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Magazine Chapter 1: Problem 8
Evaluate: (a) \(\frac{9}{2}-\frac{4}{5} \div\left(\frac{2}{3}\right)^{2} \times \frac{3}{11}\) (b) \(\frac{\frac{3}{4}+\frac{7}{5} \div \frac{2}{9} \times \frac{1}{3}}{\frac{7}{3}-\frac{11}{2} \times \frac{2}{5}+\frac{4}{9}}\) (c) \(\left(\frac{3}{4}+\frac{7}{5}\right)^{2} \div\left(\frac{7}{3}-\frac{11}{5}\right)^{2}\) (d) \(\frac{\left(\frac{5}{2}\right)^{3}-\frac{2}{9} \div\left(\frac{2}{3}\right)^{2} \times \frac{3}{2}}{\frac{3}{11}+\left(\frac{11}{2} \times \frac{2}{5}\right)^{2}-\frac{7}{5}}\)
Short Answer
Expert verified
(a) \(\frac{63}{10}\), (b) \(\frac{717}{130}\), (c) \(\frac{83205}{320}\), (d) \(\frac{6555}{1633}\).
Step by step solution
01
Simplifying Expression in Part (a)
For expression (a), recall the order of operations: division first, followed by multiplication and subtraction. First, compute the exponent:\[\left(\frac{2}{3}\right)^{2} = \frac{4}{9}\]Next, perform the division:\[\frac{4}{5} \div \frac{4}{9} = \frac{4}{5} \times \frac{9}{4} = \frac{9}{5}\]Then, perform the multiplication:\[\frac{9}{5} \times \frac{3}{11} = \frac{27}{55}\]Finally, subtract from the first fraction:\[\frac{9}{2} - \frac{27}{55}\]To subtract, find a common denominator (110):\[\frac{9}{2} = \frac{495}{110}, \quad \frac{27}{55} = \frac{54}{110}\]Subtract the fractions:\[\frac{495}{110} - \frac{54}{110} = \frac{441}{110} = \frac{63}{10}\]So, the result of part (a) is \(\frac{63}{10}\).
02
Simplifying Expression in Part (b)
For expression (b), first simplify the numerator. Compute the division and multiplication:\[\frac{7}{5} \div \frac{2}{9} = \frac{7}{5} \times \frac{9}{2} = \frac{63}{10}\]Now, multiply by \(\frac{1}{3}\):\[\frac{63}{10} \times \frac{1}{3} = \frac{21}{10}\]Add \(\frac{3}{4}\):\[\frac{3}{4} + \frac{21}{10} = \frac{75}{40} + \frac{84}{40} = \frac{159}{40}\]Now the denominator: compute multiplication:\[\frac{11}{2} \times \frac{2}{5} = \frac{11}{5}\]Subtract and add fractions:\[\frac{7}{3} - \frac{11}{5} + \frac{4}{9}\]Convert to common denominator (45):\[\frac{7}{3} = \frac{105}{45}, \quad \frac{11}{5} = \frac{99}{45}, \quad \frac{4}{9} = \frac{20}{45}\]Compute the result:\[\frac{105}{45} - \frac{99}{45} + \frac{20}{45} = \frac{26}{45}\]Finally divide the simplified numerator by denominator:\[\frac{159}{40} \div \frac{26}{45} = \frac{159}{40} \times \frac{45}{26} = \frac{7161}{1040} = \frac{717}{130}\]So the result of part (b) is \(\frac{717}{130}\).
03
Simplifying Expression in Part (c)
For expression (c), first simplify the numerator:\[\frac{3}{4} + \frac{7}{5} = \frac{15}{20} + \frac{28}{20} = \frac{43}{20}\]Compute the square:\[\left(\frac{43}{20}\right)^{2} = \frac{1849}{400}\]Next, the denominator:\[\frac{7}{3} - \frac{11}{5} = \frac{35}{15} - \frac{33}{15} = \frac{2}{15}\]Compute the square:\[\left(\frac{2}{15}\right)^{2} = \frac{4}{225}\]Finally, divide the squares:\[\frac{1849}{400} \div \frac{4}{225} = \frac{1849}{400} \times \frac{225}{4} = \frac{416025}{1600} = \frac{83205}{320}\]Therefore, the result of part (c) is \(\frac{83205}{320}\).
04
Simplifying Expression in Part (d)
For expression (d), start with the numerator:\[\left(\frac{5}{2}\right)^{3} = \frac{125}{8}\]Divide and multiply:\[\frac{2}{9} \div \left(\frac{2}{3}\right)^{2} = \frac{2}{9} \div \frac{4}{9} = \frac{1}{2}\]Multiply by \(\frac{3}{2}\):\[\frac{1}{2} \times \frac{3}{2} = \frac{3}{4}\]Subtract from the exponent:\[\frac{125}{8} - \frac{3}{4} = \frac{250}{16} - \frac{12}{16} = \frac{238}{16} = \frac{119}{8}\]For the denominator:\[\left(\frac{11}{2} \times \frac{2}{5}\right)^{2} = \left(\frac{11}{5}\right)^{2} = \frac{121}{25}\]Add and subtract:\[\frac{3}{11} + \frac{121}{25} - \frac{7}{5}\]Converting to common denominator (275):\[\frac{3}{11} = \frac{75}{275}, \quad \frac{121}{25} = \frac{1331}{275}, \quad \frac{7}{5} = \frac{385}{275}\]Computing:\[\frac{75}{275} + \frac{1331}{275} - \frac{385}{275} = \frac{1021}{275}\]Finally divide:\[\frac{119}{8} \div \frac{1021}{275} = \frac{119}{8} \times \frac{275}{1021} = \frac{32725}{8168} = \frac{6555}{1633}\]Therefore, the result of part (d) is \(\frac{6555}{1633}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Order of Operations
When dealing with algebraic fractions, it's crucial to apply the correct order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). This ensures calculations are performed in the proper sequence, which affects the final result.
Consider the algebraic expression (a) \( \frac{9}{2}-\frac{4}{5} \div\left(\frac{2}{3}\right)^{2} \times \frac{3}{11} \). First, evaluate the exponent \( (\frac{2}{3})^2 = \frac{4}{9} \).
Next, handle multiplication and division from left to right:
Understanding and mastering the order of operations means you can confidently approach any algebraic expression without mistakes.
Consider the algebraic expression (a) \( \frac{9}{2}-\frac{4}{5} \div\left(\frac{2}{3}\right)^{2} \times \frac{3}{11} \). First, evaluate the exponent \( (\frac{2}{3})^2 = \frac{4}{9} \).
Next, handle multiplication and division from left to right:
- Perform the division: \( \frac{4}{5} \div \frac{4}{9} \).
- Multiply the result by \( \frac{3}{11} \).
Understanding and mastering the order of operations means you can confidently approach any algebraic expression without mistakes.
Fraction Division
Dividing fractions is an essential skill when dealing with complex algebraic expressions. It's all about flipping and multiplying the second fraction, a process commonly referred to as "multiply by the reciprocal." For example, when you need to divide two fractions such as \( \frac{4}{5} \div \frac{4}{9} \), you rewrite the division as a multiplication: \( \frac{4}{5} \times \frac{9}{4} \).
It's important to follow this step correctly to ensure accurate results.
It's important to follow this step correctly to ensure accurate results.
- First, flip the divisor: \( \frac{9}{4} \).
- Then, multiply the numerators together: \( 4 \times 9 = 36 \).
- Finally, multiply the denominators: \( 5 \times 4 = 20 \).
Exponentiation
Exponentiation is a mathematical operation involving numbers or expressions called bases raised to a power or exponent. In algebraic fractions, exponentiation needs special attention, especially in algebraic exercises where it may appear in compound expressions. For instance, when given \( (\frac{2}{3})^2 \), calculate by squaring both the numerator and the denominator separately.
This example yields \( \frac{2^2}{3^2} = \frac{4}{9} \).
Remember the rules:
This example yields \( \frac{2^2}{3^2} = \frac{4}{9} \).
Remember the rules:
- Simplify the base expression first if necessary.
- Then apply the exponent to both parts of the fraction.
Fraction Addition and Subtraction
Adding and subtracting fractions can be tricky when the denominators differ, as a common base must be found. This technique is known as finding a "common denominator." In algebraic expressions, managing this carefully is key. Take, for example, the subtraction in exercise (a): \( \frac{9}{2} - \frac{27}{55} \). To subtract, convert both fractions to the same denominator.
The least common denominator of 2 and 55 is 110. Thus:
The least common denominator of 2 and 55 is 110. Thus:
- Convert \( \frac{9}{2} \) to \( \frac{495}{110} \).
- Convert \( \frac{27}{55} \) to \( \frac{54}{110} \).
- Once aligned, subtract the numerators: \( 495 - 54 = 441 \)
