Problem 19 - Elementary Algebra

Question

Perform the indicated multiplications and divisions and express your answers in simplest form. $$\frac{18 r s}{34} \div(9 r)$$

Step-by-Step Solution

Answer

The simplest form is \(\frac{2s}{17}\).

Step 1: Rewrite the Division as Multiplication

The problem involves division of fractions, which can be rewritten as multiplication by the reciprocal. Rewriting the problem, we get: \[\frac{18rs}{34} \div (9r) = \frac{18rs}{34} \times \frac{1}{9r}.\]

Step 2: Multiply the Fractions

Multiply the numerators and the denominators of the fractions: \[\frac{18rs \times 1}{34 \times 9r} = \frac{18rs}{306r}.\]

Step 3: Cancel Common Factors

Cancel any common factors that appear in both the numerator and the denominator. Notice that the common factor is \(9r\). Simplifying: \[\frac{18rs}{306r} = \frac{2s}{17}.\]

Step 4: Simplify the Fraction

The fraction \(\frac{2s}{17}\) is already in its simplest form since 2 and 17 have no common factors and so does \(s\).

Key Concepts

Fraction Simplification

Simplifying a fraction means reducing it to its simplest form. This involves dividing both the numerator (the top part) and the denominator (the bottom part) by their greatest common factor (GCF). For instance, if we have a fraction \(\frac{18}{34}\), the GCF of 18 and 34 is 2. By dividing both parts by 2, we simplify it to \(\frac{9}{17}\).

• Always search for the largest number that divides both the numerator and the denominator without a remainder.
• Simplify each time you perform an operation on a fraction.
• A fraction is fully simplified when there are no common factors left except 1.

Practicing fraction simplification helps in making calculations easier and results clearer.

Division of Fractions

Dividing fractions can initially seem complex, but there's a handy trick: multiply by the reciprocal. The reciprocal of a fraction is simply flipping the numerator and the denominator. In the expression \(\frac{18rs}{34} \div 9r\), taking the reciprocal of \(9r\), we get \(\frac{1}{9r}\).

• Change division to multiplication by using the reciprocal.
• This approach simplifies the calculation process and avoids division directly on fractions.
• Follow this method by multiplying numerators together and denominators together.

By understanding the concept of reciprocals, dividing fractions becomes a straightforward task.

Common Factors

Identifying common factors is crucial for both simplifying fractions and making calculations efficient. Common factors are the numbers or variables that appear both in the numerator and the denominator. For example, in the fraction \(\frac{18rs}{306r}\), components like \(9r\) are common factors.

• Check both the numerator and the denominator for numbers or variables that appear in both.
• Dividing these out can significantly reduce the fraction.
• In algebraic expressions, focus on variables as well as numbers for simplification.

Recognizing and cancelling common factors ensures that expressions are as simple as possible, making them easier to understand and use.