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Chapter 3: Problem 27

Add and simplify. $$ \frac{5}{12}+\frac{7}{24} $$

Short Answer

Expert verified
The sum is \( \frac{17}{24} \).

Step by step solution

01

- Find a Common Denominator

Identify the least common denominator (LCD) for the fractions involved. The denominators are 12 and 24. The LCD of 12 and 24 is 24.
02

- Convert Fractions to Common Denominator

Convert each fraction to have the common denominator of 24. For \( \frac{5}{12} \): Multiply both numerator and denominator by 2 to get \( \frac{5 \times 2}{12 \times 2} = \frac{10}{24} \).The fraction \(\frac{7}{24}\) already has 24 as a denominator.
03

- Add the Fractions

Add the fractions with the common denominator. \[\frac{10}{24} + \frac{7}{24} = \frac{10 + 7}{24} = \frac{17}{24} \]
04

- Simplify the Result

Check if the resulting fraction \( \frac{17}{24} \) can be simplified. In this case, 17 and 24 have no common factors other than 1, so \( \frac{17}{24} \) is already in its simplest form.

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Key Concepts

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

least common denominator
When adding fractions, we need a common basis to combine them effectively. This common basis is known as the Least Common Denominator (LCD). The LCD is the smallest number that both denominators can divide into evenly. For example, if you have fractions with denominators of 12 and 24, you need to find the smallest number that both 12 and 24 can divide into without a remainder. Here鈥檚 how you find the LCD in this case:
  • List the multiples of 12: 12, 24, 36, 48, ...
  • List the multiples of 24: 24, 48, 72, ...
The smallest number that appears in both lists is 24, so 24 is the LCD. Now that we have a common denominator, we can move on to convert the fractions.
fraction conversion
To add fractions, we need to convert them to have the same denominator. Conversion ensures that we are comparing and adding parts of the same size. Here鈥檚 how you do it:
  • For \( \frac{5}{12} \): Multiply both the numerator (top part) and the denominator (bottom part) by 2 to get \( \frac{10}{24} \). This is because 12 multiplied by 2 equals 24, our LCD.
  • For \( \frac{7}{24} \): The fraction is already in terms of our common denominator, so no changes are needed.
Once both fractions have the same denominator (24 in this case), they are ready to be added.
simplifying fractions
After adding fractions, it鈥檚 essential to simplify the result. Simplifying a fraction means reducing it to its simplest form where the numerator and denominator are as small as possible. Here鈥檚 what you do: \[ \frac{10}{24} + \frac{7}{24} = \frac{17}{24} \] Check if 17 and 24 have any common factors other than 1. Since they don鈥檛, \( \frac{17}{24} \) is already in its simplest form. Simplifying fractions makes numbers easier to understand and work with. In conclusion:
  • Always find the least common denominator first.
  • Convert fractions to have the same denominator.
  • Add the fractions and simplify the result if possible.
Understanding these steps will help you master adding fractions quickly and accurately.

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Most popular questions from this chapter

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