Math Problem Statement

Simplify 3 4/9 ÷ (5 1/3 - 2 3/4) + 5 9/10

Solution

To simplify the given expression, let's break down the steps involved. The expression is:

349÷(513234)+59103 \frac{4}{9} \div \left( 5 \frac{1}{3} - 2 \frac{3}{4} \right) + 5 \frac{9}{10}

Step 1: Convert Mixed Numbers to Improper Fractions

  1. 349=279+49=3193 \frac{4}{9} = \frac{27}{9} + \frac{4}{9} = \frac{31}{9}
  2. 513=153+13=1635 \frac{1}{3} = \frac{15}{3} + \frac{1}{3} = \frac{16}{3}
  3. 234=84+34=1142 \frac{3}{4} = \frac{8}{4} + \frac{3}{4} = \frac{11}{4}
  4. 5910=5010+910=59105 \frac{9}{10} = \frac{50}{10} + \frac{9}{10} = \frac{59}{10}

Step 2: Simplify the Expression Inside the Parentheses

163114\frac{16}{3} - \frac{11}{4}

To subtract these fractions, find a common denominator:

  • Common denominator of 3 and 4 is 12.
  • Convert each fraction: 163=6412,114=3312\frac{16}{3} = \frac{64}{12}, \quad \frac{11}{4} = \frac{33}{12}

Subtract the fractions: 64123312=3112\frac{64}{12} - \frac{33}{12} = \frac{31}{12}

Step 3: Divide the First Fraction by the Result of the Parentheses

319÷3112\frac{31}{9} \div \frac{31}{12}

Division of fractions means multiplying by the reciprocal: 319×1231=31×129×31=129=43\frac{31}{9} \times \frac{12}{31} = \frac{31 \times 12}{9 \times 31} = \frac{12}{9} = \frac{4}{3}

Step 4: Add the Result to the Last Fraction

43+5910\frac{4}{3} + \frac{59}{10}

Find a common denominator for 3 and 10, which is 30: 43=4030,5910=17730\frac{4}{3} = \frac{40}{30}, \quad \frac{59}{10} = \frac{177}{30}

Add these fractions: 4030+17730=21730\frac{40}{30} + \frac{177}{30} = \frac{217}{30}

Final Answer

The simplified expression is: 21730\frac{217}{30}

Or, as a mixed number: 77307 \frac{7}{30}

Would you like any further details or have any questions about the steps? Here are some related questions:

  1. How do you convert mixed numbers to improper fractions?
  2. What is the process to find a common denominator for fractions?
  3. How do you multiply and divide fractions?
  4. How can you simplify complex fractions more efficiently?
  5. Can this result be expressed in decimal form?

Tip: Always perform operations inside parentheses first and convert mixed numbers to improper fractions for easier calculation.

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Math Problem Analysis

Mathematical Concepts

Fractions
Mixed Numbers
Division
Subtraction
Addition
Improper Fractions

Formulas

Convert mixed numbers to improper fractions
Fraction subtraction: a/b - c/d = (ad - bc) / bd
Fraction division: a/b ÷ c/d = a/b × d/c
Find a common denominator for addition or subtraction

Theorems

Basic arithmetic operations on fractions

Suitable Grade Level

Grades 6-8