Mixed Numbers Questions with Solutions

Questions on fractions and mixed numbers are presented along with with answers and detailed solutions
NOTE: In what follows a mixed number with a whole part \( n \) and a fractional part \( \dfrac{a}{b} \) is written as \( n \dfrac{a}{b} \).
For example the mixed number \( 3 \dfrac{1}{5} \) has a whole part \( 3 \) and a fractional part \( \dfrac{1}{5} \) and has the mathematical meaning: \( 3 \dfrac{1}{5} = 3 + \dfrac{1}{5}\) .
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1. Write the improper fraction \( \dfrac{21}{5} \) as a mixed number.
   A) \( \quad 1 \dfrac{1}{5} \)
   B) \( \quad 2 \dfrac{1}{5} \)
   C) \( \quad 4 \dfrac{1}{5} \)
   D) \( \quad 4 \dfrac{1}{4} \)
   E) \( \quad \dfrac{5}{21} \)
   
   
2. Write the mixed \( 4 \dfrac{1}{3} \) number as an improper fraction.
   A) \( \quad \dfrac{4}{3} \)
   B) \( \quad \dfrac{13}{3} \)
   C) \( \quad \dfrac{5}{3} \)
   D) \( \quad \dfrac{3}{13} \)
   E) \( \quad \dfrac{1}{3} \)
   
   
3. Add the mixed numbers and simplify the expression
   \[ 3 \dfrac{2}{5} + 1 \dfrac{3}{5} \]
   A) \( \quad 4 \)
   B) \( \quad 1 \)
   C) \( \quad 4 \dfrac{2}{5} \)
   D) \( \quad 4 \dfrac{3}{5} \)
   E) \( \quad 5 \)
   
   
4. Simplify the expression by adding the mixed numnbers
   \[ 5 \dfrac{2}{3} + 6 \dfrac{3}{4} \]
   A) \( \quad 12 \dfrac{5}{12} \)
   B) \( \quad 12 \)
   C) \( \quad 11 \dfrac{5}{12} \)
   D) \( \quad 11 \)
   E) \( \quad \dfrac{17}{12} \)
   
   
5. Subtract the mixed numbers and simplify the expression
   \[ 7 \dfrac{2}{3} - 4 \dfrac{1}{5} \]
   A) \( \quad 3 \)
   B) \( \quad 3 \dfrac{1}{2} \)
   C) \( \quad 3 \dfrac{7}{15} \)
   D) \( \quad - 3 \dfrac{7}{15} \)
   E) \( \quad 4 \)
   
   
6. Simplify the expression by subtracting the mixed numbers
   \[ 9 \dfrac{1}{4} - 5 \dfrac{3}{4} \]
   A) \( \quad 4 \dfrac{1}{2} \)
   B) \( \quad 4 \)
   C) \( \quad 2 \dfrac{1}{2} \)
   D) \( \quad 3 \dfrac{1}{2} \)
   E) \( \quad 4 \)
   
   
7. Multiply the mixed numbers and simplify the expression
   \[ (1 \dfrac{1}{3}) \times (2 \dfrac{2}{3}) \]
   A) \( \quad 3 \dfrac{5}{9} \)
   B) \( \quad 2 \dfrac{2}{9} \)
   C) \( \quad \dfrac{8}{9} \)
   D) \( \quad \dfrac{2}{9} \)
   E) \( \quad 2 \)
   
   
8. Divide the mixed numbers and simplify the expression
   \[ (3 \dfrac{1}{2}) \div (2 \dfrac{1}{2}) \]
   A) \( \quad \dfrac{3}{2} \)
   B) \( \quad \dfrac{2}{3} \)
   C) \( \quad \dfrac{2}{5} \)
   D) \( \quad 3 \dfrac{2}{5} \)
   E) \( \quad 1 \dfrac{2}{5} \)
   
   
9. What number should be added to the expression \( 1 \dfrac{1}{3} - 2 \dfrac{1}{2} \) to obtain \( 2 \)?
   A) \( \quad 3 \)
   B) \( \quad 2 \dfrac{1}{2} \)
   C) \( \quad 3 \dfrac{1}{6} \)
   D) \( \quad1 \dfrac{1}{3} \)
   E) \( \quad 1 \dfrac{1}{2} \)
   
   
10. What number should be subtracted from the expression \( 2 \dfrac{1}{5} + 5 \dfrac{1}{3} \) to obtain \( 0 \)?
    A) \( \quad 7 \dfrac{1}{5} \)
    B) \( \quad 7 \dfrac{1}{3} \)
    C) \( \quad 7 \)
    D) \( \quad 7 \dfrac{8}{15} \)
    E) \( \quad 6 \dfrac{8}{15} \)
    
    
11. What is the reciprocal of the mixed number \( 2 \dfrac{1}{8} \)?
    A) \( \dfrac{8}{17} \)
    B) \( \quad 2 \dfrac{8}{17} \)
    C) \( \quad \dfrac{1}{2} \)
    D) \( \quad \dfrac{4}{5} \)
    E) \( \quad \dfrac{8}{21} \)

1. Answer C
   Solution
   Given \( \dfrac{21}{5} \)
   The above is an improper fraction and may therefore be a mixed number.
   Use division of whole numbers to write: \( \quad \dfrac{21}{5} = 4 + \dfrac{1}{5} \) where \( 4 \) is the quotient and \( 1 \) is the remainder of the division of whole numbers.
   Hence the improper fraction \( \dfrac{21}{5} \) may be written as the mixed number \( \quad 4 \dfrac{1}{5} \)
   
   
2. Answer B
   Solution
   Given \( 4 \dfrac{1}{3} \)
   Write as a sum of a whole number and a fraction: \( \quad 4 \dfrac{1}{3} = 4 + \dfrac{1}{3} \)
   Rewrite the whole number \( 4 \) as a fraction with denominator \( 3 \):\( \quad \dfrac{4 \times 3}{3} + \dfrac{1}{3} = \dfrac{12}{3} + \dfrac{1}{3} = \dfrac{13}{3} \)
   Hence the mixed number \( 4 \dfrac{1}{3} \) may be written as the improper fraction \( \quad \dfrac{13}{3} \)
   
   
3. Answer E
   Solution
   In order to add mixed numbers, we need to write each mixed number as sum of a whole number and a fraction.
   Hence: \( \quad 3 \dfrac{2}{5} + 1 \dfrac{3}{5} = 3 + \dfrac{2}{5} + 1 + \dfrac{3}{5} \)
   We now add the whole parts and the fractional parts and simplify: \( \quad = (3 + 1) + (\dfrac{2}{5} + \dfrac{3}{5}) \)
   Simplify: \( \quad = 4 + \dfrac{5}{5} = 4 + 1 = 5 \)
   
   
4. Answer A
   Solution
   Add whole numbers together and fractions together: \( \quad 5 \dfrac{2}{3} + 6 \dfrac{3}{4} = (5 + 6) + (\dfrac{2}{3} + \dfrac{3}{4}) \)
   Simplify: \( \quad = 11 + (\dfrac{2}{3} + \dfrac{3}{4}) \)
   Find the lowest common multiple of the denominators \( 3 \) and \( 4 \) which is \( 12 \) and rewrite the fraction with common denominator \( 12\)
   \( \quad = 11 + (\dfrac{2 \times 4}{3 \times 4} + \dfrac{3 \times 3}{4 \times 3}) \)
   Simplify: \( \quad = 11 + \dfrac{8+9}{12} = 11 + \dfrac{17}{12} \quad \) (I)
   Divide \( 17 \) by \( 12 \) and write the improper fraction \( \dfrac{17}{12} \) as a mixed number as follows: \( \quad \dfrac{17}{12} = 1 + \dfrac{5}{12} \)
   We now substitute in expression (II) and simplify
   \( \quad 5 \dfrac{2}{3} + 6 \dfrac{3}{4} = 11 + 1 + \dfrac{5}{12} = 12 \dfrac{5}{12}\)
   
   
5. Answer C
   Solution
   Subtract whole numbers separately and fractions separately: \( \quad 7 \dfrac{2}{3} - 4 \dfrac{1}{5} = (7 - 4) + (\dfrac{2}{3} - \dfrac{1}{5}) \)
   Simplify: \( \quad = 3 + (\dfrac{2}{3} - \dfrac{1}{5}) \)
   The lowest common multiple of the denominators \( 3 \) and \( 5 \) is \( 15 \); rewrite the fraction with common denominator \( 15\)
   ; \( \quad = 3 + (\dfrac{2 \times 5}{3 \times 5} - \dfrac{1 \times 3}{5 \times 3}) \)
   Simplify: \( \quad = 3 + \dfrac{10-3}{15} = 3 \dfrac{7}{15} \quad \)
   
   
6. Answer D
   Solution
   Subtract whole numbers separately and fractions separately: \( \quad 9 \dfrac{1}{4} - 5 \dfrac{3}{4} = (9 - 5) + (\dfrac{1}{4} - \dfrac{3}{4}) \)
   Simplify: \( \quad = 4 - \dfrac{2}{4} = 4 - \dfrac{1}{2} \)
   The whole part is positive and the fractional part is negative. We therefore need to make the fractional part positive by rewriting the above as: \( \quad 4 - \dfrac{1}{2} = 3 + (1 - \dfrac{1}{2}) \)
   Simplify: \( \quad = 3 + \dfrac{1}{2} = 3 \dfrac{1}{2} \quad \)
   
   
7. Answer A
   Solution
   In order to multiply mixed numbers, we first rewrite them as improper fractions: \( \quad (1 \dfrac{1}{3}) \times (2 \dfrac{2}{3}) = \dfrac{4}{3} \times \dfrac{8}{3} = \dfrac{32}{9} \)
   Rewrite the improper fraction \( \dfrac{32}{9} \) as a mixed number: \( \quad \dfrac{32}{9} = 3 \dfrac{5}{9} \)
   
   
8. Answer E
   Solution
   In order to divide mixed numbers, we first rewrite them as improper fractions: \( \quad (3 \dfrac{1}{2}) \div (2 \dfrac{1}{2}) = \dfrac{7}{2} \div \dfrac{5}{2} \)
   Use the rule of division of two fractions (multiply by the reciprocal): \( \quad = \dfrac{7}{2} \times \dfrac{2}{5} \)
   Simplify: \( \quad = \dfrac{7}{5} \)
   Rewrite the improper fraction \( \dfrac{7}{5} \) as a mixed number: \( \quad \dfrac{7}{5} = 1 \dfrac{2}{5} \)
   
   
   
   
9. Answer C
   Solution
   Let \( x \) be the number to be added to \( 1 \dfrac{1}{3} - 2 \dfrac{1}{2} \) to obtain \( 2\). Hence \( \quad 1 \dfrac{1}{3} - 2 \dfrac{1}{2} + x = 2\)
   Solve for \( x \): \( \quad x = 2 - 1 \dfrac{1}{3} + 2 \dfrac{1}{2} \)
   Group the terms on the right: \( \quad x = (2 - 1 + 2 ) - \dfrac{1}{3} + \dfrac{1}{2} \)
   Simplify to obtain: \( \quad x = 3 - \dfrac{2}{6} + \dfrac{3}{6} = 3 \dfrac{1}{6} \)
   
   
10. Answer D
    Solution
    Let \( x \) be the number to be subtracted from \( 2 \dfrac{1}{5} + 5 \dfrac{1}{3} \) to obtain \( 0 \). Hence \( \quad 2 \dfrac{1}{5} + 5 \dfrac{1}{3} - x = 0\)
    Solve for \( x \): \( \quad x = 2 \dfrac{1}{5} + 5 \dfrac{1}{3} \)
    Simplify the right side to obtain: \( \quad x = 7\dfrac{8}{15} \)
    
    
11. Answer A
    Solution
    We first need to rewrite the given mixed number as an improper fraction: \( 2 \dfrac{1}{8} = \dfrac{17}{8} \)
    The reciprocal of the given number is given by: \( \quad \dfrac{1}{\dfrac{17}{8}} = \dfrac{8}{17} \)