Problem 120 - Prealgebra and Introductory Algebra

Question

Evaluate the expression \(x \div y\) for the given values of \(x\) and \(y.\) $$x=-5 \frac{2}{5}, y=-9$$

Step-by-Step Solution

Answer

-5 2/5 divided by -9 equals 3/5. So, \(x \div y = \frac{3}{5}\).

Step 1: Convert mixed number into simple fraction

First, let's represent the mixed fraction \(-5 \frac{2}{5}\) as an improper fraction. A mixed number consists of a whole number and a proper fraction. We can convert it to an improper fraction with the formula \(whole number \times denominator + numerator\) over the denominator. So, this gives us \(-1 \times \(\frac{27}{5}\). The negative sign has been put outside the brackets because the entire value of \(x\) is negative.

Step 2: Perform the operation

Next, divide \(\frac{-27}{5}\) by \(-9\). When we perform this operation, we divide the numerator of the first fraction by the numerator of the second fraction (which is the value of y) and put the result in the numerator of the answer, we divide the denominator of the first fraction by the denominator of the second fraction (which is 1, as y is an integer, not a fraction) and put the result in the denominator of the answer. So the expression becomes \(\frac{-27}{5} \div -9 = \frac{-27 \div -9}{5 \div 1}.\) Now, simplify this expression.

Step 3: Simplify the fraction

-27 divided by -9 gives 3. And 5 divided by 1 gives 5. So, \(\frac{-27 \div -9}{5 \div 1}= \frac{3}{5}\). Therefore, \(x \div y = \frac{3}{5}\).

Key Concepts

Converting Mixed Numbers

Converting a mixed number to an improper fraction is a fundamental skill in mathematics. A mixed number, like \(-5 \frac{2}{5}\), is a number that combines a whole number with a fraction. To convert, multiply the whole number by the fraction's denominator and then add the numerator. For example, with \(-5 \frac{2}{5}\), multiply 5 (the whole number) by 5 (the denominator), and add 2 (the numerator). This gives you \(25 + 2 = 27\). So, the improper fraction is\(-\frac{27}{5}\). The negative sign indicates the value is less than zero.Just remember:

• Multiply the whole number by the denominator.
• Add the numerator to this product.
• Place the total over the original denominator.

These steps help in converting any mixed number to an improper fraction efficiently.

Improper Fractions

Improper fractions are fractions where the numerator is greater than or equal to the denominator, like \(\frac{27}{5}\) in our example. They're useful in division and multiplication because they eliminate the mix of whole and fraction parts, making calculations simpler.An improper fraction can be seen as another way of expressing a number without separating it into whole parts and fraction parts. This means that all fractional calculations can be completed without fussing over whole numbers separately.While improper fractions might initially seem daunting, they are a normal part of different mathematical operations, including fraction multiplication and division. By converting mixed numbers into improper fractions, you can streamline your calculations, especially in algebraic evaluations.

Fraction Division

The division of fractions may sound complex, but it's quite straightforward once you understand it. You divide fractions by multiplying by the reciprocal. In our example, we are dividing \(-\frac{27}{5}\) by \(-9\). Since \(-9\) can be written as \(-\frac{9}{1}\), we multiply \(-\frac{27}{5}\) by the reciprocal of \(-\frac{9}{1}\), which is \(-\frac{1}{9}\).Here's a quick guide:

• Write the divisor as a fraction, if needed.
• Find the reciprocal of the divisor and multiply.
• Perform the multiplication across the numerators and denominators.

For \(-\frac{27}{5} \times -\frac{1}{9}\), simplify to get the end result. This is sometimes described as "flip and multiply."

Simplifying Fractions

Simplifying fractions makes them easier to work with or interpret. After performing operations such as division or multiplication, you might get a fraction that can be simplified. Simplifying involves breaking down both the numerator and the denominator into the smallest numbers possible.Consider \(\frac{3}{5}\), which resulted from our calculation. If the numerator and denominator don't share any common factors (besides 1), the fraction is already in simplest form.To simplify any fraction effectively, follow these steps:

• Identify the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.

Being able to reduce fractions is a valuable skill, especially as it declutters expressions, making them much more manageable.