraction \(\frac{75}{50}\), we first find the GCD of 75 and 50, which is 25.

• Divide 75 by 25 to get 3.
• Divide 50 by 25 to get 2.

This results in the fraction \(\frac{3}{2}\), which is in its simplest form. This step is vital because it reduces the fraction to its lowest terms, making it easier to interpret or compare with other fractions.

Simplifying fractions not only helps in basic arithmetic but is also essential in algebra, probability, and many other areas of mathematics.

Reciprocal of Fractions

The reciprocal of a fraction is simply what you get when you swap its numerator and denominator. Understanding this is crucial for fraction division. To divide by a fraction, multiply by its reciprocal. This is because dividing by a fraction is equivalent to multiplying by its inverse.

For example, the reciprocal of \(\frac{5}{25}\) is \(\frac{25}{5}\). When solving \(\frac{3}{10} \div \frac{5}{25}\), you flip \(\frac{5}{25}\) to get \(\frac{25}{5}\). Now, change the division into multiplication: \(\frac{3}{10} \times \frac{25}{5}\).

Multiplying by the reciprocal transforms the division problem into a multiplication problem, which is usually easier to solve. Understanding reciprocals is important not only in fractions but also in solving equations and understanding ratios.

Greatest Common Divisor

The greatest common divisor (GCD), sometimes called the greatest common factor (GCF), is the largest number that divides two numbers with no remainder. Finding the GCD is a critical step in simplifying fractions because it helps to reduce the numbers to their simplest form.

To find the GCD of two numbers, list out the factors of each number and identify the largest factor that they have in common. For example:

• The factors of 75 are 1, 3, 5, 15, 25, and 75.
• The factors of 50 are 1, 2, 5, 10, 25, and 50.

Both numbers share the factor 25, which is the greatest among them. Therefore, the GCD of 75 and 50 is 25.

Using the GCD allows you to divide both the numerator and the denominator of a fraction to simplify it effectively. Understanding GCDs is important not only in simplifying fractions but also in solving problems involving ratios, percentages, and number theory.