Problems 1-11 : Solve for x.
Problem 1 :
ˣ⁄₁₂ = ⅔
Problem 2 :
²ˣ⁄₁₅ = ⁶⁄₅
Problem 3 :
ˣ⁄₂ = ˣ⁄₃ - 1
Problem 4 :
⁵ˣ⁄₄ - ˣ⁄₂ = ⁻³⁄₄
Problem 5 :
ˣ⁄₂ - ³ˣ⁄₄ = ¾ - x
Problem 6 :
⁽ˣ ⁺ ¹⁾⁄₃ = ⁽ˣ ⁻ ¹⁾⁄₅
Problem 7 :
ˣ⁄₂ - ⅕ = ˣ⁄₃ + ¼
Problem 8 :
ˣ⁄₂ - ³ˣ⁄₄ + ⁵ˣ⁄₆ = 21
Problem 9 :
x + 7 - ⁸ˣ⁄₃ = ¹⁷⁄₆ - ⁵ˣ⁄₂
Problem 10 :
⁽ˣ ⁺ ⁴⁾⁄₁₂ - ⁽ˣ ⁻ ⁵⁾⁄₁₈ = 1
Problem 11 :
⁽³ˣ ⁻ ²⁾⁄₄ - ⁽²ˣ ⁺ ³⁾⁄₃ = ⅔ - x
Problem 12 :
Subtracting two-third of a number from 5 results 3. Find the number.
1. Answer :
ˣ⁄₁₂ = ⅔
Least common multiple of the denominators 12 and 3 is 12.
Multiply both sides of the equation by 12 to get rid of the denominators 12 and 3.
12(ˣ⁄₁₂) = 12(⅔)
x = 4(2)
x = 8
2. Answer :
²ˣ⁄₁₅ = ⁶⁄₅
Least common multiple of the denominators 15 and 5 is 15.
Multiply both sides of the equation by 15 to get rid of the denominators 15 and 5.
15(²ˣ⁄₁₅) = 15(⁶⁄₅)
2x = 3(6)
2x = 18
Divide both sides by 2.
x = 9
3. Answer :
ˣ⁄₂ = ˣ⁄₃ - 1
Least common multiple of the denominators 2 and 3 is 6.
Multiply both sides of the equation by 6 to get rid of the denominators 2 and 3.
6(ˣ⁄₂) = 6(ˣ⁄₃ - 1)
3x = 6(ˣ⁄₃) - 6(1)
3x = 2x - 6
Subtract 2x from both sides.
x = -6
4. Answer :
⁵ˣ⁄₄ - ˣ⁄₂ = ⁻³⁄₄
Least common multiple of the denominators 4 and 2 is 4.
Multiply both sides of the equation by 4 to get rid of the denominators 4 and 2.
4(⁵ˣ⁄₄ - ˣ⁄₂) = 4(⁻³⁄₄)
4(⁵ˣ⁄₄) - 4(ˣ⁄₂) = -3
5x - 2x = -3
3x = -3
Divide both sides by 3.
x = -1
5. Answer :
ˣ⁄₂ - ³ˣ⁄₄ = ¾ - x
Least common multiple of the denominators 2 and 4 is 4.
Multiply both sides of the equation by 4 to get rid of the denominators.
4(ˣ⁄₂ - ³ˣ⁄₄) = 4(¾ - x)
4(ˣ⁄₂) - 4(³ˣ⁄₄) = 4(¾) - 4(x)
2x - 3x = 3 - 4x
-x = 3 - 4x
Add 4x to both sides.
3x = 3
Divide both sides by 3.
x = 1
6. Answer :
⁽ˣ ⁺ ¹⁾⁄₃ = ⁽ˣ ⁻ ¹⁾⁄₅
Least common multiple of the denominators 3 and 5 is 15.
Multiply both sides of the equation by 15 to get rid of the denominators 3 and 5.
15[⁽ˣ ⁺ ¹⁾⁄₃] = 15[⁽ˣ ⁻ ¹⁾⁄₅]
5(x + 1) = 3(x - 1)
5x + 5 = 3x - 3
Subtract 3x from both sides.
2x + 5 = -3
Subtract 5 from both sides.
2x = -8
Divide both sides by 4.
x = -4
7. Answer :
ˣ⁄₂ - ⅕ = ˣ⁄₃ + ¼
Least common multiple of the denominators 2, 5, 3 and 4 is 60.
Multiply each side of the above equation by 60 to get rid of the denominators.
60(ˣ⁄₂ - ⅕) = 60(ˣ⁄₃ + ¼)
Using Distributive Property,
60(ˣ⁄₂) - 60(⅕) = 60(ˣ⁄₃) + 60(¼)
30x - 12 = 20x + 15
Subtract 20x from both sides.
10x - 12 = 15
Add 12 to each side.
10x = 27
Divide each side by 10.
x = ²⁷⁄₁₀
8. Answer :
ˣ⁄₂ - ³ˣ⁄₄ + ⁵ˣ⁄₆ = 21
Least common multiple of the denominators 2, 4 and 6 is 12.
Multiply each side of the above equation by 12 to get rid of the denominators.
12(ˣ⁄₂ - ³ˣ⁄₄ + ⁵ˣ⁄₆) = 12(21)
Using Distributive Property,
12(ˣ⁄₂) - 12(³ˣ⁄₄) + 12(⁵ˣ⁄₆) = 252
6x - 9x + 10x = 252
7x = 252
x = 36
9. Answer :
x + 7 - ⁸ˣ⁄₃ = ¹⁷⁄₆ - ⁵ˣ⁄₂
Least common multiple of the denominators 3, 6 and 2 is 6.
Multiply each side of the above equation by 6 to get rid of the denominators.
6(x + 7 - ⁸ˣ⁄₃) = 6(¹⁷⁄₆ - ⁵ˣ⁄₂)
Using Distributive Property,
6(x) + 6(7) - 6(⁸ˣ⁄₃) = 6(¹⁷⁄₆) - 6(⁵ˣ⁄₂)
6x + 42 - 16x = 17 - 15x
-10x + 42 = 17 - 15x
Add 15x to each side.
5x + 42 = 17
Subtract 42 from each side.
5x = -25
Divide each side by 5.
x = -5
10. Answer :
⁽ˣ ⁺ ⁴⁾⁄₁₂ - ⁽ˣ ⁻ ⁵⁾⁄₁₈ = 1
Least common multiple of the denominators 12 and 18 is 36.
Multiply each side of the above equation by 36 to get rid of the denominators.
36[⁽ˣ ⁺ ⁴⁾⁄₁₂ - ⁽ˣ ⁻ ⁵⁾⁄₁₈] = 36(1)
Using Distributive Property,
36[⁽ˣ ⁺ ⁴⁾⁄₁₂] - 36[⁽ˣ ⁻ ⁵⁾⁄₁₈] = 36
3(x + 4) - 2(x - 5) = 36
3x + 12 - 2x + 10 = 36
x + 22 = 36
Subtract 22 from each side.
x = 14
11. Answer :
⁽³ˣ ⁻ ²⁾⁄₄ - ⁽²ˣ ⁺ ³⁾⁄₃ = ⅔ - x
Least common multiple of the denominators 4 and 3 is 12.
Multiply each side of the above equation by 12 to get rid of the denominators.
12[⁽³ˣ ⁻ ²⁾⁄₄ - ⁽²ˣ ⁺ ³⁾⁄₃] = 12(⅔ - x)
Distribute.
12[⁽³ˣ ⁻ ²⁾⁄₄] - 12[⁽²ˣ ⁺ ³⁾⁄₃] = 12(⅔) - 12(x)
3(3x - 2) - 4(2x + 3) = 4(2) - 12x
9x - 6 - 8x - 12 = 8 - 12x
x - 18 = 8 - 12x
Add 12x to each side.
13x - 18 = 8
Add 18 to each side.
13x = 26
Divide each side by 13.
x = 2
12. Answer :
Let x be the number.
It is given that subtracting two-third of a number from 5 results 3.
5 - (⅔)x = 3
5 - ²ˣ⁄₃ = 3
Multiply both sides by 3 to get rid of the denominator.
3(5 - ²ˣ⁄₃) = 3(3)
3(5) - 3(²ˣ⁄₃) = 9
15 - 2x = 9
Subtract 15 from both sides.
-2x = -6
Divide both sides by -2.
x = 3
The number is 3.
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