Problem 48 - Elementary Algebra

Question

For exercises 39-82, simplify. $$ \frac{9 h k}{40 n^{2}} \div \frac{3 h^{2}}{8 n} $$

Step-by-Step Solution

Answer

\(\frac{3k}{5h}\)

Step 1 - Rewrite the division as multiplication

To simplify \(\frac{9hk}{40n^2} \div \frac{3h^2}{8n}\), rewrite the division as multiplication by the reciprocal. This means \(\frac{9hk}{40n^2} \div \frac{3h^2}{8n}\) becomes \(\frac{9hk}{40n^2} \times \frac{8n}{3h^2}\).

Step 2 - Multiply the fractions

Multiply the numerators together and the denominators together: \(\frac{9hk \times 8n}{40n^2 \times 3h^2} = \frac{72hkn}{120n^2 h^2}\).

Step 3 - Simplify the fraction

To simplify \(\frac{72hkn}{120n^2 h^2}\), start by canceling out common factors in the numerator and the denominator. Here, both 72 and 120 share a common factor of 24. Additionally, cancel out common variables: \(\frac{72hkn}{120n^2 h^2} = \frac{6kn}{10nh} = \frac{3k}{5h}\).

Key Concepts

algebraic fractions

An algebraic fraction is simply a fraction where the numerator and/or the denominator contain algebraic expressions instead of just numbers. Think of it just like a regular fraction but with variables and constants mixed in. For example, in the exercise given: \( \frac{9hk}{40n^2} \), the numerator \(9hk\) and the denominator \(40n^2\) are both algebraic expressions.
To work with algebraic fractions, you often need to follow the same basic rules that you use with numerical fractions:

• Adding or subtracting requires a common denominator
• Multiplication is done across numerators and denominators
• Division involves multiplying by the reciprocal

Always remember these rules as they help simplify your work with algebraic fractions!

multiplication of fractions

Multiplying fractions, whether they're numerical or algebraic, follows a straightforward rule: multiply the numerators together and the denominators together. In our given problem, after rewriting the division as multiplication, we got: \( \frac{9hk}{40n^2} \times \frac{8n}{3h^2} \).

This means we multiply the numerators (\(9hk \times 8n\)) and the denominators (\(40n^2 \times 3h^2\)). To do the math:

• Numerators: \(9hk \times 8n = 72hkn\)
• Denominators: \(40n^2 \times 3h^2 = 120n^2 h^2\)

This gives us \( \frac{72hkn}{120n^2 h^2} \). It looks more complex now, but don't worry, we can simplify it next!

simplifying fractions

Simplifying algebraic fractions follows the same principle as simplifying numerical fractions: find common factors in the numerator and the denominator and cancel them out. For our fraction \(\frac{72hkn}{120n^2 h^2}\):

• Notice both 72 and 120 share a common factor of 24. Think of it like reducing a fraction like 24/48 to 1/2.
• The variables can also be simplified by canceling out common factors. For example, \(h\) in the numerator and \(h^2\) in the denominator can be simplified to remove one \(h\).

Breaking it down step-by-step, we get:
1. Cancel numerical common factors: \(\frac{72}{120} = \frac{6}{10}\).
2. Simplify variables: \(\frac{6hk n}{10n^2 h^2} = \frac{6k}{10nh} = \frac{3k}{5h}\).
And that's our simplified fraction: \( \frac{3k}{5h} \)!

reciprocal

A reciprocal is what you get when you flip a fraction upside down. For example, the reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \).
When dealing with dividing fractions, you multiply by the reciprocal of the divisor. In our problem, we started with:

• \( \frac{9hk}{40n^2} \times \frac{8n}{3h^2} \)
• The reciprocal of \( \frac{3h^2}{8n} \) is \( \frac{8n}{3h^2} \)

Thus, we effectively turned the division problem into a multiplication one. Multiplying by the reciprocal simplifies the fraction and makes solving the problem much easier. Make sure to always double-check the reciprocal to avoid errors!