Problem 328 - Intermediate Algebra

Question

In the following exercises, simplify. $$ 20\left(\frac{3}{5} q\right) $$

Step-by-Step Solution

Answer

12q

Step 1 - Distribute the 20

Multiply 20 by the term inside the parentheses. So, 20 is multiplied by the fraction \(\frac{3}{5} q\).

Step 2 - Multiply 20 with \( \frac{3}{5} \)

To multiply 20 by the fraction \( \frac{3}{5} \), multiply 20 by 3 and then divide by 5. This gives \( 20 \times \frac{3}{5} = \frac{20 \times 3}{5} = 12 \).

Step 3 - Include the variable q

Since the term inside the parentheses includes q, the result after multiplication is \( 12q \).

Key Concepts

distributive property

The distributive property is a fundamental principle in algebra. It lets you multiply a sum by multiplying each addend separately and then adding the products. In simple terms, if you have a multiplication of a number outside a parenthesis with an expression inside, you distribute the multiplication to each term inside. For example, in the exercise, we had 20 multiplied by \(\frac{3}{5} q\). We distributed the 20 to the \(\frac{3}{5}\) first. \(

\)This is expressed as: \[a(b + c) = ab + ac\]. By applying this property, you simplify complex multiplication tasks and make calculations easier.

multiplying fractions

Multiplying fractions might seem tricky, but it’s easier than you think. Basically, you multiply the numerators (top numbers) together and the denominators (bottom numbers) together. In the exercise step, we multiplied 20 with \(\frac{3}{5}\). Step-by-step, it looks like this:

• First, you multiply the whole number 20 by 3, giving you 60.


• Then, you take that result and divide by 5.

So
<\[ 20 \times \frac{3}{5} = \frac{20 \times 3}{5} = \frac{60}{5} = 12 \]. This example shows that when multiplying fractions with whole numbers, breaking it down into parts makes it simple to solve.

variables in algebra

Variables are symbols used to represent numbers in algebra, often letters like x, y, or q. They allow you to write expressions that can represent different values. In our exercise, the variable q was included in the parenthesis. When we simplified 20 times \(\frac{3}{5} q\), we ended up with 12 multiplied by q.

This means our final answer is 12q. Understanding that variables hold the place of numbers helps you manipulate and solve equations more easily. Always remember to include them in your final simplified expressions.