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Chapter 5: Problem 66

Find each sum or difference. Write in simplest form. $$\frac{13}{14}+\frac{3}{14}$$

Short Answer

Expert verified
\( \frac{13}{14} + \frac{3}{14} = \frac{8}{7} = 1\frac{1}{7} \).

Step by step solution

01

Identify the Problem

We need to find the sum of the two fractions \( \frac{13}{14} \) and \( \frac{3}{14} \).
02

Check the Denominators

Both fractions \( \frac{13}{14} \) and \( \frac{3}{14} \) have the same denominator, which is 14, so we can proceed to add them directly.
03

Add the Numerators

Since the denominators are the same, add the numerators: \( 13 + 3 = 16 \). The result of the addition is \( \frac{16}{14} \).
04

Simplify the Fraction

We simplify \( \frac{16}{14} \) by finding the greatest common divisor (GCD) of 16 and 14, which is 2. Divide both the numerator and denominator by 2: \( \frac{16 \div 2}{14 \div 2} = \frac{8}{7} \).
05

Check if Simplified Fraction is Proper

\( \frac{8}{7} \) is an improper fraction. It cannot be simplified further, but it can be written as a mixed number: \( 1 \frac{1}{7} \).

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Key Concepts

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

Addition of Fractions
Adding fractions is a fundamental concept in mathematics, and it's crucial to understand how to do it correctly. The key is making sure the denominators are the same. When fractions have the same denominator, you can add them directly by adding their numerators and keeping the common denominator.
For example, with the fractions \( \frac{13}{14} \) and \( \frac{3}{14} \), both fractions have 14 as the denominator. This means you can directly add the numerators 13 and 3 together, which gives 16.
Thus, the sum is \( \frac{16}{14} \).
  • Keep the denominator the same when adding.
  • Add only the numerators.
This straightforward rule simplifies the process and ensures the result is accurate.
Simplifying Fractions
Simplifying fractions involves reducing the fraction to its lowest terms. It makes the fraction easier to understand and use.
After adding two fractions, you might end up with a result like \( \frac{16}{14} \). This fraction is not in its simplest form since both the numerator (16) and the denominator (14) can be divided by a common number.
To simplify \( \frac{16}{14} \), perform the following steps:
  • Identify a number that divides both the numerator and the denominator evenly.
  • In this case, the number is 2.
  • Divide the numerator by 2: \( 16 \div 2 = 8 \).
  • Divide the denominator by 2: \( 14 \div 2 = 7 \).
The result is \( \frac{8}{7} \), which is the simplest form of the fraction.
Greatest Common Divisor
Finding the greatest common divisor (GCD) helps simplify fractions effectively. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder.
To find the GCD of 16 and 14, list the divisors of each number:
  • Divisors of 16: 1, 2, 4, 8, 16
  • Divisors of 14: 1, 2, 7, 14
The greatest common divisor that appears in both lists is 2, which allows us to simplify \( \frac{16}{14} \) to \( \frac{8}{7} \).
This technique ensures fractions are expressed in their simplest form, making calculations easier in future steps.

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Most popular questions from this chapter

Find each sum. $$24+(-12)+15$$

Solve each equation. Check your solution. $$\frac{5}{8}=-\frac{1}{2} r$$

Write each number in standard form. $$2 \times 10^{3}$$

Write each fraction or mixed number as a decimal. Use a bar to show a repeating decimal. $$\frac{2}{5}$$

Estimate each product. $$1 \frac{2}{3} \cdot 4 \frac{1}{8}$$
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