Problem 119 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. $$ 60\left(\frac{3}{20} r-\frac{4}{15}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(9r - 16\).

Step 1: Distribute the 60

Use the distributive property to multiply 60 by each term inside the parentheses. This means distributing the 60 to \( \frac{3}{20} r \) and \( -\frac{4}{15} \). Write the expression as: \[ 60 \cdot \frac{3}{20} r - 60 \cdot \frac{4}{15} \]

Step 2: Simplify Each Multiplication

Calculate each operation separately. For the first term, calculate: \[ 60 \times \frac{3}{20} r = \frac{60 \times 3}{20} r = \frac{180}{20} r = 9r \]For the second term, calculate: \[ 60 \times -\frac{4}{15} = -\frac{60 \times 4}{15} = -\frac{240}{15} = -16 \]

Step 3: Combine the Terms

Combine the results from Step 2 to write the simplified form of the expression. Combine \(9r\) and \(-16\) to get: \[ 9r - 16 \]

Key Concepts

Distributive Property

The distributive property is a simple but powerful tool in algebra. It helps you to multiply a single term with terms inside a set of parentheses. In mathematical terms, it looks like this: a(b + c) = ab + acThis property allows you to distribute, or "hand out," the term outside the parentheses to each term within the parentheses separately.
In our example, we applied the distributive property to the expression 60(\(\frac{3}{20}r - \frac{4}{15}\)). This meant that we multiplied 60 by \(\frac{3}{20}r\) and then 60 by \(-\frac{4}{15}\).
Remember that every number and variable within the expression is treated as a separate entity when distributed. This step makes it easier to simplify the expression further. Using the distributive property correctly is crucial in algebra when simplifying complex expressions.

Expression Simplification

Expression simplification in algebra involves reducing an expression to its simplest form. Every step should aim to clear away complexities while retaining the expression's original value.
In our exercise, after the distributive property was used, there were two expressions to simplify:

• 60 \(\times \frac{3}{20}r\)
• 60 \(\times -\frac{4}{15}\)

By simplifying these calculations, the expression becomes much easier to work with. For the term \(60 \times \frac{3}{20}r\), dividing 60 by 20 first simplifies the multiplication. Similarly, for \(60 \times -\frac{4}{15}\), dividing simplifies the solution further.
This process is essential in making mathematical expressions manageable and accurate. Simplification allows you to focus on the core value of an expression without unnecessary complications.

Multiplying Fractions

Multiplying fractions may look intimidating at first, but it's quite straightforward once you understand the basic principle. It involves multiplying the numerators together and the denominators together.
For example, when we multiplied 60 by \(\frac{3}{20}\), the process was to multiply 60 by the numerator 3, then divide by the denominator 20:

• Calculate 60 \(\times 3\) = 180
• Divide 180 by 20 to get 9

Similarly, when multiplying 60 by \(-\frac{4}{15}\), you multiply as follows:

• 60 \(\times (-4) = -240\)
• Then, \(-240\) divided by 15 gives \(-16\)

Understanding this process helps in quickly figuring out the multiplication results when dealing with fractions. It breaks things down so you can solve complex-looking problems with ease.