Problem 31 - Algebra 1: Concepts and Skills

Question

Find the product. \(15\left(-\frac{2}{15}\right)\left(\frac{3}{4}\right)\)

Step-by-Step Solution

Answer

The result of the product is -1.5

Step 1: Pair and Multiply 15 with -2/15

Pair and multiply 15 with -2/15 (since -2/15 and 15 are reciprocal). This gets: \(15 * (-\frac{2}{15}) = - 2 \)

Step 2: Multiply Result with 3/4

Take the result of the first multiplication (-2) and multiply it with the third number which is 3/4. This gives: \(-2 * (\frac{3}{4}) = -\frac{3}{2} = -1.5 \)

Key Concepts

Understanding Reciprocals

Reciprocals play a crucial role in fraction multiplication. A reciprocal of a number is essentially 1 divided by that number.
For example, the reciprocal of a fraction like \(-\frac{2}{15}\) is obtained by flipping the numerator and denominator, giving \(-\frac{15}{2}\).
Notice in our problem, multiplying \(15\) with \(-\frac{2}{15}\), the reciprocal helps us simplify the multiplication process.

• By definition, two numbers are reciprocals if their product is 1.
• In our exercise, when multiplying \(15\) by \(-\frac{2}{15}\), we essentially find their "pair" or inverse by virtue of reciprocal property.
• The denominator of one is cancelled by the numerator of the other, leading to a simplified product.

Understanding reciprocals minimizes the complexity in multiplying fractions and whole numbers, making calculations more intuitive.

Product of Fractions

When dealing with the product of fractions, the basic operation involves multiplying the numerators and the denominators separately.
In our example, after simplifying \(15 \times -\frac{2}{15}\) to \(-2\), we multiply \(-2\) by \(\frac{3}{4}\).
Here is how it's done neatly:

• Numerator: Multiplying \(-2\) (or \(-2/1\) as a fraction) by \(3\) results in \(-6\).
• Denominator: Multiplying the implied 1 in \(-2\) by 4 gives 4.
• Our result, therefore, is \(-\frac{6}{4}\), which simplifies to \(-\frac{3}{2}\) or \(-1.5\).

Simplification is often necessary in fractional multiplication, ensuring the end result is in its reduced form.

Handling Negative Numbers

Working with negative numbers requires careful attention, especially when combined with fractions.
In multiplication, the rule of thumb is that multiplying two negatives yields a positive result, whereas a negative multiplied by a positive remains negative.
Let’s break it down:

• In our solution, \(15\) multiplied by \(-\frac{2}{15}\) results in \(-2\) because there is only one negative sign involved.
• While further multiplying \(-2\) by positive \(\frac{3}{4}\), the product remains negative. Hence, the outcome is \(-\frac{3}{2}\).
• Keeping track of signs in each step is crucial, as it directly affects the end value.

Attention to the rules about negative numbers ensures precision in mathematical operations and correct interpretation of the final answer.