What is the value of \(\rm \frac{192}{64}\times \left(\frac{47}{94}+\frac{42}{84}\right)+\frac{182}{14}?\)
16
Understanding the Mathematical Expression
We need to find the value of the given mathematical expression: \(\rm \frac{192}{64}\times \left(\frac{47}{94}+\frac{42}{84}\right)+\frac{182}{14}\). To solve this, we will follow the order of operations (often remembered by acronyms like BODMAS or PEMDAS), which dictates the sequence of operations: Parentheses/Brackets, Orders/Exponents, Division and Multiplication (from left to right), and Addition and Subtraction (from left to right).
Step-by-Step Solution to Evaluate the Expression
Let's break down the evaluation of the expression into manageable steps.
Step 1: Simplify the fractions
First, we will simplify each fraction in the expression before performing any other operations. Simplifying fractions makes the calculation easier.
- Simplify \(\rm \frac{192}{64}\): We can see that 192 is a multiple of 64. \(64 \times 3 = 192\). So, \(\rm \frac{192}{64} = 3\).
- Simplify \(\rm \frac{47}{94}\): We notice that 94 is twice 47. \(47 \times 2 = 94\). So, \(\rm \frac{47}{94} = \frac{1}{2}\).
- Simplify \(\rm \frac{42}{84}\): Similar to the previous fraction, 84 is twice 42. \(42 \times 2 = 84\). So, \(\rm \frac{42}{84} = \frac{1}{2}\).
- Simplify \(\rm \frac{182}{14}\): We can perform division. \(182 \div 14 = 13\). So, \(\rm \frac{182}{14} = 13\).
Now, substitute these simplified values back into the original expression:
\(\rm 3 \times \left(\frac{1}{2}+\frac{1}{2}\right)+13\)
Step 2: Solve the operation inside the parentheses
Next, we perform the addition inside the parentheses.
\(\rm \frac{1}{2}+\frac{1}{2} = \frac{1+1}{2} = \frac{2}{2} = 1\)
Substitute this result back into the expression:
\(\rm 3 \times 1 + 13\)
Step 3: Perform the multiplication
According to the order of operations, multiplication comes before addition. We multiply 3 by 1.
\(\rm 3 \times 1 = 3\)
Substitute this result back into the expression:
\(\rm 3 + 13\)
Step 4: Perform the addition
Finally, we perform the addition.
\(\rm 3 + 13 = 16\)
Thus, the value of the expression \(\rm \frac{192}{64}\times \left(\frac{47}{94}+\frac{42}{84}\right)+\frac{182}{14}\) is 16.
Final Result
The value of the given expression is 16.
Revision Table: Key Steps in Evaluating the Expression
| Step | Operation | Calculation | Expression Status |
|---|---|---|---|
| 1 | Simplify Fractions | \(\rm \frac{192}{64}=3\), \(\rm \frac{47}{94}=\frac{1}{2}\), \(\rm \frac{42}{84}=\frac{1}{2}\), \(\rm \frac{182}{14}=13\) | \(\rm 3 \times \left(\frac{1}{2}+\frac{1}{2}\right)+13\) |
| 2 | Parentheses | \(\rm \frac{1}{2}+\frac{1}{2}=1\) | \(\rm 3 \times 1 + 13\) |
| 3 | Multiplication | \(\rm 3 \times 1 = 3\) | \(\rm 3 + 13\) |
| 4 | Addition | \(\rm 3 + 13 = 16\) | \(\rm 16\) |
Additional Information on Order of Operations
The order of operations is crucial for evaluating mathematical expressions correctly. Following a standard order ensures everyone gets the same result for the same expression.
- Parentheses/Brackets: Operations inside parentheses or brackets are always performed first. This groups parts of the expression that should be treated as a single value.
- Exponents/Orders: Next, evaluate any powers or roots.
- Multiplication and Division: These operations are performed from left to right as they appear in the expression. They have equal priority.
- Addition and Subtraction: These operations are performed last, also from left to right as they appear. They also have equal priority.
Understanding and applying the order of operations is fundamental to solving mathematical problems involving multiple operations.
Important Questions from Simplification
Simplify the following expression.
\(\left(\frac{7}{16} \div \frac{1}{2}\:of\: \frac{1}{5}\right)\times \frac{4}{5}-\frac{1}{3}\times\frac{5}{8}\div \frac{1}{2}+\frac{3}{4}\)
The value of \(\left( {2\frac{6}{7}of4\frac{1}{5} \div \frac{2}{3}} \right) \times 5\frac{1}{9} \div \left( {\frac{3}{4} \times 2\frac{2}{3}of\frac{1}{2} \div \frac{1}{4}} \right)\) is:
The value of \(\left[ {\frac{4}{7}\rm \;of\;2\frac{4}{5} \times 1\frac{2}{3} - \left( {3\frac{1}{2} - 2\frac{1}{6}} \right)} \right] \div \left( {3\frac{1}{5} \div 4\frac{1}{2}\;\rm of\;\;5\frac{1}{3}} \right)\) is:
The value of \(\frac{{0.0203 \times 2.92}}{{0.7 \times 0.0365 \times 2.9}} \div \frac{{{{\left( {12.12} \right)}^2} - {{\left( {8.12} \right)}^2}}}{{{{\left( {0.25} \right)}^2} + \left( {0.25} \right)\left( {19.99} \right)}}\) is:
The value of 4 ÷ 12 of [3 ÷ 4 of {(4 - 2) × 6 ÷ 2}] - 2 × 6 ÷ 8 + 3 is:
