Problem 106 - Prealgebra and Introductory Algebra

Question

What is the product of \(-\frac{5}{13},-\frac{26}{75},\) and \(\frac{5}{8} ?\)

Step-by-Step Solution

Answer

The product of \(-\frac{5}{13}, -\frac{26}{75}\), and \(\frac{5}{8} is \frac{13}{156}

Step 1: Multiply the Numerators

The three numerators are -5, -26, and 5. Multiply these three numbers together: \(-5 \times -26 \times 5 = 650\). The product is positive since the multiplication of two negative numbers is a positive number.

Step 2: Multiply the Denominators

The three denominators are 13, 75, and 8. Multiply these three numbers together: \(13 \times 75 \times 8 = 7800\)

Step 3: Write the Result as a Fraction

The final step is to express the result as a fraction. So the product of \(-\frac{5}{13},-\frac{26}{75}, and \frac{5}{8} is \frac{650}{7800}\), which simplifies to \(\frac{13}{156}\).

Key Concepts

Multiplication of Numerators

When multiplying fractions, the first important step is to focus on their numerators. Just like in our exercise, you take all the numerators of the fractions involved. In this scenario, we have \(-5\), \(-26\), and \(5\). Multiplying them seems daunting at first, but it's simpler if broken down.

• Start with the first two: \(-5 \times -26 = 130\). Remember, multiplying two negative numbers always results in a positive product.
• Then, multiply this result by the third numerator: \(130 \times 5 = 650\).

This way, you get the cumulative product of the numerators. The sign of your product depends on the number of negative numerators you're multiplying. An even number of negative signs yields a positive result, while an odd amount would lead to a negative result. In this particular exercise, our product is positive because we dealt with two negative numbers followed by one positive.

Multiplication of Denominators

Next, let's look at the denominators. Handling them is very similar to working with numerators. The denominators in our fractions are \(13\), \(75\), and \(8\). The process remains just as straightforward:

• Multiply the first two: \(13 \times 75 = 975\).
• Now, take this result and multiply by the third denominator: \(975 \times 8 = 7800\).

This approach ensures you've multiplied all parts of your fractions correctly. You'll notice that when multiplying denominators, the operation doesn't require attention to positive or negative signs, because denominators are typically positive in fraction operations. This process gives us the total product of the denominators, forming the bottom of our resulting fraction.

Simplifying Fractions

Once you have the product of the numerators and the denominators, the final part of solving fraction multiplication is simplification. Our product from the exemplified fractions is \(\frac{650}{7800}\). Simplifying essentially means finding equivalent fractions by reducing the fraction to its simplest form.Begin by identifying the greatest common divisor (GCD) of both the numerator and the denominator:

• Here, the GCD of \(650\) and \(7800\) is \(50\).

Use this to divide both parts of the fraction:

• \(\frac{650 \div 50}{7800 \div 50} = \frac{13}{156}\).

In this more reduced form, \(\frac{13}{156}\), you've completed the simplification. This fraction can't be reduced any further because the greatest common divisor between \(13\) and \(156\) is \(1\). Simplifying fractions not only makes your answers cleaner but often helps in spotting any errors in earlier calculations.