Problem 20 - Elementary Algebra

Question

For the following problems, use the order of operations to find each value. $$\frac{8(6+20)}{8}+\frac{3(6+16)}{22}$$

Step-by-Step Solution

Answer

Question: Following the order of operations, solve the expression: $(6+20)\cdot\frac{8}{8}+(6+16)\cdot\frac{3}{22}$. Answer: 29

Step 1: Solve the expressions inside the parentheses

First, we'll focus on the expressions inside the parentheses: $$6+20=26$$ $$6+16=22$$ Now, the given expression becomes: $$\frac{8(26)}{8}+\frac{3(22)}{22}$$

Step 2: Simplify the numerators

Next, we will multiply the numbers outside the parentheses with the results from the previous step: $$8 \cdot 26 = 208$$ $$3 \cdot 22 = 66$$ Now, the expression becomes: $$\frac{208}{8}+\frac{66}{22}$$

Step 3: Divide the fractions

Now, we will divide the fractions: $$\frac{208}{8}=26$$ $$\frac{66}{22}=3$$ So, the expression now becomes: $$26+3$$

Step 4: Perform the final addition

Finally, we will add the two numbers together: $$26+3=29$$ This is the final result: $$\boxed{29}$$

Key Concepts

Parenthetical Expressions

Parenthetical expressions, also known as expressions within parentheses, are crucial in determining the sequence of solving mathematical problems. In the order of operations, these expressions are addressed first before moving on to other operations. For instance, in the exercise discussed, we start by simplifying the expressions within the parentheses \(6+20\) and \(6+16\), obtaining 26 and 22, respectively. Handling these expressions at the very beginning is important because it sets the stage for the rest of the problem-solving process. It's key to remember that failing to compute these values first could lead to incorrect answers since the order of operations dictates that operations within parentheses override multiplication, division, addition, and subtraction.

When you encounter parenthetical expressions in mathematics, always adhere to this sequence: Simplify expressions within the parentheses or brackets first, before moving on to other operations. This straightforward step is the foundation you need to accurately navigate through more complex arithmetic operations.

Simplifying Numerators

Simplifying numerators is an essential step that follows after resolving any parenthetical expressions. In our problem, once we've determined the values within the parentheses to be 26 and 22, the next operation is multiplication. Here, we aim to simplify the numerators before proceeding to the division. The numerators are simplified by multiplying the number outside of the parentheses by the simplified parenthetical expression: \(8 \cdot 26\) equals 208, and \(3 \cdot 22\) equals 66. This multiplication must be performed prior to dividing by the denominator for each fraction.

Understanding how to simplify numerators effectively is essentially about following the order of operations correctly—after addressing values in parentheses, multiplication and division come next, from left to right. This step ensures that the numerators are accurately formed, which in turn guarantees that the subsequent steps, such as dividing fractions, can be carried out correctly.

Dividing Fractions

Dividing fractions can often be a stumbling block in arithmetic. After simplifying the numerators in our given exercise to 208 and 66, respectively, we then proceed to divide each by its denominator. The division is straightforward: divide the numerator by the denominator. We find that \(\frac{208}{8} = 26\) and \(\frac{66}{22} = 3\).

The concept of dividing fractions is simplified further by remembering that when you divide by a fraction, it is equivalent to multiplying by its reciprocal. However, in the context of this exercise, since we are dealing with whole numbers, the process is a matter of simple division. This fundamental of dividing fractions helps us to understand the way to separate a quantity into smaller, equal parts, which is a core aspect of arithmetic.

Arithmetic Operations

Arithmetic operations include addition, subtraction, multiplication, and division. In our problem-solving process, after addressing the parenthetical expressions, simplifying the numerators, and dividing the fractions, we arrive at the simple arithmetic operation of addition. We add the two numbers 26 and 3 to get the final result of 29.

Mastering arithmetic operations is about understanding not only how to perform each operation, but also the order in which they should be performed—known as the order of operations or PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). The successful application of this rule ensures that mathematical expressions are solved accurately and consistently. Arithmetic operations are the building blocks for more complex mathematical concepts, so a solid grasp of these fundamentals is essential for any student's progression in mathematics.