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Chapter 7: Problem 77

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

Short Answer

Expert verified
12q

Step by step solution

01

Distribute the 20

Multiply 20 by the fraction inside the parentheses. This is done by multiplying 20 by the numerator of the fraction.
02

Perform the multiplication

Calculate the product of 20 and 3: \[ 20 \times \frac{3}{5} = \frac{20 \times 3}{5} \]
03

Simplify the fraction

Simplify \( \frac{20 \times 3}{5} \): \[ \frac{60}{5} \]
04

Final simplification

Divide 60 by 5 to obtain the final answer: \[ \frac{60}{5} = 12q \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
To solve algebraic expressions like the given problem, we often use the distributive property.
The distributive property helps in simplifying expressions by distributing the multiplication over addition or subtraction inside parentheses.
This property is defined as: \texttt{a(b + c) = ab + ac}.
Notice how we distributed the number 20 to the term inside the parentheses \(\frac{3}{5} q\).
This is a critical first step in simplifying the expression.
Fractions Multiplication
Understanding fractions multiplication is the next crucial concept. Multiplying fractions involves straightforward steps:
  • Multiply the numerators (the numbers on top).
  • Multiply the denominators (the numbers on the bottom).

In our example, when we multiplied 20 by the entire fraction \(\frac{3}{5}\), we essentially performed two multiplications:
20 times 3, and then dividing the result by 5. This gave us \( \frac{20 \times 3}{5} \).
Fraction Simplification
Simplifying a fraction involves reducing it to its simplest form.
To simplify, look for the greatest common divisor (GCD) of the numerator and the denominator and divide both by that number.
In our example, \( \frac{60}{5} \), the fraction reduced directly to 12 because 60 divided by 5 equals 12.
Simplification makes the final answers cleaner and easier to understand.
Algebraic Expressions
Solving algebraic expressions involves operations on variables and constants together.
For our exercise, the variable was q, and we focused on manipulating the coefficients (numbers) attached to it.
The goal is to keep our variable intact while simplifying the numerical part. The solution led us to finalize the expression as 12q.
This process of breaking down and simplifying helps in understanding and solving more complex algebraic equations in the future.

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