Evaluate (3.5*2/7-2/3)/16
The question presents a numerical expression that requires evaluation. The expression involves a series of mathematical operations: multiplication, division, subtraction, and a final division by 16. The task is to perform the correct order of operations—parentheses first, followed by multiplication and division from left to right, and then addition and subtraction from left to right—to determine the value of the entire expression.
$\frac{\left(\right. 3.5 \cdot \frac{2}{7} - \frac{2}{3} \left.\right)}{16}$
Answer
Solution:
Step 1: Simplify the numerator.
Step 1.1: Eliminate the common factor in the first term of the numerator.
Step 1.1.1: Extract the factor of $3.5$ from the term involving $7$.
$\frac{3.5 \cdot \frac{2}{3.5 \cdot 2} - \frac{2}{3}}{16}$
Step 1.1.2: Simplify by removing the common factor.
$\frac{\cancel{3.5} \cdot \frac{2}{\cancel{3.5} \cdot 2} - \frac{2}{3}}{16}$
Step 1.1.3: Rewrite the simplified expression.
$\frac{\frac{2}{2} - \frac{2}{3}}{16}$
Step 1.2: Simplify the fraction $\frac{2}{2}$.
$\frac{1 - \frac{2}{3}}{16}$
Step 1.3: Express $1$ as a fraction with a denominator of $3$.
$\frac{\frac{3}{3} - \frac{2}{3}}{16}$
Step 1.4: Combine the fractions over the same denominator.
$\frac{\frac{3 - 2}{3}}{16}$
Step 1.5: Perform the subtraction in the numerator.
$\frac{\frac{1}{3}}{16}$
Step 2: Multiply the numerator by the reciprocal of the denominator.
$\frac{1}{3} \cdot \frac{1}{16}$
Step 3: Perform the multiplication.
Step 3.1: Multiply the numerators and denominators.
$\frac{1}{3 \cdot 16}$
Step 3.2: Calculate the product of the denominators.
$\frac{1}{48}$
Step 4: Present the result in various formats.
Exact Form: $\frac{1}{48}$ Decimal Form: $0.02083$
Knowledge Notes:
The problem involves simplifying a complex fraction and converting it to its simplest form. The process includes several mathematical operations and concepts:
Simplifying Fractions: This involves reducing fractions to their simplest form by canceling common factors in the numerator and denominator.
Arithmetic Operations: Basic operations such as subtraction and multiplication are used.
Reciprocal of a Number: The reciprocal of a number is 1 divided by that number. It is used when dividing fractions, essentially multiplying by the reciprocal.
Combining Fractions: When fractions have the same denominator, their numerators can be combined directly.
Decimal Conversion: Converting a fraction to decimal form involves division of the numerator by the denominator.
In the given solution, the process starts by simplifying the numerator of the complex fraction, followed by multiplying the simplified numerator by the reciprocal of the denominator. The final step is to multiply the fractions to get the result, which is then presented in both exact (fractional) and decimal forms.
