Evaluate pi(3125/5-6250/3+3125)
This question asks for the evaluation of a mathematical expression that involves the arithmetic operations of division, multiplication, and addition. Specifically, the value of pi (π) is to be multiplied by the result of the arithmetic operations on the numbers given: 3125 divided by 5, minus 6250 divided by 3, plus 3125. The question requires the correct order of operations to be applied — typically parentheses first (if any), exponents (none are present in this question), multiplication and division (from left to right), and addition and subtraction (from left to right) — to find the numerical value of the expression.
$\pi \left(\right. \frac{3125}{5} - \frac{6250}{3} + 3125 \left.\right)$
Answer
Solution:
Step 1: Establish a common denominator
Step 1.1: Adjust the first fraction
Multiply $\frac{3125}{5}$ by $\frac{3}{3}$ to get $\pi \left( \frac{3125 \cdot 3}{5 \cdot 3} - \frac{6250}{3} + 3125 \right)$.
Step 1.2: Adjust the first fraction further
The result is $\pi \left( \frac{9375}{15} - \frac{6250}{3} + 3125 \right)$.
Step 1.3: Adjust the second fraction
Multiply $\frac{6250}{3}$ by $\frac{5}{5}$ to get $\pi \left( \frac{9375}{15} - \frac{6250 \cdot 5}{3 \cdot 5} + 3125 \right)$.
Step 1.4: Adjust the second fraction further
The result is $\pi \left( \frac{9375}{15} - \frac{31250}{15} + 3125 \right)$.
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 $\pi \left( \frac{9375}{15} - \frac{31250}{15} + \frac{3125 \cdot 15}{1 \cdot 15} \right)$.
Step 1.7: Adjust the whole number fraction further
The result is $\pi \left( \frac{9375}{15} - \frac{31250}{15} + \frac{46875}{15} \right)$.
Step 1.8: Simplify the denominator
Reorder the factors of $15$ to $\pi \left( \frac{9375}{15} - \frac{31250}{15} + \frac{46875}{15} \right)$.
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: $5235.98775598 \ldots$
Knowledge Notes:
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.
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.
Simplifying Fractions: Fractions are simplified by canceling out common factors in the numerator and the denominator.
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.
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.
