NTS Quantitative Reasoning: Algebra — Questions and Answers
Question 1: If 2x + 5 = 17, what is the value of x?
Correct answer: 6
2x + 5 = 17 → 2x = 12 → x = 6.
Solving step by step: 2x + 5 = 17. Subtract 5 from both sides: 2x = 12. Divide both sides by 2: x = 6. Verification: 2(6) + 5 = 12 + 5 = 17. Linear equations in one variable are fundamental to NTS quantitative reasoning.
Question 2: What is the value of (a + b)² if a = 3 and b = 4?
- 25
- 36
- 49 (Correct answer)
- 64
Correct answer: 49
(a + b)² = (3 + 4)² = 7² = 49.
(a + b)² = (3 + 4)² = 7² = 49. Alternatively, using the algebraic identity: (a+b)² = a² + 2ab + b² = 9 + 24 + 16 = 49. Both methods give the same answer. Algebraic identities and their applications are regularly tested in NTS.
Question 3: Solve for y: 3y - 9 = 2y + 4
- 11
- 12
- 13 (Correct answer)
- 14
Correct answer: 13
3y - 9 = 2y + 4 → 3y - 2y = 4 + 9 → y = 13.
Solving: 3y - 9 = 2y + 4. Subtract 2y from both sides: y - 9 = 4. Add 9 to both sides: y = 13. Verification: 3(13) - 9 = 39 - 9 = 30; 2(13) + 4 = 26 + 4 = 30. Both sides equal 30. Equations with variables on both sides are common in NTS algebra.
Question 4: If x² - 9 = 0, what are the possible values of x?
- ±2
- ±3 (Correct answer)
- ±4
- ±9
Correct answer: ±3
x² - 9 = 0 → x² = 9 → x = ±3.
x² - 9 = 0 can be solved by: Method 1: x² = 9, so x = ±√9 = ±3. Method 2: Factor as difference of squares: (x+3)(x-3) = 0, giving x = -3 or x = 3. The difference of squares factoring pattern (a² - b² = (a+b)(a-b)) is frequently tested in NTS algebra.
Question 5: What is the sum of the first 10 natural numbers?
- 45
- 50
- 55 (Correct answer)
- 60
Correct answer: 55
Sum = n(n+1)/2 = 10(11)/2 = 55.
The sum of the first n natural numbers is given by the formula S = n(n+1)/2. For n = 10: S = 10 × 11 / 2 = 110/2 = 55. This can also be verified by adding: 1+2+3+4+5+6+7+8+9+10 = 55. This formula, attributed to Gauss, is a standard NTS mathematics topic.
Question 6: Simplify: 5(2x - 3) + 4x
- 14x - 15 (Correct answer)
- 14x - 3
- 10x - 15
- 14x + 15
Correct answer: 14x - 15
5(2x - 3) + 4x = 10x - 15 + 4x = 14x - 15.
Step 1: Distribute 5 across (2x - 3): 5 × 2x = 10x, 5 × (-3) = -15. So 5(2x-3) = 10x - 15. Step 2: Add 4x: 10x - 15 + 4x = 14x - 15. Combining like terms (10x + 4x = 14x) gives the final answer: 14x - 15. Simplification using distribution is a core NTS algebra skill.
Question 7: Find the value of x: x/4 + 3 = 7
- 12
- 14
- 16 (Correct answer)
- 18
Correct answer: 16
x/4 + 3 = 7 → x/4 = 4 → x = 16.
Solving step by step: x/4 + 3 = 7. Subtract 3 from both sides: x/4 = 4. Multiply both sides by 4: x = 16. Verification: 16/4 + 3 = 4 + 3 = 7. Equations involving fractions are common in NTS quantitative reasoning and require careful manipulation.
Question 8: If a = 2, b = -3, what is the value of a² + b² - 2ab?
- 25 (Correct answer)
- 20
- 15
- 10
Correct answer: 25
a² + b² - 2ab = (a-b)² = (2-(-3))² = 5² = 25.
The expression a² + b² - 2ab equals (a - b)². Substituting a = 2, b = -3: (2 - (-3))² = (2 + 3)² = 5² = 25. Direct calculation also works: a² = 4, b² = 9, 2ab = 2(2)(-3) = -12. So 4 + 9 - (-12) = 4 + 9 + 12 = 25. Recognizing algebraic identities speeds up problem-solving in NTS.
Question 9: What is the product of (x + 3)(x - 3)?
- x² + 9
- x² - 9 (Correct answer)
- x² + 6x - 9
- x² - 6x + 9
Correct answer: x² - 9
Using the identity (a+b)(a-b) = a² - b²: (x+3)(x-3) = x² - 9.
(x + 3)(x - 3) follows the algebraic identity (a + b)(a - b) = a² - b². Here a = x and b = 3, so the product = x² - 3² = x² - 9. Using FOIL: x² - 3x + 3x - 9 = x² - 9 (the middle terms cancel). The difference of squares identity is one of the most important algebra concepts tested in NTS.
Question 10: Solve the simultaneous equations: x + y = 10 and x - y = 4
- x=6, y=4
- x=7, y=3 (Correct answer)
- x=8, y=2
- x=5, y=5
Correct answer: x=7, y=3
Adding both equations: 2x = 14, so x = 7. Substituting: 7 + y = 10, so y = 3.
Adding the equations: (x + y) + (x - y) = 10 + 4 → 2x = 14 → x = 7. Substituting x = 7 into x + y = 10: 7 + y = 10 → y = 3. Verification: 7 + 3 = 10 and 7 - 3 = 4. Both equations satisfied. Simultaneous equations (elimination method) are frequently tested in NTS algebra.
Question 11: If 4^x = 64, what is the value of x?
Correct answer: 3
64 = 4³, so 4^x = 4³, therefore x = 3.
To solve 4^x = 64, express 64 as a power of 4: 4¹ = 4, 4² = 16, 4³ = 64. So 4^x = 4³, which means x = 3. Alternatively, using logarithms: x = log₄(64) = log(64)/log(4) = 3. Exponential equations are a standard topic in NTS quantitative reasoning.
Question 12: What is the LCM of 12 and 18?
- 24
- 30
- 36 (Correct answer)
- 48
Correct answer: 36
LCM of 12 and 18: 12 = 2² × 3, 18 = 2 × 3². LCM = 2² × 3² = 36.
Prime factorization: 12 = 2² × 3¹, 18 = 2¹ × 3². LCM takes the highest power of each prime factor: LCM = 2² × 3² = 4 × 9 = 36. Verification: 36/12 = 3 (integer) and 36/18 = 2 (integer). LCM and HCF calculations are fundamental NTS quantitative reasoning topics.
