What is $\frac{1}{2}+\frac{5}{6}+\frac{4}{3}$ ?

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teamplayer4848 teamplayer4848

06-11-2026
Mathematics

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What is $\frac{1}{2}+\frac{5}{6}+\frac{4}{3}$ ?

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-06T20:45:11-05:00

Identify the least common denominator (LCD) for 2, 6, and 3, which is 6.
Convert each fraction to an equivalent fraction with the LCD: $\frac{3}{6}, \frac{5}{6}, \frac{8}{6}$.
Sum the fractions to obtain $\frac{16}{6}$.
Simplify and convert the final fraction: $\frac{16}{6} = \frac{8}{3} = \boxed{2 \frac{2}{3}}$.

Explanation

Finding the Common Denominator

To find the sum of $\frac{1}{2} + \frac{5}{6} + \frac{4}{3}$, we need to add these three fractions. The first step is to ensure that the fractions have a common denominator. The denominators are 2, 6, and 3. We find the least common multiple (LCM) of these numbers. The multiples of 2 are 2, 4, 6, 8... The multiples of 6 are 6, 12, 18... The multiples of 3 are 3, 6, 9... The smallest common multiple among these is 6.

Converting to Equivalent Fractions

Now, we convert each fraction to an equivalent fraction with a denominator of 6:

For $\frac{1}{2}$: Multiply both numerator and denominator by 3, so $\frac{1 \times 3}{2 \times 3} = \frac{3}{6}$.
For $\frac{5}{6}$: The denominator is already 6, so it remains $\frac{5}{6}$.
For $\frac{4}{3}$: Multiply both numerator and denominator by 2, so $\frac{4 \times 2}{3 \times 2} = \frac{8}{6}$.

Adding the Fractions

Now that all fractions have the same denominator, we add the numerators:
\[\frac{3}{6} + \frac{5}{6} + \frac{8}{6} = \frac{3 + 5 + 8}{6}\]
Summing the numerators: $3 + 5 + 8 = 16$.
So the sum is $\frac{16}{6}$.

Simplifying and Converting the Result

To simplify $\frac{16}{6}$, we find the greatest common divisor of 16 and 6, which is 2. Dividing both the numerator and the denominator by 2 gives:
\[\frac{16 \div 2}{6 \div 2} = \frac{8}{3}\]
To express this as a mixed number, we divide 8 by 3. Since 3 goes into 8 two times with a remainder of 2, the mixed number is $2 \frac{2}{3}$.

Examples

Adding fractions with different denominators is a fundamental skill used in many daily tasks, such as cooking or construction. For instance, if you are measuring ingredients for a recipe, you might need to add $\frac{1}{2}$ cup of flour, $\frac{5}{6}$ cup of sugar, and $\frac{4}{3}$ cups of oats. By finding a common denominator, you can easily determine the total volume of dry ingredients required, which is $2 \frac{2}{3}$ cups, ensuring your measurements remain precise for your cooking needs.