Problem 123 - Beginning and Intermediate Algebra

Question

Write a numerical expression for each phrase, and simplify the expression. The product of \(-\frac{1}{2}\) and \(\frac{3}{4},\) divided by \(-\frac{2}{3}\)

Step-by-Step Solution

Answer

\( \frac{9}{16} \)

Step 1: Write the expression

Translate the given phrase into a numerical expression. The phrase 'The product of \(-\frac{1}{2}\) and \(\frac{3}{4}\)' can be written as \(\left( -\frac{1}{2} \times \frac{3}{4} \right)\). Then, this product is divided by \(-\frac{2}{3}\), so the full expression becomes \(\left( -\frac{1}{2} \times \frac{3}{4} \right) \div (-\frac{2}{3})\).

Step 2: Calculate the product

First, calculate the product of \(-\frac{1}{2}\) and \(\frac{3}{4}\). \( -\frac{1}{2} \times \frac{3}{4} = \frac{-1 \times 3}{2 \times 4} = \frac{-3}{8} \).

Step 3: Rewrite the division as a multiplication

Instead of dividing by \(-\frac{2}{3}\), multiply by its reciprocal. So, \( \frac{-3}{8} \div ( -\frac{2}{3} ) = \frac{-3}{8} \times -\frac{3}{2} \).

Step 4: Perform the multiplication

Now, perform the multiplication. \( \frac{-3}{8} \times -\frac{3}{2} = \frac{(-3 \times -3)}{(8 \times 2)} = \frac{9}{16} \).

Key Concepts

Multiplying Fractions

When multiplying fractions, follow these simple steps:

First, multiply the numerators (top numbers) together. Then, multiply the denominators (bottom numbers) together.

For example, if we have \(-\dfrac{1}{2}\) and \(\dfrac{3}{4}\), we simply multiply like this:

\[ -\dfrac{1}{2} \times \dfrac{3}{4} = \dfrac{-1 \times 3}{2 \times 4} = \dfrac{-3}{8} \]

Remember to keep the negative sign with you during the entire process.

• Numerators multiplied together: \(-1 \times 3 = -3\)
• Denominators multiplied together: \(2 \times 4 = 8\)

Division of Fractions

To divide fractions, follow a key rule: change the division into multiplication by flipping the second fraction (divisor). This flipped fraction is called the reciprocal.

So, if we have to divide \( \dfrac{-3}{8} \div \left(-\dfrac{2}{3}\right) \), we'll convert division into multiplication like this:

\[ \dfrac{-3}{8} \div \left( -\dfrac{2}{3} \right) = \dfrac{-3}{8} \times -\dfrac{3}{2} \]

Remember to not change the first fraction, only flip (reciprocal) the second one. Now, we just multiply these fractions as learned previously:

• \(-3\) and \(-3\) are the numerators forming \( -3 \times -3 = 9 \)
• \(8\) and \(2\) are the denominators forming \(8 \times 2 = 16\)

So, \[ \dfrac{-3}{8} \times -\dfrac{3}{2} = \dfrac{9}{16} \]

Numerical Expressions

Numerical expressions are mathematical phrases involving numbers and operations. Here, the phrase is: 'The product of \( -\dfrac{1}{2} \) and \( \dfrac{3}{4} \), divided by \( -\dfrac{2}{3} \)'.

We break it down step by step:

• First, we find the product of \( -\dfrac{1}{2} \) and \( \dfrac{3}{4} \): \( -\dfrac{1}{2} \times \dfrac{3}{4} = \dfrac{-3}{8} \)
• Then, divide this result by \( -\dfrac{2}{3} \).

In a single numerical expression, it reads like:
\[ \left( -\dfrac{1}{2} \times \dfrac{3}{4} \right) \div \left( -\dfrac{2}{3} \right) = \dfrac{9}{16} \]

By following the step-by-step process: changing division into multiplication by reciprocal technique of the second fraction and then performing the basic fraction multiplication, you simplify it correctly.