Problem 18 - Prealgebra

Question

Find sum or difference. Write in simplest form. \(-\frac{13}{16}+\left(-\frac{9}{16}\right)\)

Step-by-Step Solution

Answer

The sum is \(-\frac{11}{8}\) in simplest form.

Step 1: Understand the Problem

We need to find the sum of two negative fractions: \(-\frac{13}{16}\) and \(-\frac{9}{16}\). Since both fractions have the same denominator, it will be a straightforward addition of their numerators.

Step 2: Add the Numerators

Since the denominators are the same (16), we can add the numerators directly. The operation becomes: \(-13 + (-9) = -13 - 9 = -22\). This gives us the fraction \(-\frac{22}{16}\).

Step 3: Simplify the Fraction

Now, we simplify \(-\frac{22}{16}\). Both numerator and denominator can be divided by their greatest common divisor, which is 2. Dividing gives us \(-\frac{22 \div 2}{16 \div 2} = -\frac{11}{8}\).

Step 4: Conclude the Solution

Since \(-\frac{11}{8}\) is in its simplest form, we have found the sum of the fractions as a simplified improper fraction.

Key Concepts

Adding Fractions

Adding fractions might seem tricky at first, but it's manageable with practice.
When adding fractions, like \(-\frac{13}{16}\) and \(-\frac{9}{16}\), it's essential they have the same denominator.
A shared denominator allows you to simply add or subtract the numerators.
Think of it as having the same kind of parts; like pieces of a pizza.

• With common denominators, add the numerators: \(-13 + (-9) = -22\)
  The result becomes your new numerator, keeping the denominator the same.

It's always a good idea to check if your final fraction can be simplified. If you find a common factor for both the numerator and the denominator, use it to reduce your fraction.
For example, with \(-\frac{22}{16}\), dividing top and bottom by 2 gives \(-\frac{11}{8}\).

Negative Fractions

Handling negative fractions requires a solid understanding of both fractions and negative numbers. Adding two negative fractions, like \(-\frac{13}{16}\) and \(-\frac{9}{16}\), follows the same rules as working with positive fractions, just consider the negative signs.

• Adding negative numbers is like combining debts. You sum the absolute values, and the result keeps the negative sign.
• For example, \(-13 + (-9)\) becomes \(-22\). It's the same principle for numerators.

Remember, when adding or subtracting any numbers:- If they have the same sign, add their absolute values and keep the sign.- If they have different signs, subtract the smaller absolute value from the larger. Keep the sign of the larger number.
When you practice with negative fractions, you'll find them as straightforward as positive ones.

Improper Fractions

An improper fraction is simply a fraction where the numerator is larger than the denominator. They can often lead to confusion at first, but they're nothing to fear.
In the exercise mentioned, \(-\frac{11}{8}\) is an improper fraction. This means there are more numerators parts than a whole (denominator).

• To simplify or better understand improper fractions, you might convert them into mixed numbers, though it's not always necessary. \(\frac{11}{8}\) (ignoring the negative sign at first) converts to \(1\frac{3}{8}\).

To convert: Divide the numerator by the denominator. The whole number is the quotient, and the remainder becomes the new numerator over the original denominator.
Although you often leave answers in improper form in more advanced math because they're easier to work with in equations.