Problem 38 - Fundamentals of Mathematics

Question

Find each value. Check each result with a calculator. \(\frac{8 \cdot(6+20)}{8}+\frac{3 \cdot(6+16)}{22}\)

Step-by-Step Solution

Answer

The answer is 29.

Step 1: Calculate the Expression Inside Parentheses

Start by simplifying the expressions inside the parentheses: \[ (6 + 20) = 26 \] and \[ (6 + 16) = 22 \]. Substitute these values back into the original problem.

Step 2: Substitute Simplified Expressions

Rewrite the expression with the simplified terms: \[ \frac{8 \cdot 26}{8} + \frac{3 \cdot 22}{22} \].

Step 3: Simplify Fractions

Simplify each term in the expression: \[ \frac{8 \cdot 26}{8} = 26 \], since the \(8\)s cancel out, and \[ \frac{3 \cdot 22}{22} = 3 \], since the \(22\)s cancel out.

Step 4: Add the Simplified Results

Add the simplified results from Step 3 together: \[ 26 + 3 = 29 \].

Step 5: Verify Solution with a Calculator

Use a calculator to check the initial expression is equal to \(29\). Input as originally: \[ \frac{8 \cdot (6 + 20)}{8} + \frac{3 \cdot (6 + 16)}{22} \] and confirm that the result is indeed 29.

Key Concepts

Parentheses Simplification

Understanding how to simplify expressions within parentheses is fundamental in arithmetic. In arithmetic operations, you often first need to focus on what's inside parentheses. This is because arithmetic rules dictate that parentheses should be addressed before multiplication, division, addition, or subtraction. In the example exercise, we first simplify expressions like

• \( (6 + 20) \), which results in \( 26 \), and
• \( (6 + 16) \), which results in \( 22 \).

By substituting these results back into the original expression, the problem becomes far simpler to manage. Parentheses act as a guide to focus on and complete specific arithmetic before moving to the next step.
Always remember: completing and simplifying expressions within parentheses can significantly simplify complex arithmetic problems.

Fraction Simplification

Simplifying fractions is a key mathematical skill that makes working with fractions much easier. When you simplify a fraction, you convert it into its simplest form without changing its value. This means reducing the numerator (top number) and the denominator (bottom number) such that they have no common divisors other than 1.
For example, in our exercise:

• \( \frac{8 \cdot 26}{8} = 26 \) because the \(8\)s can be cancelled out, leaving just the coefficient \(26\).
• \( \frac{3 \cdot 22}{22} = 3 \) because the \(22\)s cancel out, simplifying to \(3\).

By simplifying fractions, calculations become more straightforward and less error-prone. Simplification comes in handy, especially when dealing with complex arithmetic involving multiple fractions.

Using Calculators

Calculators are powerful tools that help verify arithmetic calculations, ensuring accuracy. While manual calculation is essential to understand arithmetic concepts, using a calculator can be a quick way to check your work.
In the exercise, after calculating manually, it's recommended to input the entire original expression into a calculator: \\[ \frac{8 \cdot (6 + 20)}{8} + \frac{3 \cdot (6 + 16)}{22} \]You should expect to see the result \(29\). By doing this, you cross-check the manual arithmetic operations with technologically assisted accuracy. Calculators help confirm results and avoid simple arithmetic errors, thereby enhancing your confidence in solving problems.

Addition of Fractions

Adding fractions might seem daunting at first, but it becomes a breeze once you know the techniques involved. Fractions represent parts of a whole, and when adding them, you are bringing those parts together.
In our exercise solution, notice how simplified whole numbers resulted from the initial fractions:

• \(26\) from \(\frac{8 \cdot 26}{8}\)
• \(3\) from \(\frac{3 \cdot 22}{22}\)

Adding these simplified values was straightforward:
\[ 26 + 3 = 29 \]When adding fractions with common denominators, or when they're simplified to integers, it's simply a matter of summing these numbers. Understanding the basics of fraction addition is essential for handling more complex expressions that involve multiple fractions.