Problem 11 - Prealgebra

Question

Find each product. Use an area model if necessary. $$-5 \frac{1}{3} \cdot-3 \frac{3}{8}$$

Step-by-Step Solution

Answer

The product is 14.

Step 1: Convert Mixed Numbers to Improper Fractions

To simplify the multiplication, we first convert the mixed numbers into improper fractions. For \(-5 \frac{1}{3}\), multiply the whole number \(-5\) by the denominator \(3\) and add the numerator \(1\). This gives \(-15 + 1 = -16\), and the fraction is \(-\frac{16}{3}\). Similarly, for \(-3 \frac{3}{8}\), multiply \(-3\) by \(8\), and add \(3\) to get \(-24 + 3 = -21\), so the fraction is \(-\frac{21}{8}\).

Step 2: Multiply the Fractions

Now, multiply the two improper fractions: \(-\frac{16}{3} \times -\frac{21}{8}\). Multiply the numerators to get \((-16) \times (-21) = 336\), and multiply the denominators to get \(3 \times 8 = 24\). Therefore, the product is \(\frac{336}{24}\).

Step 3: Simplify the Fraction

The fraction \(\frac{336}{24}\) can be simplified by finding the greatest common divisor (GCD) of 336 and 24, which is 24. Divide the numerator and denominator by 24: \(\frac{336 \div 24}{24 \div 24} = \frac{14}{1}\). So the simplified product is 14.

Key Concepts

Mixed Numbers

Mixed numbers are a combination of a whole number and a fraction. They are useful in representing quantities that are not whole numbers. For example, in a recipe, you might need 2 1/2 cups of flour. Here, "2 1/2" is a mixed number. To operate mathematically with mixed numbers, you often need to convert them into improper fractions. This is done by multiplying the whole number by the denominator and then adding the numerator.

In the given problem, we have mixed numbers \(-5 \frac{1}{3}\) and \(-3 \frac{3}{8}\). Both these numbers are negative, which only affects the overall sign of the result in multiplication.

Improper Fractions

Improper fractions are fractions where the numerator is greater than or equal to the denominator. When converting mixed numbers, you'll often turn them into improper fractions.

For instance, to convert \(-5 \frac{1}{3}\) into an improper fraction, multiply the whole number \(-5\) by the denominator 3, giving \(-15\), and then add the numerator 1 to get \(-16\). This transforms the mixed number into \(-\frac{16}{3}\).

This step is crucial as it simplifies the multiplication process, making it straightforward to multiply across fractions rather than dealing with mixed numbers directly.

Fraction Multiplication

Multiplying fractions is simpler than it may first appear. You multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. In our problem, we have \(-\frac{16}{3} \times -\frac{21}{8}\).

Multiplying the numerators, \((-16) \times (-21)\), gives 336. Remember, multiplying two negative numbers results in a positive number. Meanwhile, for the denominators, \(3 \times 8\) equals 24.

As fractions are essentially another form of division, this entire operation results in a new fraction: \(\frac{336}{24}\). This fraction is not in its simplest form yet, which brings us to our next step.

Simplifying Fractions

Simplifying fractions involves reducing them to their lowest terms so they are easier to understand and work with. To simplify a fraction, you divide the numerator and the denominator by their greatest common divisor (GCD).

For the fraction \(\frac{336}{24}\), calculate the GCD, which is 24. Next, divide both the numerator and the denominator by 24, yielding \(\frac{14}{1}\). Since any number over 1 remains itself, the simplified product is simply 14.

Simplifying is important as it makes the result more comprehensible and aesthetically pleasing.