The value of \(\frac{52-1170\div26+13\times2}{2+1\frac{1}{8}\ \rm of\ 2-1\frac{1}{4}}\) is:
11
Simplifying Mathematical Expressions
To find the value of the given expression, we need to follow the order of operations, often remembered by acronyms like BODMAS or PEMDAS.
The expression is:
\(\frac{52-1170\div26+13\times2}{2+1\frac{1}{8}\ \rm of\ 2-1\frac{1}{4}}\)
We will evaluate the numerator and the denominator separately.
Evaluating the Numerator
The numerator is \(52-1170\div26+13\times2\).
According to the order of operations:
- First, perform division: \(1170 \div 26\)
- Second, perform multiplication: \(13 \times 2\)
- Third, perform addition and subtraction from left to right.
Let's calculate the division:
\(1170 \div 26 = 45\)
Now, let's calculate the multiplication:
\(13 \times 2 = 26\)
Substitute these values back into the numerator expression:
\(52 - 45 + 26\)
Now, perform subtraction and addition from left to right:
\(52 - 45 = 7\)
\(7 + 26 = 33\)
So, the value of the numerator is 33.
Evaluating the Denominator
The denominator is \(2+1\frac{1}{8}\ \rm of\ 2-1\frac{1}{4}\).
First, convert the mixed numbers to improper fractions:
- \(1\frac{1}{8} = \frac{(1 \times 8) + 1}{8} = \frac{9}{8}\)
- \(1\frac{1}{4} = \frac{(1 \times 4) + 1}{4} = \frac{5}{4}\)
The term "of" means multiplication. According to the order of operations, 'of' (or orders/exponents) is done before basic multiplication/division, but after brackets. In simple arithmetic expressions like this, it's often grouped with multiplication.
Perform the 'of' operation:
\(1\frac{1}{8}\ \rm of\ 2 = \frac{9}{8} \times 2 = \frac{9 \times 2}{8} = \frac{18}{8} = \frac{9}{4}\)
Substitute the values back into the denominator expression:
\(2 + \frac{9}{4} - \frac{5}{4}\)
To add and subtract these fractions, we need a common denominator, which is 4. Convert 2 to a fraction with denominator 4:
\(2 = \frac{2 \times 4}{4} = \frac{8}{4}\)
The expression becomes:
\(\frac{8}{4} + \frac{9}{4} - \frac{5}{4}\)
Now, perform addition and subtraction from left to right:
\(\frac{8}{4} + \frac{9}{4} = \frac{8+9}{4} = \frac{17}{4}\)
\(\frac{17}{4} - \frac{5}{4} = \frac{17-5}{4} = \frac{12}{4} = 3\)
So, the value of the denominator is 3.
Final Calculation
Now we divide the numerator by the denominator:
\(\frac{\text{Numerator}}{\text{Denominator}} = \frac{33}{3}\)
\(\frac{33}{3} = 11\)
The value of the given expression is 11.
| Part | Expression | Steps | Value |
|---|---|---|---|
| Numerator | \(52-1170\div26+13\times2\) | \(52 - 45 + 26\) | 33 |
| Denominator | \(2+1\frac{1}{8}\ \rm of\ 2-1\frac{1}{4}\) | \(2 + \frac{9}{4} - \frac{5}{4} = \frac{8}{4} + \frac{9}{4} - \frac{5}{4} = \frac{17}{4} - \frac{5}{4} = \frac{12}{4}\) | 3 |
| Final Value | \(\frac{\text{Numerator}}{\text{Denominator}}\) | \(\frac{33}{3}\) | 11 |
Comparing this value with the given options, we find that 11 matches one of the options.
Revision Table: Order of Operations
| Order | Operation Type | Description |
|---|---|---|
| 1 | Brackets/Parentheses | Calculations inside brackets first. |
| 2 | Orders/Exponents/'Of' | Powers, roots, and fractions introduced by 'of'. |
| 3 | Division and Multiplication | From left to right. |
| 4 | Addition and Subtraction | From left to right. |
Additional Information: Working with Mixed Numbers and Fractions
A mixed number combines a whole number and a fraction, like \(1\frac{1}{8}\). To convert a mixed number \(a\frac{b}{c}\) to an improper fraction, use the formula: \(\frac{(a \times c) + b}{c}\).
When adding or subtracting fractions, they must have a common denominator. If they don't, find the least common multiple (LCM) of the denominators and rewrite each fraction with this new denominator.
The word "of" in mathematical context, especially with fractions or percentages, typically means multiplication. For example, "half of 10" means \(\frac{1}{2} \times 10\).
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Column I
Column II
a.
Equivalent fraction of \(\frac{7}{12}\) is
i.
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b.
Equivalent fraction of \(\frac{9}{15}\) is
ii.
Improper fraction
c.
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d.
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iv.
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