Problem 97 - Algebra 1: Concepts and Skills

Question

Divide. Write the answer in simplest form. $$ 1 \frac{1}{2} \div \frac{3}{4} $$

Step-by-Step Solution

Answer

The answer in simplest form is 2.

Step 1: Convert the Mixed Fraction to an Improper Fraction

The mixed number 1 1/2 can be converted into an improper fraction. An improper fraction is a fraction where the numerator is greater than the denominator. The mixed number 1 1/2 can be converted to the improper fraction 3/2 by multiplying the whole number part by the denominator of the fractional part and adding the numerator of the fractional part, i.e. (1*2)+1 = 3.

Step 2: Change Division into Multiplication and Flip the Second Fraction

In fraction division, the problem is solved by changing the division operation into a multiplication and then flipping (finding the reciprocal of) the second fraction. For the given exercise, change 1 1/2 / 3/4 to 3/2 * 4/3.

Step 3: Multiply the Fractions and Simplify

Multiplication of fractions is done by multiplying the numerators together and the denominators together. So 3/2 * 4/3 becomes (3*4)/(2*3) which equals 12/6. We simplify the fraction to its lowest terms by dividing both the numerator and the denominator by their greatest common divisor. Simplifying 12/6 leads to 2.

Key Concepts

Improper Fraction

When working with fractions, you might encounter a type called an **improper fraction**. This is when the numerator (the top number) is greater than the denominator (the bottom number). But why do we use them?

• Improper fractions make calculations, like division or multiplication, more straightforward.
• They're especially useful when dealing with mixed numbers, which include both a whole number and a fraction part, such as \(1 \frac{1}{2}\).

To convert a mixed number into an improper fraction, you multiply the whole number by the denominator of the fraction and then add the numerator. For example, with \(1 \frac{1}{2}\), you calculate \((1 \times 2) + 1\) to get the improper fraction \(\frac{3}{2}\). This transformation simplifies operations, making it easier to handle the fraction in mathematical problems.

Simplifying Fractions

Simplifying fractions is all about reducing the fraction to its simplest form, where the numerator and denominator have no common factors other than 1. But how is it done?

• First, you need to find the greatest common divisor (GCD) of the numerator and the denominator.
• Then, you divide both the numerator and the denominator by this GCD.

For instance, if you end up with a fraction like \(\frac{12}{6}\), you check which number is the largest that can divide both 12 and 6. Here, the GCD is 6. Therefore, divide both 12 and 6 by 6 to get \(\frac{2}{1}\), which simplifies to 2. This process ensures that the fraction is in its simplest form, easy to read and work with.

Reciprocal of a Fraction

Understanding the **reciprocal of a fraction** is crucial for operations like fraction division. The reciprocal of a fraction \(\frac{a}{b}\) is just \(\frac{b}{a}\), essentially flipping the numerator and the denominator. But why is this important?

• It's particularly useful because dividing by a fraction is the same as multiplying by its reciprocal.
• Using reciprocals transforms a division operation into a multiplication, simplifying the task since multiplication is usually more straightforward.

For example, with the fraction division problem \(1 \frac{1}{2} \div \frac{3}{4}\), you convert \(1 \frac{1}{2}\) to \(\frac{3}{2}\), then take the reciprocal of \(\frac{3}{4}\) to get \(\frac{4}{3}\). Instead of division, you now multiply: \(\frac{3}{2} \times \frac{4}{3}\). This method is consistent and reliable, especially when dealing with more complex fraction operations.