Here are the step-by-step solutions for each problem on the worksheet.
1) $\frac{3x}{7} + \frac{x + 3}{7}$
Since the denominators are the same (7), we can combine the numerators directly.
$$ \frac{3x + x + 3}{7} $$
Combine like terms ($3x + x = 4x$):
$$ \frac{4x + 3}{7} $$
2) $\frac{8x}{3} + \frac{x - 3}{12}$
Find a common denominator for 3 and 12, which is 12. Multiply the first fraction by $\frac{4}{4}$.
$$ \frac{8x \cdot 4}{3 \cdot 4} + \frac{x - 3}{12} $$
$$ \frac{32x}{12} + \frac{x - 3}{12} $$
Combine the numerators:
$$ \frac{32x + x - 3}{12} $$
$$ \frac{33x - 3}{12} $$
Simplify by dividing the numerator and denominator by their greatest common divisor, which is 3:
$$ \frac{11x - 1}{4} $$
3) $\frac{9x}{8} - \frac{x - 5}{16}$
Find a common denominator for 8 and 16, which is 16. Multiply the first fraction by $\frac{2}{2}$.
$$ \frac{9x \cdot 2}{8 \cdot 2} - \frac{x - 5}{16} $$
$$ \frac{18x}{16} - \frac{x - 5}{16} $$
Combine the numerators. Be careful to distribute the negative sign to both terms in the second numerator:
$$ \frac{18x - (x - 5)}{16} $$
$$ \frac{18x - x + 5}{16} $$
$$ \frac{17x + 5}{16} $$
4) $\frac{x + 7}{3} + \frac{3x - 2}{18}$
Find a common denominator for 3 and 18, which is 18. Multiply the first fraction by $\frac{6}{6}$.
$$ \frac{6(x + 7)}{18} + \frac{3x - 2}{18} $$
Distribute the 6 in the first numerator:
$$ \frac{6x + 42}{18} + \frac{3x - 2}{18} $$
Combine the numerators:
$$ \frac{6x + 42 + 3x - 2}{18} $$
Combine like terms ($6x + 3x = 9x$ and $42 - 2 = 40$):
$$ \frac{9x + 40}{18} $$
5) $\frac{7x}{8} + \frac{3x}{10} - \frac{x}{5}$
Find the least common multiple for 8, 10, and 5, which is 40.
Convert each fraction:
* $\frac{7x}{8} \cdot \frac{5}{5} = \frac{35x}{40}$
* $\frac{3x}{10} \cdot \frac{4}{4} = \frac{12x}{40}$
* $\frac{x}{5} \cdot \frac{8}{8} = \frac{8x}{40}$
Combine them:
$$ \frac{35x + 12x - 8x}{40} $$
Calculate the numerator ($35 + 12 - 8 = 39$):
$$ \frac{39x}{40} $$
6) $\frac{2x}{11} - \frac{5(x - 2)}{11}$
The denominators are already the same. Combine the numerators, being careful with the subtraction.
$$ \frac{2x - 5(x - 2)}{11} $$
Distribute the $-5$:
$$ \frac{2x - 5x + 10}{11} $$
Combine like terms ($2x - 5x = -3x$):
$$ \frac{-3x + 10}{11} $$
This can also be written as $\frac{10 - 3x}{11}$.
7) $\frac{2}{x} - \frac{3}{x} + \frac{7}{2x}$
Find a common denominator for $x$, $x$, and $2x$, which is $2x$.
Convert the fractions:
* $\frac{2}{x} \cdot \frac{2}{2} = \frac{4}{2x}$
* $\frac{3}{x} \cdot \frac{2}{2} = \frac{6}{2x}$
* The last term is already $\frac{7}{2x}$.
Combine them:
$$ \frac{4 - 6 + 7}{2x} $$
Calculate the numerator ($4 - 6 = -2$, and $-2 + 7 = 5$):
$$ \frac{5}{2x} $$
8) $\frac{4x}{7} - \frac{x - 3}{8}$
Find a common denominator for 7 and 8, which is 56.
Convert the fractions:
* $\frac{4x}{7} \cdot \frac{8}{8} = \frac{32x}{56}$
* $\frac{x - 3}{8} \cdot \frac{7}{7} = \frac{7(x - 3)}{56}$
Combine them:
$$ \frac{32x - 7(x - 3)}{56} $$
Distribute the $-7$:
$$ \frac{32x - 7x + 21}{56} $$
Combine like terms ($32x - 7x = 25x$):
$$ \frac{25x + 21}{56} $$
Final Answer:
1) $\frac{4x + 3}{7}$
2) $\frac{11x - 1}{4}$
3) $\frac{17x + 5}{16}$
4) $\frac{9x + 40}{18}$
5) $\frac{39x}{40}$
6) $\frac{10 - 3x}{11}$
7) $\frac{5}{2x}$
8) $\frac{25x + 21}{56}$
Parent Tip: Review the logic above to help your child master the concept of operations with fractions worksheet pdf.
