Evaluating Complex Fraction Expressions

We need to find the value of the given mathematical expression: \[ \frac{132}{11} \times \frac{13}{12} \div \frac{169}{14}-\frac{1}{13} \]

To solve this, we must follow the order of operations, often remembered by acronyms like BODMAS or PEMDAS.

• B/P: Brackets/Parentheses first
• O/E: Orders/Exponents
• D/M: Division and Multiplication (from left to right)
• A/S: Addition and Subtraction (from left to right)

In our expression, we have multiplication, division, and subtraction. We'll perform multiplication and division first, from left to right, and then handle the subtraction.

Step-by-Step Evaluation of the Expression

Let's break down the calculation:

First, evaluate the multiplication and division part: \(\frac{132}{11} \times \frac{13}{12} \div \frac{169}{14}\)

Step 1: Simplify the first fraction. \[ \frac{132}{11} = 12 \] The expression becomes: \[ 12 \times \frac{13}{12} \div \frac{169}{14} \]

Step 2: Perform the multiplication. \[ 12 \times \frac{13}{12} \] The '12' in the numerator and denominator cancel out: \[ \cancel{12} \times \frac{13}{\cancel{12}} = 13 \] The expression now is: \[ 13 \div \frac{169}{14} \]

Step 3: Perform the division. Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of \(\frac{169}{14}\) is \(\frac{14}{169}\). \[ 13 \div \frac{169}{14} = 13 \times \frac{14}{169} \]

We know that \(169 = 13 \times 13 = 13^2\). So, we can write \(169\) as \(13 \times 13\): \[ 13 \times \frac{14}{13 \times 13} \] One '13' in the numerator cancels out with one '13' in the denominator: \[ \cancel{13} \times \frac{14}{\cancel{13} \times 13} = \frac{14}{13} \]

So, the value of the multiplication and division part is \(\frac{14}{13}\).

Step 4: Perform the subtraction. Now we take the result from Step 3 and subtract \(\frac{1}{13}\): \[ \frac{14}{13} - \frac{1}{13} \] Since the denominators are the same, we just subtract the numerators: \[ \frac{14 - 1}{13} = \frac{13}{13} \]

Step 5: Simplify the final fraction. \[ \frac{13}{13} = 1 \]

Therefore, the value of the expression \(\frac{132}{11} \times \frac{13}{12} \div \frac{169}{14}-\frac{1}{13}\) is 1.

Summary of Calculation

The final value of the expression is 1.

Revision Table: Fraction Operations & BODMAS

Additional Information on Evaluating Expressions

When evaluating mathematical expressions with multiple operations, strictly following the order of operations is crucial to get the correct result. Misinterpreting the order can lead to significant errors.

Simplifying fractions before performing multiplication or division can often make the calculations much easier by cancelling out common factors between numerators and denominators. This was evident in our step-by-step solution where we simplified \(\frac{132}{11}\) and cancelled out terms like 12 and 13.

Remember that the division and multiplication operations have the same priority, as do addition and subtraction. When they appear together, you should perform them from left to right in the expression.

Understanding how to work with fractions, including multiplication, division (using reciprocals), and subtraction (finding a common denominator if necessary, though not needed in this specific problem), is fundamental for solving such problems.