Calculate The Sum Of $3 \frac{2}{3} + 2 \frac{1}{4}$.

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Understanding Mixed Numbers

In mathematics, a mixed number is a combination of a whole number and a fraction. It is written in the form of $a \frac{b}{c}$, where $a$ is the whole number part, and $\frac{b}{c}$ is the fractional part. Mixed numbers are commonly used to represent quantities that have both a whole and a fractional part.

The Problem: Calculating the Sum of Mixed Numbers

In this article, we will focus on calculating the sum of two mixed numbers: $3 \frac{2}{3} + 2 \frac{1}{4}$. To solve this problem, we need to follow a step-by-step approach.

Step 1: Convert Mixed Numbers to Improper Fractions

To add mixed numbers, we need to convert them to improper fractions first. An improper fraction is a fraction where the numerator is greater than or equal to the denominator.

To convert a mixed number to an improper fraction, we multiply the whole number part by the denominator and add the numerator. The result is then written as the new numerator over the denominator.

For the first mixed number, $3 \frac{2}{3}$, we multiply the whole number part (3) by the denominator (3) and add the numerator (2). This gives us:

3×3+2=113 \times 3 + 2 = 11

So, the improper fraction equivalent of $3 \frac{2}{3}$ is $\frac{11}{3}$.

For the second mixed number, $2 \frac{1}{4}$, we multiply the whole number part (2) by the denominator (4) and add the numerator (1). This gives us:

2×4+1=92 \times 4 + 1 = 9

So, the improper fraction equivalent of $2 \frac{1}{4}$ is $\frac{9}{4}$.

Step 2: Find a Common Denominator

To add fractions, we need to have a common denominator. The common denominator is the least common multiple (LCM) of the denominators of the two fractions.

In this case, the denominators are 3 and 4. The LCM of 3 and 4 is 12.

Step 3: Convert Fractions to Have a Common Denominator

We need to convert both fractions to have a denominator of 12.

For the first fraction, $\frac{11}{3}$, we multiply the numerator and denominator by 4 to get:

11×43×4=4412\frac{11 \times 4}{3 \times 4} = \frac{44}{12}

For the second fraction, $\frac{9}{4}$, we multiply the numerator and denominator by 3 to get:

9×34×3=2712\frac{9 \times 3}{4 \times 3} = \frac{27}{12}

Step 4: Add the Fractions

Now that both fractions have a common denominator, we can add them.

4412+2712=7112\frac{44}{12} + \frac{27}{12} = \frac{71}{12}

Step 5: Convert the Result to a Mixed Number

To convert the improper fraction $\frac{71}{12}$ to a mixed number, we divide the numerator (71) by the denominator (12).

71÷12=5 with a remainder of 1171 \div 12 = 5 \text{ with a remainder of } 11

So, the mixed number equivalent of $\frac{71}{12}$ is $5 \frac{11}{12}$.

Conclusion

In this article, we calculated the sum of two mixed numbers: $3 \frac{2}{3} + 2 \frac{1}{4}$. We followed a step-by-step approach to convert the mixed numbers to improper fractions, find a common denominator, convert the fractions to have a common denominator, add the fractions, and finally convert the result to a mixed number.

The final answer is: 511125 \frac{11}{12}

Additional Tips and Examples

  • When adding mixed numbers, it is essential to follow the order of operations (PEMDAS) and convert the mixed numbers to improper fractions first.
  • To find a common denominator, you can use the least common multiple (LCM) of the denominators.
  • When converting fractions to have a common denominator, you can multiply the numerator and denominator by the same number.
  • When adding fractions, you can add the numerators and keep the common denominator.
  • When converting the result to a mixed number, you can divide the numerator by the denominator and write the remainder as the new numerator over the denominator.

Practice Problems

  • Calculate the sum of $2 \frac{3}{4} + 1 \frac{1}{2}$.
  • Calculate the sum of $4 \frac{2}{3} + 3 \frac{1}{4}$.
  • Calculate the sum of $1 \frac{1}{2} + 2 \frac{3}{4}$.

Real-World Applications

  • Mixed numbers are commonly used in cooking and recipes to measure ingredients.
  • Mixed numbers are used in construction and architecture to measure lengths and widths of buildings.
  • Mixed numbers are used in finance and accounting to calculate interest rates and investments.

Conclusion

In conclusion, calculating the sum of mixed numbers requires a step-by-step approach. By following the order of operations, converting mixed numbers to improper fractions, finding a common denominator, converting fractions to have a common denominator, adding the fractions, and finally converting the result to a mixed number, we can accurately calculate the sum of mixed numbers.

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Q: What is the difference between a mixed number and an improper fraction?

A: A mixed number is a combination of a whole number and a fraction, written in the form of $a \frac{b}{c}$. An improper fraction is a fraction where the numerator is greater than or equal to the denominator, written in the form of $\frac{a}{b}$.

Q: How do I convert a mixed number to an improper fraction?

A: To convert a mixed number to an improper fraction, you multiply the whole number part by the denominator and add the numerator. The result is then written as the new numerator over the denominator.

For example, to convert $3 \frac{2}{3}$ to an improper fraction, you would multiply the whole number part (3) by the denominator (3) and add the numerator (2), resulting in $\frac{11}{3}$.

Q: How do I find a common denominator for two fractions?

A: To find a common denominator for two fractions, you can use the least common multiple (LCM) of the denominators. The LCM is the smallest number that both denominators can divide into evenly.

For example, to find a common denominator for $\frac{1}{2}$ and $\frac{1}{3}$, you would find the LCM of 2 and 3, which is 6.

Q: How do I add fractions with different denominators?

A: To add fractions with different denominators, you need to find a common denominator and then convert both fractions to have that common denominator.

For example, to add $\frac{1}{2}$ and $\frac{1}{3}$, you would find a common denominator (6) and then convert both fractions to have that common denominator:

12=1×32×3=36\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}

13=1×23×2=26\frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}

Then, you can add the fractions:

36+26=56\frac{3}{6} + \frac{2}{6} = \frac{5}{6}

Q: How do I convert an improper fraction to a mixed number?

A: To convert an improper fraction to a mixed number, you divide the numerator by the denominator and write the remainder as the new numerator over the denominator.

For example, to convert $\frac{11}{3}$ to a mixed number, you would divide the numerator (11) by the denominator (3), resulting in 3 with a remainder of 2. The mixed number equivalent of $\frac{11}{3}$ is $3 \frac{2}{3}$.

Q: What are some real-world applications of mixed numbers?

A: Mixed numbers are commonly used in cooking and recipes to measure ingredients. They are also used in construction and architecture to measure lengths and widths of buildings. Additionally, mixed numbers are used in finance and accounting to calculate interest rates and investments.

Q: Can I use a calculator to calculate the sum of mixed numbers?

A: Yes, you can use a calculator to calculate the sum of mixed numbers. However, it's essential to understand the underlying math and be able to convert mixed numbers to improper fractions and vice versa.

Q: How do I simplify a mixed number?

A: To simplify a mixed number, you can convert it to an improper fraction and then simplify the fraction.

For example, to simplify $3 \frac{2}{4}$, you would convert it to an improper fraction:

324=3×4+24=1443 \frac{2}{4} = \frac{3 \times 4 + 2}{4} = \frac{14}{4}

Then, you can simplify the fraction:

144=72\frac{14}{4} = \frac{7}{2}

The simplified mixed number equivalent of $\frac{7}{2}$ is $3 \frac{1}{2}$.

Q: Can I use mixed numbers in algebraic expressions?

A: Yes, you can use mixed numbers in algebraic expressions. However, it's essential to understand that mixed numbers are not always the most convenient or efficient way to represent quantities in algebraic expressions.

Q: How do I graph a mixed number on a number line?

A: To graph a mixed number on a number line, you can start by graphing the whole number part and then adding the fractional part.

For example, to graph $3 \frac{2}{3}$ on a number line, you would start by graphing the whole number part (3) and then adding the fractional part ($\frac{2}{3}$).

Q: Can I use mixed numbers in geometry?

A: Yes, you can use mixed numbers in geometry to measure lengths and widths of shapes and figures.

Q: How do I calculate the area of a shape using mixed numbers?

A: To calculate the area of a shape using mixed numbers, you can use the formula for the area of a shape and substitute the mixed numbers into the formula.

For example, to calculate the area of a rectangle with a length of $3 \frac{2}{3}$ and a width of $2 \frac{1}{2}$, you would use the formula:

Area = length x width

Substituting the mixed numbers into the formula, you would get:

Area = $3 \frac{2}{3}$ x $2 \frac{1}{2}$

To calculate the product of the mixed numbers, you would convert them to improper fractions and then multiply:

323=1133 \frac{2}{3} = \frac{11}{3}

212=522 \frac{1}{2} = \frac{5}{2}

Multiplying the fractions, you would get:

113×52=556\frac{11}{3} \times \frac{5}{2} = \frac{55}{6}

The area of the rectangle is $\frac{55}{6}$ square units.

Q: Can I use mixed numbers in trigonometry?

A: Yes, you can use mixed numbers in trigonometry to measure angles and sides of triangles.

Q: How do I calculate the sine, cosine, and tangent of an angle using mixed numbers?

A: To calculate the sine, cosine, and tangent of an angle using mixed numbers, you can use the definitions of these trigonometric functions and substitute the mixed numbers into the formulas.

For example, to calculate the sine of an angle with a measure of $3 \frac{2}{3}$ degrees, you would use the formula:

sin(angle) = opposite side / hypotenuse

Substituting the mixed number into the formula, you would get:

sin($3 \frac{2}{3}$) = opposite side / hypotenuse

To calculate the sine of the angle, you would convert the mixed number to an improper fraction and then substitute it into the formula:

323=1133 \frac{2}{3} = \frac{11}{3}

sin($\frac{11}{3}$) = opposite side / hypotenuse

The sine of the angle is $\frac{11}{3}$.

Q: Can I use mixed numbers in calculus?

A: Yes, you can use mixed numbers in calculus to measure rates of change and accumulation.

Q: How do I calculate the derivative of a function using mixed numbers?

A: To calculate the derivative of a function using mixed numbers, you can use the definition of the derivative and substitute the mixed numbers into the formula.

For example, to calculate the derivative of a function with a mixed number coefficient, you would use the formula:

f'(x) = d/dx (f(x))

Substituting the mixed number into the formula, you would get:

f'(x) = d/dx ($3 \frac{2}{3}$x^2)

To calculate the derivative, you would convert the mixed number to an improper fraction and then substitute it into the formula:

323=1133 \frac{2}{3} = \frac{11}{3}

f'(x) = d/dx ($\frac{11}{3}$x^2)

The derivative of the function is $\frac{22}{3}$x.

Q: Can I use mixed numbers in statistics?

A: Yes, you can use mixed numbers in statistics to measure means and standard deviations.

Q: How do I calculate the mean of a dataset using mixed numbers?

A: To calculate the mean of a dataset using mixed numbers, you can use the formula for the mean and substitute the mixed numbers into the formula.

For example, to calculate the mean of a dataset with a mixed number mean, you would use the formula:

mean = (sum of data points) / (number of data points)

Substituting the mixed number into the formula, you would get:

mean = ($3 \frac{2}{3}$ + $2 \frac{1}{2}$) / 2

To calculate the mean, you would convert the mixed numbers to improper fractions and then substitute them into the formula:

3 \frac{2}{3} = \frac{11}{3