Simplify The Following Expressions:a. $3 \frac{3}{4} - 1 \frac{3}{3} + \frac{1}{6}$b. $2 \frac{1}{5} - 1 \frac{1}{6} + 2 \frac{1}{2}$c. $6 \frac{16}{3} + 12 \frac{1}{2} - 5 \frac{5}{12}$

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Introduction

In this article, we will simplify three given expressions involving mixed numbers and fractions. Mixed numbers are a combination of a whole number and a fraction, while fractions are a part of a whole. To simplify these expressions, we will first convert the mixed numbers to improper fractions, then perform the required operations, and finally convert the result back to a mixed number if necessary.

Simplifying Expression a

Expression a: 334−133+163 \frac{3}{4} - 1 \frac{3}{3} + \frac{1}{6}

To simplify this expression, we will first convert the mixed numbers to improper fractions.

  • 334=(3×4)+34=12+34=1543 \frac{3}{4} = \frac{(3 \times 4) + 3}{4} = \frac{12 + 3}{4} = \frac{15}{4}
  • 133=(1×3)+33=3+33=63=21 \frac{3}{3} = \frac{(1 \times 3) + 3}{3} = \frac{3 + 3}{3} = \frac{6}{3} = 2

Now, we can rewrite the expression as:

154−2+16\frac{15}{4} - 2 + \frac{1}{6}

To add and subtract fractions, we need to have the same denominator. The least common multiple (LCM) of 4 and 6 is 12. So, we will convert the fractions to have a denominator of 12.

  • 154=15×34×3=4512\frac{15}{4} = \frac{15 \times 3}{4 \times 3} = \frac{45}{12}
  • 2=2×1212=24122 = \frac{2 \times 12}{12} = \frac{24}{12}
  • 16=1×26×2=212\frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12}

Now, we can rewrite the expression as:

4512−2412+212\frac{45}{12} - \frac{24}{12} + \frac{2}{12}

To add and subtract fractions, we will combine the numerators and keep the denominator the same.

45−24+212=2312\frac{45 - 24 + 2}{12} = \frac{23}{12}

To convert the improper fraction back to a mixed number, we will divide the numerator by the denominator.

2312=11112\frac{23}{12} = 1 \frac{11}{12}

Therefore, the simplified expression is:

111121 \frac{11}{12}

Simplifying Expression b

Expression b: 215−116+2122 \frac{1}{5} - 1 \frac{1}{6} + 2 \frac{1}{2}

To simplify this expression, we will first convert the mixed numbers to improper fractions.

  • 215=(2×5)+15=10+15=1152 \frac{1}{5} = \frac{(2 \times 5) + 1}{5} = \frac{10 + 1}{5} = \frac{11}{5}
  • 116=(1×6)+16=6+16=761 \frac{1}{6} = \frac{(1 \times 6) + 1}{6} = \frac{6 + 1}{6} = \frac{7}{6}
  • 212=(2×2)+12=4+12=522 \frac{1}{2} = \frac{(2 \times 2) + 1}{2} = \frac{4 + 1}{2} = \frac{5}{2}

Now, we can rewrite the expression as:

115−76+52\frac{11}{5} - \frac{7}{6} + \frac{5}{2}

To add and subtract fractions, we need to have the same denominator. The least common multiple (LCM) of 5, 6, and 2 is 30. So, we will convert the fractions to have a denominator of 30.

  • 115=11×65×6=6630\frac{11}{5} = \frac{11 \times 6}{5 \times 6} = \frac{66}{30}
  • 76=7×56×5=3530\frac{7}{6} = \frac{7 \times 5}{6 \times 5} = \frac{35}{30}
  • 52=5×152×15=7530\frac{5}{2} = \frac{5 \times 15}{2 \times 15} = \frac{75}{30}

Now, we can rewrite the expression as:

6630−3530+7530\frac{66}{30} - \frac{35}{30} + \frac{75}{30}

To add and subtract fractions, we will combine the numerators and keep the denominator the same.

66−35+7530=10630\frac{66 - 35 + 75}{30} = \frac{106}{30}

To simplify the fraction, we will divide the numerator and denominator by their greatest common divisor (GCD).

10630=5315\frac{106}{30} = \frac{53}{15}

To convert the improper fraction back to a mixed number, we will divide the numerator by the denominator.

5315=3815\frac{53}{15} = 3 \frac{8}{15}

Therefore, the simplified expression is:

38153 \frac{8}{15}

Simplifying Expression c

Expression c: 6163+1212−55126 \frac{16}{3} + 12 \frac{1}{2} - 5 \frac{5}{12}

To simplify this expression, we will first convert the mixed numbers to improper fractions.

  • 6163=(6×3)+163=18+163=3436 \frac{16}{3} = \frac{(6 \times 3) + 16}{3} = \frac{18 + 16}{3} = \frac{34}{3}
  • 1212=(12×2)+12=24+12=25212 \frac{1}{2} = \frac{(12 \times 2) + 1}{2} = \frac{24 + 1}{2} = \frac{25}{2}
  • 5512=(5×12)+512=60+512=65125 \frac{5}{12} = \frac{(5 \times 12) + 5}{12} = \frac{60 + 5}{12} = \frac{65}{12}

Now, we can rewrite the expression as:

343+252−6512\frac{34}{3} + \frac{25}{2} - \frac{65}{12}

To add and subtract fractions, we need to have the same denominator. The least common multiple (LCM) of 3, 2, and 12 is 12. So, we will convert the fractions to have a denominator of 12.

  • 343=34×43×4=13612\frac{34}{3} = \frac{34 \times 4}{3 \times 4} = \frac{136}{12}
  • 252=25×62×6=15012\frac{25}{2} = \frac{25 \times 6}{2 \times 6} = \frac{150}{12}
  • 6512\frac{65}{12}

Now, we can rewrite the expression as:

13612+15012−6512\frac{136}{12} + \frac{150}{12} - \frac{65}{12}

To add and subtract fractions, we will combine the numerators and keep the denominator the same.

136+150−6512=22112\frac{136 + 150 - 65}{12} = \frac{221}{12}

To simplify the fraction, we will divide the numerator and denominator by their greatest common divisor (GCD).

22112=221÷1312÷4=173\frac{221}{12} = \frac{221 \div 13}{12 \div 4} = \frac{17}{3}

To convert the improper fraction back to a mixed number, we will divide the numerator by the denominator.

173=523\frac{17}{3} = 5 \frac{2}{3}

Therefore, the simplified expression is:

Introduction

In this article, we will simplify three given expressions involving mixed numbers and fractions. Mixed numbers are a combination of a whole number and a fraction, while fractions are a part of a whole. To simplify these expressions, we will first convert the mixed numbers to improper fractions, then perform the required operations, and finally convert the result back to a mixed number if necessary.

Simplifying Expression a

Expression a: 334−133+163 \frac{3}{4} - 1 \frac{3}{3} + \frac{1}{6}

To simplify this expression, we will first convert the mixed numbers to improper fractions.

  • 334=(3×4)+34=12+34=1543 \frac{3}{4} = \frac{(3 \times 4) + 3}{4} = \frac{12 + 3}{4} = \frac{15}{4}
  • 133=(1×3)+33=3+33=63=21 \frac{3}{3} = \frac{(1 \times 3) + 3}{3} = \frac{3 + 3}{3} = \frac{6}{3} = 2

Now, we can rewrite the expression as:

154−2+16\frac{15}{4} - 2 + \frac{1}{6}

To add and subtract fractions, we need to have the same denominator. The least common multiple (LCM) of 4 and 6 is 12. So, we will convert the fractions to have a denominator of 12.

  • 154=15×34×3=4512\frac{15}{4} = \frac{15 \times 3}{4 \times 3} = \frac{45}{12}
  • 2=2×1212=24122 = \frac{2 \times 12}{12} = \frac{24}{12}
  • 16=1×26×2=212\frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12}

Now, we can rewrite the expression as:

4512−2412+212\frac{45}{12} - \frac{24}{12} + \frac{2}{12}

To add and subtract fractions, we will combine the numerators and keep the denominator the same.

45−24+212=2312\frac{45 - 24 + 2}{12} = \frac{23}{12}

To convert the improper fraction back to a mixed number, we will divide the numerator by the denominator.

2312=11112\frac{23}{12} = 1 \frac{11}{12}

Therefore, the simplified expression is:

111121 \frac{11}{12}

Simplifying Expression b

Expression b: 215−116+2122 \frac{1}{5} - 1 \frac{1}{6} + 2 \frac{1}{2}

To simplify this expression, we will first convert the mixed numbers to improper fractions.

  • 215=(2×5)+15=10+15=1152 \frac{1}{5} = \frac{(2 \times 5) + 1}{5} = \frac{10 + 1}{5} = \frac{11}{5}
  • 116=(1×6)+16=6+16=761 \frac{1}{6} = \frac{(1 \times 6) + 1}{6} = \frac{6 + 1}{6} = \frac{7}{6}
  • 212=(2×2)+12=4+12=522 \frac{1}{2} = \frac{(2 \times 2) + 1}{2} = \frac{4 + 1}{2} = \frac{5}{2}

Now, we can rewrite the expression as:

115−76+52\frac{11}{5} - \frac{7}{6} + \frac{5}{2}

To add and subtract fractions, we need to have the same denominator. The least common multiple (LCM) of 5, 6, and 2 is 30. So, we will convert the fractions to have a denominator of 30.

  • 115=11×65×6=6630\frac{11}{5} = \frac{11 \times 6}{5 \times 6} = \frac{66}{30}
  • 76=7×56×5=3530\frac{7}{6} = \frac{7 \times 5}{6 \times 5} = \frac{35}{30}
  • 52=5×152×15=7530\frac{5}{2} = \frac{5 \times 15}{2 \times 15} = \frac{75}{30}

Now, we can rewrite the expression as:

6630−3530+7530\frac{66}{30} - \frac{35}{30} + \frac{75}{30}

To add and subtract fractions, we will combine the numerators and keep the denominator the same.

66−35+7530=10630\frac{66 - 35 + 75}{30} = \frac{106}{30}

To simplify the fraction, we will divide the numerator and denominator by their greatest common divisor (GCD).

10630=5315\frac{106}{30} = \frac{53}{15}

To convert the improper fraction back to a mixed number, we will divide the numerator by the denominator.

5315=3815\frac{53}{15} = 3 \frac{8}{15}

Therefore, the simplified expression is:

38153 \frac{8}{15}

Simplifying Expression c

Expression c: 6163+1212−55126 \frac{16}{3} + 12 \frac{1}{2} - 5 \frac{5}{12}

To simplify this expression, we will first convert the mixed numbers to improper fractions.

  • 6163=(6×3)+163=18+163=3436 \frac{16}{3} = \frac{(6 \times 3) + 16}{3} = \frac{18 + 16}{3} = \frac{34}{3}
  • 1212=(12×2)+12=24+12=25212 \frac{1}{2} = \frac{(12 \times 2) + 1}{2} = \frac{24 + 1}{2} = \frac{25}{2}
  • 5512=(5×12)+512=60+512=65125 \frac{5}{12} = \frac{(5 \times 12) + 5}{12} = \frac{60 + 5}{12} = \frac{65}{12}

Now, we can rewrite the expression as:

343+252−6512\frac{34}{3} + \frac{25}{2} - \frac{65}{12}

To add and subtract fractions, we need to have the same denominator. The least common multiple (LCM) of 3, 2, and 12 is 12. So, we will convert the fractions to have a denominator of 12.

  • 343=34×43×4=13612\frac{34}{3} = \frac{34 \times 4}{3 \times 4} = \frac{136}{12}
  • 252=25×62×6=15012\frac{25}{2} = \frac{25 \times 6}{2 \times 6} = \frac{150}{12}
  • 6512\frac{65}{12}

Now, we can rewrite the expression as:

13612+15012−6512\frac{136}{12} + \frac{150}{12} - \frac{65}{12}

To add and subtract fractions, we will combine the numerators and keep the denominator the same.

136+150−6512=22112\frac{136 + 150 - 65}{12} = \frac{221}{12}

To simplify the fraction, we will divide the numerator and denominator by their greatest common divisor (GCD).

22112=221÷1312÷4=173\frac{221}{12} = \frac{221 \div 13}{12 \div 4} = \frac{17}{3}

To convert the improper fraction back to a mixed number, we will divide the numerator by the denominator.

173=523\frac{17}{3} = 5 \frac{2}{3}

Therefore, the simplified expression is:

5235 \frac{2}{3}

Q&A

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, while an improper fraction is a fraction where the numerator is greater than or equal to the denominator.

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

A: To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator. Then, write the result as a fraction with the denominator.

Q: How do I add and subtract fractions?

A: To add and subtract fractions, you need to have the same denominator. The least common multiple (LCM) of the denominators is the common denominator. Then, combine the numerators and keep the denominator the same.

Q: How do I simplify a fraction?

A: To simplify a fraction, divide the numerator and denominator by their greatest common divisor (GCD).

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

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

Q: What is the least common multiple (LCM) of two or more numbers?

A: The least common multiple (LCM) of two or more numbers is the smallest number that is a multiple of each of the numbers.

Q: What is the greatest common divisor (GCD) of two or more numbers?

A: The greatest common divisor (GCD) of two or more numbers is the largest number that divides each of the numbers without leaving a remainder.

Q: How do I use the least common multiple (LCM) and greatest common divisor (GCD) to add and subtract fractions?

A: To add and subtract fractions, use the least common multiple (LCM) of the denominators as the common denominator. Then, combine the numerators and keep the denominator the same. To simplify the