Problem 57 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$2 \div \frac{18}{5}$$

Step-by-Step Solution

Answer

The result of \( 2 \div \frac{18}{5} \) simplifies to \( \frac{5}{9} \).

Step 1: Convert the Operation

To perform division, convert the operation 2 divided by \( \frac{18}{5} \) into multiplication by the reciprocal. The reciprocal of a fraction is achieved by swapping the numerator and denominator. Therefore, the reciprocal of \( \frac{18}{5} \) is \( \frac{5}{18} \). The new operation is thus: \(2 \times \frac{5}{18}\).

Step 2: Multiply

Perform the multiplication operation \(2 \times \frac{5}{18}\). Here, consider 2 as a fraction \( \frac{2}{1} \). So, now the operation becomes \( \frac{2}{1} \times \frac{5}{18} \). Now multiply the numerators together and the denominators together. So, \( \frac{2 \times 5}{1 \times 18} \) which equals \( \frac{10}{18} \).

Step 3: Reduce to Lowest Terms

Now, reduce the fraction \( \frac{10}{18} \) to its lowest possible terms. Both 10 and 18 are divisible by 2. Dividing both numbers by 2 gives \( \frac{5}{9} \).

Key Concepts

Reciprocal

When dealing with fractions, the concept of the reciprocal is crucial, especially in division operations. The reciprocal of a number is simply what you multiply that number by to get the product of 1. In terms of fractions, the reciprocal involves flipping the numerator and the denominator.
For example, if you have a fraction \( \frac{18}{5} \), its reciprocal is \( \frac{5}{18} \). It’s like taking the inverse of the fraction.
This concept is vital because when you divide by a fraction, you're actually multiplying by its reciprocal. In our operation \( 2 \div \frac{18}{5} \), we discovered that we can swap it to \( 2 \times \frac{5}{18} \). This makes the calculation much more straightforward.

Multiplication of Fractions

Multiplying fractions might seem tricky at first, but once you get the hang of it, it's quite simple. You basically need to multiply the numerators (the top parts of the fractions) together and the denominators (the bottom parts) together.
When you are multiplying a whole number by a fraction, like in our problem \( 2 \times \frac{5}{18} \), you can think of the whole number \( 2 \) as \( \frac{2}{1} \) for the calculation. This ensures you get a fraction at the end of the operation.

• Numerators Multiplication: Multiply the top numbers: \( 2 \times 5 = 10 \).
• Denominators Multiplication: Multiply the bottom numbers: \( 1 \times 18 = 18 \).

Hence, you get a new fraction, \( \frac{10}{18} \), which represents the product of the original multiplication.

Reducing Fractions

Reducing fractions to their lowest terms means simplifying them so that the numerator and the denominator are as small as possible, while still keeping the fraction the same in value.
To reduce the fraction \( \frac{10}{18} \), we look for the greatest common divisor (GCD) of the numbers 10 and 18. The GCD is the largest number that divides both without leaving a remainder.

• Notice if both numbers are even, they have \( 2 \) as a common divisor.
• When we divide 10 by 2, we get 5.
• When we divide 18 by 2, we get 9.

So, \( \frac{10}{18} \) reduces to \( \frac{5}{9} \). This is the smallest and simplest form of the fraction, making it easy to understand and work with. Reducing fractions not only makes the numbers smaller, it also makes calculations easier in future problems!