Math Problem Statement

Evaluate the expression (-4/5)^2 ÷ (-1/4 - 1/2 × 3/4). Write your answer as a fraction or as a whole or mixed number.

Solution

Let's solve the problem step by step. Here's the given expression:

(45)2(1412×34)\frac{\left(-\frac{4}{5}\right)^2}{\left(-\frac{1}{4} - \frac{1}{2} \times \frac{3}{4}\right)}

Step 1: Simplify the numerator

The numerator is (45)2\left(-\frac{4}{5}\right)^2. Squaring a negative fraction makes it positive:

(45)2=1625.\left(-\frac{4}{5}\right)^2 = \frac{16}{25}.

Step 2: Simplify the denominator

The denominator involves two terms: 1412×34.-\frac{1}{4} - \frac{1}{2} \times \frac{3}{4}.

Step 2.1: Simplify 12×34\frac{1}{2} \times \frac{3}{4}

Multiply the fractions: 12×34=38.\frac{1}{2} \times \frac{3}{4} = \frac{3}{8}.

Step 2.2: Subtract from 14-\frac{1}{4}

Now compute: 1438.-\frac{1}{4} - \frac{3}{8}.

To subtract, find a common denominator. The least common denominator of 4 and 8 is 8: 14=28.-\frac{1}{4} = -\frac{2}{8}.

Thus: 2838=58.-\frac{2}{8} - \frac{3}{8} = -\frac{5}{8}.

The denominator is 58-\frac{5}{8}.

Step 3: Compute the division

Now divide the numerator by the denominator: 162558.\frac{\frac{16}{25}}{-\frac{5}{8}}.

Dividing by a fraction is the same as multiplying by its reciprocal: 1625÷58=1625×85.\frac{16}{25} \div -\frac{5}{8} = \frac{16}{25} \times -\frac{8}{5}.

Multiply the numerators and denominators: 1625×85=168255=128125.\frac{16}{25} \times -\frac{8}{5} = \frac{16 \cdot -8}{25 \cdot 5} = \frac{-128}{125}.

Final Answer:

128125\boxed{-\frac{128}{125}}

Would you like further clarifications or explanations?


Follow-Up Questions:

  1. Can you simplify 128125\frac{-128}{125} into a mixed number?
  2. How do you find the reciprocal of a fraction?
  3. What is the significance of squaring a fraction like (45)2\left(-\frac{4}{5}\right)^2?
  4. Can this problem be solved using a calculator, and if so, how do you input fractions?
  5. What happens if one term in the denominator is zero?

Tip:

Always simplify each component of a complex fraction separately before combining them—it reduces errors!

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Math Problem Analysis

Mathematical Concepts

Arithmetic with Fractions
Order of Operations
Simplifying Expressions

Formulas

Division of Fractions
Multiplication of Fractions
Squaring a Fraction

Theorems

Reciprocal Rule for Division

Suitable Grade Level

Grades 6-8