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Chapter 5: Problem 33

Find the sums and differences. Be sure to reduce. $$ \frac{5}{2}+\frac{16}{2}-\frac{1}{2} $$

Short Answer

Expert verified
The simplified result is 10.

Step by step solution

01

Identify the Problem

We need to find the sum and the difference of the fractions given in the expression \( \frac{5}{2} + \frac{16}{2} - \frac{1}{2} \). All fractions have the same denominator.
02

Add the Fractions

Since the fractions \( \frac{5}{2} \) and \( \frac{16}{2} \) have the same denominator, add their numerators: \( 5 + 16 = 21 \). So, the result of the addition is \( \frac{21}{2} \).
03

Subtract the Fraction

Subtract \( \frac{1}{2} \) from \( \frac{21}{2} \). Since the denominators are the same, subtract the numerators: \( 21 - 1 = 20 \). So, the result is \( \frac{20}{2} \).
04

Simplify the Result

Simplify \( \frac{20}{2} \) by dividing the numerator and denominator by their greatest common divisor, which is 2. Doing this gives \( \frac{20 \div 2}{2 \div 2} = \frac{10}{1} \) or simply 10.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Adding Fractions
When adding fractions, the first and most crucial step is to make sure that the fractions have the same denominator. The denominator represents the total number of equal parts into which the whole is divided. If the denominators are the same, you can simply add the numerators (the top parts of the fractions) directly.

In our example, we have \( \frac{5}{2} \) and \( \frac{16}{2} \). Both fractions share the denominator 2, so we can easily proceed to add them:
  • Add the numerators: \( 5 + 16 = 21 \).
  • Keep the common denominator: 2.
Thus, after adding, we get \( \frac{21}{2} \). Make sure to always verify that denominators match before adding. When they don't, you'll need to find a common denominator before proceeding.
Subtracting Fractions
Subtracting fractions follows the same foundational rule as adding fractions: you need matching denominators. Only then can you directly subtract the numerators. In our case, after adding, we received \( \frac{21}{2} \) which needs to be reduced by \( \frac{1}{2} \):
  • The denominators are the same (2), so you just subtract the numerators.
  • Subtract: \( 21 - 1 = 20 \).
This yields the fraction \( \frac{20}{2} \). Remember, when the fractions have different denominators, find a common denominator before subtracting the numerators.
Simplifying Fractions
After adding or subtracting fractions, the final step should be simplifying the resulting fraction if possible. A fraction is simplified when you cannot divide the numerator and the denominator by any number other than 1 and still have whole numbers.

In our subtraction example, the result was \( \frac{20}{2} \). To simplify:
  • Identify the greatest common divisor (GCD) of the numerator (20) and the denominator (2), which is 2.
  • Divide both the numerator and the denominator by that GCD: \( \frac{20 \div 2}{2 \div 2} \).
  • This gives you \( \frac{10}{1} \).
Fraction \( \frac{10}{1} \) simplifies to the whole number 10. Simplifying fractions makes your answers clearer and often easier to understand.

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Most popular questions from this chapter

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