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Difficulty: Very hardSimplification and BODMAS Rule
Evaluate the following complex mathematical expression strictly adhering to the hierarchical order of operations (BODMAS rule):
{64% of 125+[133÷{216−(1.4×2.5−35+710‾)}]}×0.375\left\{ 64\% \text{ of } 125 + \left[ \frac{13}{3} \div \left\{ 2\frac{1}{6} - \left( 1.4 \times 2.5 - \overline{\frac{3}{5} + \frac{7}{10}} \right) \right\} \right] \right\} \times 0.375
What is the exact numerical value of the final simplified expression?

Answer: -18.75

Answer

The simplified final value of the expression is -18.75.
Following the BODMAS rule strictly: first simplify under the vinculum (3/5 + 7/10 = 1.3), then inside the round brackets (1.4 * 2.5 - 1.3 = 2.2). Next, inside the curly brackets (13/6 - 2.2 = -1/30), then the division in the square brackets ((13/3) / (-1/30) = -130). Evaluating 64% of 125 gives 80. Combining inside the main bracket gives 80 + (-130) = -50. Finally, multiplying by 0.375 yields -18.75.

Step-by-Step Solution

1
Evaluate the expression under the vinculum bar
35+710=1.3\frac{3}{5} + \frac{7}{10} = 1.3
The vinculum (bar bracket) has the highest priority under BODMAS and must be evaluated first.
2
Simplify the innermost round brackets
1.4 \times 2.5 - 1.3 = 3.5 - 1.3 = 2.2
Perform multiplication inside the round brackets before subtraction, then subtract the vinculum result.
3
Evaluate the expression inside the inner curly brackets
2\frac{1}{6} - 2.2 = \frac{13}{6} - \frac{11}{5} = -\frac{1}{30}
Convert the mixed fraction and decimal to improper fractions and subtract.
4
Evaluate the square bracket division
\frac{13}{3} \div \left( -\frac{1}{30} \right) = \frac{13}{3} \times (-30) = -130
Dividing by a fraction is equivalent to multiplying by its reciprocal.
5
Calculate the percentage term '64% of 125'
64100×125=80\frac{64}{100} \times 125 = 80
'Of' operation takes precedence over addition.
6
Combine the terms within the main outer brackets
80 + (-130) = -50
Add the result of the percentage term to the square bracket result.
7
Multiply by the outer factor 0.375
−50×0.375=−18.75-50 \times 0.375 = -18.75
Perform final multiplication to complete the simplification.

Key Concept

BODMAS order of operations with nested brackets, vinculum, fractions, decimals, and percentage operations.
Estimated Time:3m 0s
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Topics

Simplification and BODMAS Rule

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