Simplify Mathematical Expression Calculation

This solution details the step-by-step process to simplify the given mathematical expression:

$ \frac{ ( 20^{2} - 10^{2} ) +5 \times 3 +10 } { \frac{1}{3} \text{of} 27 + 10 + 2 + 1 } $

We need to evaluate the numerator and the denominator separately before performing the final division.

Numerator Calculation Steps

First, let's calculate the value of the numerator (the top part of the fraction). We must follow the order of operations (PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction).

• Handle Parentheses and Exponents:
  • First, evaluate the terms inside the parentheses: $( 20^{2} - 10^{2} )$.
  • Calculate the exponents: $20^{2} = 20 \times 20 = 400$ and $10^{2} = 10 \times 10 = 100$.
  • Perform the subtraction within the parentheses: $400 - 100 = 300$.
  • Alternatively, we could use the difference of squares formula, $a^2 - b^2 = (a-b)(a+b)$. Applying this: $20^2 - 10^2 = (20-10)(20+10) = (10)(30) = 300$.
• Perform Multiplication:
  • Next, calculate the multiplication part: $5 \times 3 = 15$.
• Perform Addition:
  • Finally, add all the calculated values together to find the numerator's total value: $300 + 15 + 10 = 325$.

The value of the numerator is 325.

Denominator Calculation Steps

Now, let's calculate the value of the denominator (the bottom part of the fraction), again following the order of operations.

• Calculate "of" (Multiplication):
  • The term $\frac{1}{3} \text{ of } 27$ means $\frac{1}{3}$ multiplied by 27.
  • Calculation: $\frac{1}{3} \times 27 = \frac{27}{3} = 9$.
• Perform Addition:
  • Add the result to the remaining numbers in the denominator: $9 + 10 + 2 + 1 = 22$.

The value of the denominator is 22.

Final Calculation: Division

With the numerator and denominator calculated, we can now perform the final division to simplify the expression.

$ \text{Result} = \frac{\text{Numerator}}{\text{Denominator}} = \frac{325}{22} $

Expressing the Answer as a Mixed Number

The result $\frac{325}{22}$ is an improper fraction (where the numerator is larger than the denominator). We can convert this into a mixed number.

• Divide the numerator (325) by the denominator (22): $325 \div 22$.
• Performing the division: $325 \div 22$ gives a quotient of 14 and a remainder of 17. ($325 = 14 \times 22 + 17$).
• The mixed number is formed by the quotient and the fraction of the remainder over the divisor.

Therefore, the simplified value of the expression as a mixed number is $14 \frac{17}{22}$.