Problem 34 - Fundamentals of Mathematics

Question

For the following problems, find each value. $$\frac{4}{9} \div \frac{6}{15}$$

Step-by-Step Solution

Answer

The result is \(\frac{10}{9}\).

Step 1: Understanding Division of Fractions

To divide two fractions, you multiply the first fraction by the reciprocal of the second fraction. The reciprocal of a fraction is obtained by swapping the numerator and the denominator.

Step 2: Finding the Reciprocal

The reciprocal of \(\frac{6}{15}\) is obtained by swapping its numerator and denominator. Therefore, the reciprocal is \(\frac{15}{6}\).

Step 3: Multiplying by the Reciprocal

Now, multiply the first fraction \(\frac{4}{9}\) by the reciprocal of the second fraction \(\frac{15}{6}\):\[\frac{4}{9} \times \frac{15}{6}\]

Step 4: Multiplying the Numerators and Denominators

Multiply the numerators and the denominators:\[\frac{4 \times 15}{9 \times 6}= \frac{60}{54}\]

Step 5: Simplifying the Fraction

To simplify \(\frac{60}{54}\), find the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 60 and 54 is 6. Divide both the numerator and the denominator by 6:\[\frac{60 \div 6}{54 \div 6} = \frac{10}{9}\]

Key Concepts

Reciprocal

Dividing fractions might feel tricky at first, but there's a handy concept that makes it easier to handle: the reciprocal.

Imagine you have a fraction like \( \frac{6}{15} \). Now, to find its reciprocal, you simply swap the places of the numerator (top number) and the denominator (bottom number). So, the reciprocal of \( \frac{6}{15} \) becomes \( \frac{15}{6} \).

Why do we do this? Well, when you divide by a fraction, you're essentially multiplying by its reciprocal. This switch makes the math a lot easier to follow! The concept of the reciprocal is crucial because it transforms the division problem into a multiplication problem, which is generally easier to solve.

Multiplying Fractions

Once the division problem transforms into a multiplication problem using the reciprocal, the next step is straightforward: multiply the fractions.

Let's look at \( \frac{4}{9} \times \frac{15}{6} \). To multiply fractions, you simply multiply the numerators with each other and the denominators with each other.

So, you take 4 and multiply it by 15, and then take 9 and multiply it by 6. This gives you a new fraction: \( \frac{4 \times 15}{9 \times 6} \). Doing the math, that turns out to be \( \frac{60}{54} \).

The process of multiplying fractions is simple because instead of worrying about common denominators (like in addition and subtraction), you directly handle the numbers as they are.

Simplifying Fractions

The final step in our journey with fractions is to simplify or reduce the fraction to its simplest form. This means making the numbers as small as possible while keeping the value the same.

Take the fraction \( \frac{60}{54} \). At first glance, these are large numbers, but they can be simplified.

To do this, look for the greatest common divisor (GCD). For 60 and 54, the GCD is 6. Now, divide both the numerator and the denominator by this number:

• \( 60 \div 6 = 10 \)
• \( 54 \div 6 = 9 \)

This gives you \( \frac{10}{9} \).

Simplifying ensures the fraction is as easy to work with as possible, and it's always good mathematical practice to present fractions in their simplest form.