Problem 16 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(1-34,\) perform the indicated multiplication. $$-\frac{5}{7} \cdot\left(-\frac{3}{8}\right)$$

Step-by-Step Solution

Answer

The result of the multiplication is \(\frac{15}{56}\)

Step 1: Simplify the signs

The multiplication involves two negative fractions. The product of two negative numbers is a positive number. Hence, we can ignore the negative signs for now. The multiplication becomes \(\frac{5}{7} \cdot \frac{3}{8}\)

Step 2: Multiply the fractions

To multiply two fractions, we need to multiply the numerators together and the denominators together. The multiplication would result in \(\frac{5 \times 3}{7 \times 8}\)

Step 3: Simplify the result

Now, multiplying the numerators and the denominators, the result is \(\frac{15}{56}\)

Key Concepts

Simplifying Signs

When multiplying fractions, handling the signs correctly is crucial. In this exercise, we have two negative fractions:

• The first fraction is \(-\frac{5}{7}\).
• The second fraction is \(-\frac{3}{8}\).

The rule for multiplying negative numbers states that a negative times a negative results in a positive. That's why, before even beginning the multiplication of the fractions, we can simplify the signs and rewrite the expression as a simple multiplication of positive fractions:\[\frac{5}{7} \cdot \frac{3}{8}\]This simplification allows us to focus on the numerical calculation without worrying about the signs.
Remember that by simplifying signs upfront, you're able to manipulate the numbers more easily and avoid compounded errors.

Numerator and Denominator

Understanding the numerators and denominators is key to mastering fraction multiplication.

• The numerator is the top part of the fraction, representing how many parts of a whole are considered.
• The denominator is the bottom part, which tells you into how many parts the whole is divided.

In the fraction \(\frac{5}{7}\):

• 5 is the numerator,
• 7 is the denominator.

Similarly, for \(\frac{3}{8}\):

• 3 is the numerator,
• 8 is the denominator.

Understanding these terms helps in the next multiplication step, where numerators and denominators are multiplied separately. It is as if the entire multiplication process happens in two separate spaces, the numerators getting multiplied together and the denominators following suit, shaping the new fraction.

Fraction Multiplication Steps

To multiply fractions, a clear series of steps is followed, ensuring accuracy and ease.
Start by lining the fractions up next to each other:\[\frac{5}{7} \cdot \frac{3}{8}\]The next move is to multiply the numerators:

• Multiply the top numbers: \(5 \times 3 = 15\).

Then, handle the denominators:

• Multiply the bottom numbers: \(7 \times 8 = 56\).

This results in the new fraction:\[\frac{15}{56}\]Always check if the resulting fraction can be simplified. Simplifying involves finding a common divisor of both the numerator and the denominator, then dividing both by that number. However, in this case, 15 and 56 have no common factors other than 1, so it's already in simplest form.
Following these multiplication steps ensures that each part of the fraction is tackled separately, helping maintain clarity and accuracy throughout.