Pre-Algebra Workbook 5: Fractions II – Adding & Subtracting

The denominators are the same, so keep the denominator and add the numerators. Then check if the result can be simplified.

The fractions have the same denominator ($8$), so add the numerators:

$$\dfrac{3}{8} + \dfrac{1}{8} = \dfrac{3 + 1}{8} = \dfrac{4}{8}$$

Simplify $\dfrac{4}{8}$ by dividing numerator and denominator by $4$

$$\dfrac{4}{8} = \dfrac{4 \div 4}{8 \div 4} = \dfrac{1}{2}$$

So the sum is $\dfrac{1}{2}$

The correct choice is C. $\dfrac{1}{2}$.

The denominators are the same, so subtract the numerators and keep the denominator. Simplify if possible.

The fractions have the same denominator ($9$), so subtract the numerators:

$$\dfrac{7}{9} - \dfrac{2}{9} = \dfrac{7 - 2}{9} = \dfrac{5}{9}$$

The fraction $\dfrac{5}{9}$ cannot be simplified further (no common factors other than $1$).

So the difference is $\dfrac{5}{9}$

The correct choice is B. $\dfrac{5}{9}$.

The denominators are different. Find the least common denominator (LCD), rewrite each fraction with that denominator, then add.

Find the least common denominator of $4$ and $6$

The least common multiple of $4$ and $6$ is $12$ so use $12$ as the common denominator.

Rewrite each fraction with denominator $12$

$$\dfrac{1}{4} = \dfrac{3}{12}, \quad \dfrac{1}{6} = \dfrac{2}{12}$$

Now add the fractions:

$$\dfrac{3}{12} + \dfrac{2}{12} = \dfrac{3 + 2}{12} = \dfrac{5}{12}$$

The fraction $\dfrac{5}{12}$ is already in simplest form.

So the sum is $\dfrac{5}{12}$

The correct choice is C. $\dfrac{5}{12}$.

The denominators are different. Find the least common denominator, rewrite each fraction with that denominator, then subtract.

Find the least common denominator of $6$ and $4$

The least common multiple of $6$ and $4$ is $12$ so use $12$ as the common denominator.

Rewrite each fraction with denominator $12$

$$\dfrac{5}{6} = \dfrac{10}{12}, \quad \dfrac{1}{4} = \dfrac{3}{12}$$

Now subtract:

$$\dfrac{10}{12} - \dfrac{3}{12} = \dfrac{10 - 3}{12} = \dfrac{7}{12}$$

The fraction $\dfrac{7}{12}$ is already in simplest form.

So the difference is $\dfrac{7}{12}$

The correct choice is D. $\dfrac{7}{12}$.

You can add the fractional parts first, then combine with the whole number. Remember that $\dfrac{1}{4} + \dfrac{3}{4}$ makes a whole.

Start with the mixed number and the fraction:

$$2 \dfrac{1}{4} + \dfrac{3}{4}$$

Add the fractional parts:

$$\dfrac{1}{4} + \dfrac{3}{4} = \dfrac{4}{4} = 1$$

Now add this $1$ to the whole number $2$

$$2 + 1 = 3$$

So the sum is $3$

The correct choice is A. $3$.

You can either convert both mixed numbers to improper fractions, or add whole numbers and fractional parts separately. For the fractions, find a common denominator.

First, add the whole-number parts:

$$1 + 2 = 3$$

Now add the fractional parts $\dfrac{1}{2}$ and $\dfrac{2}{3}$

The least common denominator of $2$ and $3$ is $6$

$$\dfrac{1}{2} = \dfrac{3}{6}, \quad \dfrac{2}{3} = \dfrac{4}{6}$$

Add the fractions:

$$\dfrac{3}{6} + \dfrac{4}{6} = \dfrac{7}{6} = 1 \dfrac{1}{6}$$

Combine this with the $3$ from earlier:

$$3 + 1 \dfrac{1}{6} = 4 \dfrac{1}{6}$$

So the sum is $4 \dfrac{1}{6}$

The correct choice is D. $4 \dfrac{1}{6}$.

Convert both mixed numbers to improper fractions, subtract, and then convert back to a mixed number. Use a common denominator for the fractional parts.

Convert each mixed number to an improper fraction.

$$3 \dfrac{1}{4} = \dfrac{3 \cdot 4 + 1}{4} = \dfrac{13}{4}, \quad 1 \dfrac{2}{3} = \dfrac{1 \cdot 3 + 2}{3} = \dfrac{5}{3}$$

Find a common denominator for $4$ and $3$ which is $12$

$$\dfrac{13}{4} = \dfrac{13 \cdot 3}{4 \cdot 3} = \dfrac{39}{12}, \quad \dfrac{5}{3} = \dfrac{5 \cdot 4}{3 \cdot 4} = \dfrac{20}{12}$$

Now subtract:

$$\dfrac{39}{12} - \dfrac{20}{12} = \dfrac{39 - 20}{12} = \dfrac{19}{12}$$

Convert $\dfrac{19}{12}$ back to a mixed number:

$$19 \div 12 = 1 \text{ remainder } 7, \quad \dfrac{19}{12} = 1 \dfrac{7}{12}$$

So the difference is $1 \dfrac{7}{12}$

The correct choice is C. $1 \dfrac{7}{12}$.

Add the two fractions. The denominators are the same, so add the numerators and keep the denominator.

Add the distances:

$$\dfrac{3}{5} + \dfrac{2}{5} = \dfrac{3 + 2}{5} = \dfrac{5}{5} = 1$$

So Freya walked $1$ mile in all.

The correct choice is B. $1$ mile.

Add the mixed number and the fraction. You can convert $1 \dfrac{1}{2}$ to an improper fraction, or add $1$ and the fractional parts separately using a common denominator.

Start with $1 \dfrac{1}{2}$ cups and add $\dfrac{3}{4}$ cup.

Separate the whole number and fraction:

$$1 \dfrac{1}{2} + \dfrac{3}{4} = 1 + \left(\dfrac{1}{2} + \dfrac{3}{4}\right)$$

Find a common denominator for $\dfrac{1}{2}$ and $\dfrac{3}{4}$ which is $4$

$$\dfrac{1}{2} = \dfrac{2}{4}, \quad \dfrac{3}{4} = \dfrac{3}{4}$$

Add the fractions:

$$\dfrac{2}{4} + \dfrac{3}{4} = \dfrac{5}{4} = 1 \dfrac{1}{4}$$

Combine with the whole number $1$

$$1 + 1 \dfrac{1}{4} = 2 \dfrac{1}{4}$$

So there are $2 \dfrac{1}{4}$ cups of sugar in the bowl.

The correct choice is C. $2 \dfrac{1}{4}$ cups.

Find the difference between $\dfrac{5}{8}$ and $\dfrac{3}{8}$ The denominators are the same, so subtract the numerators.

Subtract the fractions:

$$\dfrac{5}{8} - \dfrac{3}{8} = \dfrac{5 - 3}{8} = \dfrac{2}{8}$$

Simplify $\dfrac{2}{8}$ by dividing numerator and denominator by $2$

$$\dfrac{2}{8} = \dfrac{1}{4}$$

So Sam spent $\dfrac{1}{4}$ more of his money than Alex.

The correct choice is D. $\dfrac{1}{4}$.