Problem 15 - Elementary and Intermediate Algebra

Question

Multiply, and then simplify, if possible. \(\frac{35 n}{12} \cdot \frac{16}{7 n^{2}}\)

Step-by-Step Solution

Answer

The simplified result is \( \frac{20}{3n} \).

Step 1: Write the Problem as a Single Fraction

To multiply fractions, the first step is to write them as a single fraction by multiplying the numerators together and the denominators together. Given:\[\frac{35n}{12} \cdot \frac{16}{7n^{2}}\]This becomes:\[\frac{35n \times 16}{12 \times 7n^{2}}\]

Step 2: Multiply the Numerators and Denominators

Multiply the numerators:\[35n \times 16 = 560n\]and the denominators:\[12 \times 7n^{2} = 84n^{2}.\]Thus, the combined fraction is:\[\frac{560n}{84n^{2}}\]

Step 3: Simplify the Fraction

Simplify the fraction by canceling out common factors in the numerator and denominator as much as possible. Notice that both the numerator and the denominator have a factor of \(n\). After canceling \(n\), we get:\[\frac{560}{84n}\]

Step 4: Reduce the Fraction Further

To further simplify, find the greatest common divisor (GCD) of 560 and 84. The GCD is 28, so divide both numerator and denominator by 28:\[\frac{560 \div 28}{84 \div 28} = \frac{20}{3n}\]Thus, the simplified form of the expression is:\[\frac{20}{3n}\]

Key Concepts

Simplifying Fractions

Simplifying fractions is a crucial skill when dealing with fractions in math. When we simplify a fraction, we are reducing it to its smallest possible form without changing its value. Simplifying involves finding common factors between the numerator and the denominator and cancelling them out.
For example, consider the expression \( \frac{560n}{84n^{2}} \). We can see that both the numerator (560n) and the denominator (84n²) share common factors. Primarily, both terms included the variable \( n \). By dividing both the numerator and denominator by \( n \), we eliminate this common factor, resulting in:

• Numerator: \( 560n / n = 560 \)
• Denominator: \( 84n^{2} / n = 84n \)

This cancellation simplifies the original fraction into \( \frac{560}{84n} \). Successfully simplifying fractions not only makes the calculations easier but also clarifies the problem-solving path forward.

Greatest Common Divisor

The greatest common divisor (GCD) is a number that represents the highest factor two numbers share. In simplifying fractions, determining the GCD is an essential step to ensure the fraction is in its simplest form.
When given \( \frac{560}{84} \) from our exercise, the task is to find the GCD of 560 and 84. The GCD is obtained by determining the largest number that perfectly divides both numbers. For 560 and 84, the GCD is 28 because:

• 28 is a factor of 560
• 28 is also a factor of 84

Once the GCD is identified, divide both the numerator and the denominator by 28 to simplify:

• \( 560 \div 28 = 20 \)
• \( 84 \div 28 = 3 \)

Therefore, the simplified fraction is \( \frac{20}{3n} \). Understanding how to find the GCD provides a simple yet powerful tool to simplify numerical expressions effectively.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and arithmetic operations. In our exercise, we dealt with an algebraic expression that included the variable \( n \). Expressions like \( \frac{35n}{12} \cdot \frac{16}{7n^{2}} \) pose opportunities to apply algebraic principles and simplify.
Algebraic expressions require care when multiplying or simplifying. The key is to follow the proper arithmetic rules and factorization methods. In the problem, the goal was to multiply and simplify the fractions, ensuring to account for any variable present, like \( n \).
While multiplying the fractions:

• Calculate the numerators: \( 35n \times 16 = 560n \)
• Calculate the denominators: \( 12 \times 7n^{2} = 84n^{2} \)

To simplify, consider the separate roles that numerators and denominators play and handle variables carefully by cancelling out or simplifying the expressions. Algebraic expressions form the grounding tool for tackling more complex math problems.