Problem 6 - Fundamentals of Mathematics

Question

Perform the following multiplications. $$ \frac{5}{6} \cdot \frac{7}{8} $$

Step-by-Step Solution

Answer

The product is \( \frac{35}{48} \).

Step 1: Multiply the Numerators

To find the product of two fractions, multiply the numerators (the numbers on top) together. Here, the numerators are 5 and 7. So, multiply them: \[ 5 \times 7 = 35 \]

Step 2: Multiply the Denominators

Next, we multiply the denominators (the numbers on the bottom) of the fractions. In this problem, the denominators are 6 and 8. So, multiply them: \[ 6 \times 8 = 48 \]

Step 3: Write the Fraction Form

Now, write down the new fraction using the results from Step 1 and Step 2. The product of the fractions \( \frac{5}{6} \cdot \frac{7}{8} \) is \( \frac{35}{48} \).

Step 4: Simplify the Fraction (If Possible)

Check if the fraction \( \frac{35}{48} \) can be simplified by finding the Greatest Common Divisor (GCD) of 35 and 48. Since the GCD is 1, \( \frac{35}{48} \) is already in its simplest form.

Key Concepts

Numerator Multiplication

When you multiply fractions, your first step is to focus on the numerators, which are the top numbers of the fractions. In our example, the numerators are 5 and 7. Multiplying these together is just like multiplying any two whole numbers:

• Think of it as multiplying two separate items of your pizza order: 5 pies and 7 single slices each.
• Calculate: \[5 \times 7 = 35\]This results in the top part of our new fraction.

Remember, the numerator's product gives us part of the answer, but we can't stop here. We have to do more! When dealing with multiplying fractions, it's key to deal separately with numerators and denominators before putting it all together.

Denominator Multiplication

Once you’ve multiplied the numerators, a similar approach is needed for the denominators—the numbers on the bottom of the fractions. Think of the denominators as describing the size of each slice in a pie. In our case, they are 6 and 8.

• Consider it like determining how many pieces you're cutting each pie, so a smaller slice in essence.
• Multiply these numbers: \[6 \times 8 = 48\]This product forms the bottom portion of our new fraction.

So far, you have formed a new fraction of \(\frac{35}{48}\). This means you have described another size or measure of pieces in the pie. But wait, there's more! Before this can be our final answer, we need to simplify.

Simplifying Fractions

After finding the fraction from multiplying numerators and denominators, simplifying it is the next step to ensure clarity. Simplification involves reducing the fraction to its simplest form. The goal here is to make it as understandable as possible.To simplify \(\frac{35}{48}\), you must find a number that evenly divides both the numerator and the denominator, ideally the greatest common divisor (GCD).

• For 35 and 48, the greatest factor is 1.
• When the greatest common factor is 1, it means the fraction is already simplified.

Thus, \(\frac{35}{48}\) remains as it is, and you are done with simplifying. This last step is critical; it ensures that every fraction is presented in the neatest, most straightforward way possible, making it easier to understand the size or portion it represents.