Here are the solutions to the problems on your worksheet. I have worked through each one step-by-step to ensure accuracy.
1) Express as a single fraction
a) $\frac{5x}{8} - \frac{x}{8}$
* The denominators are the same (8), so we just subtract the numerators: $5x - x = 4x$.
* Result: $\frac{4x}{8}$
* Simplify by dividing top and bottom by 4: $\frac{x}{2}$
b) $\frac{3x}{10} + \frac{x}{10}$
* The denominators are the same (10), so we add the numerators: $3x + x = 4x$.
* Result: $\frac{4x}{10}$
* Simplify by dividing top and bottom by 2: $\frac{2x}{5}$
c) $\frac{12x}{13} - \frac{x}{26}$
* Find a common denominator. 26 is a multiple of 13 ($13 \times 2 = 26$).
* Convert the first fraction: $\frac{12x \times 2}{13 \times 2} = \frac{24x}{26}$.
* Subtract: $\frac{24x}{26} - \frac{x}{26} = \frac{23x}{26}$.
d) $\frac{x}{2} + \frac{x}{8}$
* Common denominator is 8.
* Convert the first fraction: $\frac{x \times 4}{2 \times 4} = \frac{4x}{8}$.
* Add: $\frac{4x}{8} + \frac{x}{8} = \frac{5x}{8}$.
e) $\frac{2x}{7} + \frac{x}{5}$
* Common denominator is $7 \times 5 = 35$.
* Convert fractions: $\frac{2x \times 5}{35} + \frac{x \times 7}{35} = \frac{10x}{35} + \frac{7x}{35}$.
* Add: $\frac{17x}{35}$.
f) $\frac{x}{3} + \frac{3x}{11}$
* Common denominator is $3 \times 11 = 33$.
* Convert fractions: $\frac{x \times 11}{33} + \frac{3x \times 3}{33} = \frac{11x}{33} + \frac{9x}{33}$.
* Add: $\frac{20x}{33}$.
g) $\frac{27x}{28} - \frac{7x}{12}$
* Find the Lowest Common Multiple for 28 and 12.
* $28 = 4 \times 7$
* $12 = 4 \times 3$
* LCM = $4 \times 7 \times 3 = 84$.
* Convert fractions:
* $\frac{27x \times 3}{84} = \frac{81x}{84}$
* $\frac{7x \times 7}{84} = \frac{49x}{84}$
* Subtract: $\frac{81x - 49x}{84} = \frac{32x}{84}$.
* Simplify (divide by 4): $\frac{8x}{21}$.
h) $\frac{13x}{20} - \frac{3x}{8}$
* Find the Lowest Common Multiple for 20 and 8.
* Multiples of 20: 20, 40...
* 40 is divisible by 8. So, LCM is 40.
* Convert fractions:
* $\frac{13x \times 2}{40} = \frac{26x}{40}$
* $\frac{3x \times 5}{40} = \frac{15x}{40}$
* Subtract: $\frac{26x - 15x}{40} = \frac{11x}{40}$.
i) $\frac{x}{8} - \frac{x+2}{3}$
* Common denominator is $8 \times 3 = 24$.
* Convert fractions:
* $\frac{x \times 3}{24} = \frac{3x}{24}$
* $\frac{(x+2) \times 8}{24} = \frac{8(x+2)}{24} = \frac{8x + 16}{24}$
* Subtract carefully (watch the negative sign):
* $\frac{3x - (8x + 16)}{24} = \frac{3x - 8x - 16}{24}$
* Result: $\frac{-5x - 16}{24}$ or $-\frac{5x + 16}{24}$.
j) $\frac{x-3}{6} + \frac{x+8}{7}$
* Common denominator is $6 \times 7 = 42$.
* Convert fractions:
* $\frac{7(x-3)}{42} = \frac{7x - 21}{42}$
* $\frac{6(x+8)}{42} = \frac{6x + 48}{42}$
* Add: $\frac{7x - 21 + 6x + 48}{42}$
* Combine like terms ($7x+6x=13x$ and $-21+48=27$):
* Result: $\frac{13x + 27}{42}$.
---
2) Express as a single fraction
a) $\frac{7x}{10} + \frac{9x}{10}$
* Denominators are the same. Add numerators: $7x + 9x = 16x$.
* Result: $\frac{16x}{10}$
* Simplify (divide by 2): $\frac{8x}{5}$.
b) $\frac{7x}{9} - \frac{x}{9}$
* Denominators are the same. Subtract numerators: $7x - x = 6x$.
* Result: $\frac{6x}{9}$
* Simplify (divide by 3): $\frac{2x}{3}$.
c) $\frac{3x}{4} + \frac{x}{12}$
* Common denominator is 12 (since $4 \times 3 = 12$).
* Convert first fraction: $\frac{3x \times 3}{12} = \frac{9x}{12}$.
* Add: $\frac{9x}{12} + \frac{x}{12} = \frac{10x}{12}$.
* Simplify (divide by 2): $\frac{5x}{6}$.
──────────────────────────────────────
Final Answer:
1) Express as a single fraction
a) $\frac{x}{2}$
b) $\frac{2x}{5}$
c) $\frac{23x}{26}$
d) $\frac{5x}{8}$
e) $\frac{17x}{35}$
f) $\frac{20x}{33}$
g) $\frac{8x}{21}$
h) $\frac{11x}{40}$
i) $\frac{-5x - 16}{24}$
j) $\frac{13x + 27}{42}$
2) Express as a single fraction
a) $\frac{8x}{5}$
b) $\frac{2x}{3}$
c) $\frac{5x}{6}$
Parent Tip: Review the logic above to help your child master the concept of algebra with fractions worksheet.
