Evaluate pi(3125/5-6250/3+3125)

Solution:

Step 1: Establish a common denominator

Step 1.1: Adjust the first fraction

Multiply $\frac{3125}{5}$ by $\frac{3}{3}$ to get $\pi (\frac{3125\cdot 3}{5\cdot 3}-\frac{6250}{3}+3125)$.

Step 1.2: Adjust the first fraction further

The result is $\pi (\frac{9375}{15}-\frac{6250}{3}+3125)$.

Step 1.3: Adjust the second fraction

Multiply $\frac{6250}{3}$ by $\frac{5}{5}$ to get $\pi (\frac{9375}{15}-\frac{6250\cdot 5}{3\cdot 5}+3125)$.

Step 1.4: Adjust the second fraction further

The result is $\pi (\frac{9375}{15}-\frac{31250}{15}+3125)$.

Step 1.5: Convert the whole number to a fraction

Express $3125$ as $\frac{3125}{1}$.

Step 1.6: Adjust the whole number fraction

Multiply $\frac{3125}{1}$ by $\frac{15}{15}$ to get $\pi (\frac{9375}{15}-\frac{31250}{15}+\frac{3125\cdot 15}{1\cdot 15})$.

Step 1.7: Adjust the whole number fraction further

The result is $\pi (\frac{9375}{15}-\frac{31250}{15}+\frac{46875}{15})$.

Step 1.8: Simplify the denominator

Reorder the factors of $15$ to $\pi (\frac{9375}{15}-\frac{31250}{15}+\frac{46875}{15})$.

Step 1.9: Combine the fractions

Combine the numerators over the common denominator of $15$.

Step 2: Combine the numerators

$\pi \frac{9375-31250+46875}{15}$

Step 3: Simplify each term

Step 3.1: Simplify the first term

$9375$ multiplied by $\pi$.

Step 3.2: Simplify the second term

$-6250$ times $5$ multiplied by $\pi$.

Step 3.3: Simplify the third term

$3125$ times $15$ multiplied by $\pi$.

Step 4: Combine and reduce terms

Step 4.1: Combine terms

Subtract $31250$ from $9375$ and add $46875$.

Step 4.2: Simplify the expression

Combine the terms to get $\pi \frac{25000}{15}$.

Step 4.3: Reduce the fraction

Step 4.3.1: Factor out common terms

Factor $5$ out of $25000$.

Step 4.3.2: Cancel common factors

Divide both numerator and denominator by $5$.

Step 4.3.2.1: Simplify the denominator

Reduce $15$ by factoring out $5$.

Step 4.3.2.2: Cancel out $5$

Cancel the $5$ in both numerator and denominator.

Step 4.3.2.3: Final simplification

The result is $\pi \frac{5000}{3}$.

Step 4.4: Combine $\pi$ with the fraction

Multiply $\pi$ by $\frac{5000}{3}$.

Step 4.5: Rearrange the terms

Place $5000$ to the left of $\pi$ to get $\frac{5000\pi }{3}$.

Step 5: Present the result in various forms

Exact Form: $\frac{5000\pi }{3}$ Decimal Form: $5235.98775598\dots$

Knowledge Notes:

1. Common Denominator: When adding or subtracting fractions, a common denominator is required to combine the terms. This is achieved by finding a number that each denominator can multiply into without changing the value of the fractions.

2. Multiplying Fractions: To multiply a fraction by a whole number, you can either convert the whole number to a fraction with a denominator of 1 or directly multiply the numerator by the whole number.

3. Simplifying Fractions: Fractions are simplified by canceling out common factors in the numerator and the denominator.

4. Pi ($\pi$): Pi is a mathematical constant approximately equal to 3.14159. It represents the ratio of a circle's circumference to its diameter and is used in formulas involving circles and periodic functions.

5. Exact vs. Decimal Form: The exact form of a number includes pi or other irrational numbers as symbols, representing an exact value. The decimal form approximates the exact value to a certain number of decimal places, which can be useful for practical calculations but may lose precision.