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: $3 \dfrac{1}{5} = 3 + \dfrac{1}{5}$ .

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) $\quad \dfrac{8}{21}$

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 $\quad 4 \dfrac{1}{5}$

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 $\quad \dfrac{13}{3}$

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: $\quad = 4 + \dfrac{5}{5} = 4 + 1 = 5$

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
$\quad 5 \dfrac{2}{3} + 6 \dfrac{3}{4} = 11 + 1 + \dfrac{5}{12} = 12 \dfrac{5}{12}$

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: $\quad = 3 + \dfrac{10-3}{15} = 3 \dfrac{7}{15} \quad$

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: $\quad = 3 + \dfrac{1}{2} = 3 \dfrac{1}{2} \quad$

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: $\quad \dfrac{32}{9} = 3 \dfrac{5}{9}$

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: $\quad \dfrac{7}{5} = 1 \dfrac{2}{5}$

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: $\quad x = 3 - \dfrac{2}{6} + \dfrac{3}{6} = 3 \dfrac{1}{6}$

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: $\quad x = 7\dfrac{8}{15}$
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: $\quad \dfrac{1}{\dfrac{17}{8}} = \dfrac{8}{17}$