Simplify 0.25/(-3/4)*(6/9)/(8-5x)
The given problem is a mathematical expression that requires simplification. The expression involves several arithmetic operations including division, multiplication, and subtraction. The expression includes a decimal (0.25), a negative fraction (-3/4), another fraction (6/9), and a binomial expression (8-5x) within the denominator of a complex fraction. The task is to perform the appropriate arithmetic operations while following the order of operations (PEMDAS/BODMAS) to simplify the expression to its simplest form, taking into account that the variable 'x' remains as is, since the expression is not solved for any particular value of 'x'.
$\frac{0.25}{- \frac{3}{4}} \cdot \frac{\frac{6}{9}}{8 - 5 x}$
Answer
Solution:
Simplification Process:
Step 1: Simplify the fraction within the numerator
Extract the common factor of $6$ and $9$.
$$\frac{0.25}{-\frac{3}{4}} \cdot \frac{\frac{2}{3}}{8 - 5x}$$
Step 2: Multiply by the reciprocal of the denominator
Invert the denominator and multiply.
$$0.25 \cdot \left(-\frac{4}{3}\right) \cdot \frac{\frac{2}{3}}{8 - 5x}$$
Step 3: Simplify the constant factors
Simplify the multiplication involving $0.25$ and $-\frac{4}{3}$.
$$-\frac{1}{3} \cdot \frac{\frac{2}{3}}{8 - 5x}$$
Step 4: Simplify the fraction involving $-4$ and $12$
Reduce the fraction by dividing both numerator and denominator by $4$.
$$-\frac{1}{3} \cdot \frac{\frac{2}{3}}{8 - 5x}$$
Step 5: Position the negative sign
Place the negative sign in front of the entire expression.
$$-\frac{1}{3} \cdot \frac{\frac{2}{3}}{8 - 5x}$$
Step 6: Multiply the numerators
Multiply the numerators together.
$$-\frac{1}{3} \cdot \left(\frac{2}{3} \cdot \frac{1}{8 - 5x}\right)$$
Step 7: Multiply the fractions
Perform the multiplication of the fractions.
$$-\frac{1}{3} \cdot \frac{2}{3(8 - 5x)}$$
Step 8: Final multiplication
Multiply the remaining terms.
$$-\frac{2}{9(8 - 5x)}$$
Knowledge Notes:
Fraction Simplification: When simplifying fractions, look for common factors in the numerator and denominator that can be canceled out. This can often make the rest of the problem easier to solve.
Multiplying by Reciprocal: To divide by a fraction, multiply by its reciprocal. This is the same as flipping the numerator and denominator of the fraction you are dividing by.
Negative Numbers: Keep track of negative signs throughout the problem. A negative sign can be moved in front of the fraction or distributed across the numerator or denominator.
Multiplication of Fractions: When multiplying fractions, multiply the numerators together and the denominators together. Simplify before multiplying if possible to make calculations easier.
Order of Operations: Follow the order of operations (PEMDAS/BODMAS) when solving expressions that include multiple operations. This ensures that the calculations are performed in the correct sequence.
Algebraic Expressions: When working with algebraic expressions, treat variables as you would numerical values, following the same rules for simplification and multiplication. However, do not attempt to cancel out terms that include variables unless they are common factors.
