Algebra Basics
CSAT · Quantitative Aptitude
Introduction
CSAT algebra is light: forming and solving linear equations, simple quadratics, age problems and identities. The real skill is translating words into equations correctly.
Key concepts
Linear equations
Translate carefully: 'x is 5 more than twice y' → x = 2y + 5. Two unknowns need two equations. Sum S and difference D of two numbers: larger = (S + D)/2, smaller = (S − D)/2.
Age problems
Let the present ages be x and kx. 'After t years' adds t to both ages; 't years ago' subtracts t from both. The age difference never changes.
Quadratic equations
x² − (sum)x + (product) = 0. Factorise by finding two numbers with the required sum and product. The discriminant b² − 4ac tells you whether roots are real (≥ 0) or equal (= 0).
Identities
| Identity | Use |
|---|---|
| (a + b)² = a² + 2ab + b² | squares |
| a² − b² = (a + b)(a − b) | quick products: 98 × 102 = 100² − 2² |
| a² + b² = (a + b)² − 2ab | given sum and product |
| a³ + b³ = (a + b)³ − 3ab(a + b) | given sum and product |
Consecutive numbers
Let the integers be x, x + 1, x + 2 … (consecutive even/odd: x, x + 2, x + 4 …). The middle one equals the average.
Checking with options
Substituting options back into the conditions is a valid and often faster method in an MCQ paper.
Translating words into equations
Most CSAT algebra is translation. 'Five years ago, A was twice as old as B' becomes A − 5 = 2(B − 5). 'The sum of two numbers is 40 and their difference is 6' gives x + y = 40 and x − y = 6, so x = 23 and y = 17.
Using the options
With four options, substitution is often faster than solving. Put each option into the conditions; the one that satisfies all of them is correct. Start with the middle option so that you know whether to go higher or lower.
Quadratics in CSAT
Quadratics usually factorise neatly: find two numbers whose product is c and whose sum is b. If x² − 7x + 12 = 0, the numbers are 3 and 4, so x = 3 or 4. The sum of roots is −b/a and the product is c/a – useful when the question asks only for the sum or product.
- Larger = (S + D)/2
- x² − (α + β)x + αβ = 0
- a² + b² = (a + b)² − 2ab
- a³ + b³ = (a + b)³ − 3ab(a + b)
Solved examples
Example 1
The sum of two numbers is 100 and their difference is 36. What is the larger number?
(a) 32 (b) 50 (c) 68 (d) 69
- Larger = (sum + difference) ÷ 2 = (100 + 36) ÷ 2 = 68.
Answer: (c) 68
Example 2
The sum of two numbers is 132 and their difference is 34. What is the larger number?
(a) 83 (b) 49 (c) 66 (d) 84
- Larger = (sum + difference) ÷ 2 = (132 + 34) ÷ 2 = 83.
Answer: (a) 83
Example 3
The sum of 5 consecutive integers is 275. What is the largest?
(a) 53 (b) 57 (c) 58 (d) 55
- Let the first be x: 5x + 10 = 275 → x = 53.
- Largest = 53 + 4 = 57.
Answer: (b) 57
Example 4
The sum of 3 consecutive integers is 150. What is the largest?
(a) 50 (b) 49 (c) 52 (d) 51
- Let the first be x: 3x + 3 = 150 → x = 49.
- Largest = 49 + 2 = 51.
Answer: (d) 51
Example 5
The sum of two numbers is 117 and their difference is 45. What is the larger number?
(a) 82 (b) 81 (c) 36 (d) 58
- Larger = (sum + difference) ÷ 2 = (117 + 45) ÷ 2 = 81.
Answer: (b) 81
Example 6
If a + b = 9 and ab = 23, what is a² + b²?
(a) 58 (b) 81 (c) 35 (d) 127
- a² + b² = (a + b)² − 2ab = 81 − 46 = 35.
Answer: (c) 35
Example 7
If a + b = 9 and ab = 10, what is a³ + b³?
(a) 459 (b) 468 (c) 999 (d) 729
- a³ + b³ = (a + b)³ − 3ab(a + b) = 729 − 270 = 459.
Answer: (a) 459
Example 8
What are the roots of x² + 5x − 24 = 0?
(a) -8 and 4 (b) -3 and 8 (c) 5 and -24 (d) -8 and 3
- Find two numbers with sum -5 and product -24: 3 and -8.
- (x − 3)(x + 8) = 0, so x = -8 or 3.
Answer: (d) -8 and 3
Example 9
If a + b = 13 and ab = 24, what is a² + b²?
(a) 121 (b) 145 (c) 217 (d) 169
- a² + b² = (a + b)² − 2ab = 169 − 48 = 121.
Answer: (a) 121
Example 10
The sum of 5 consecutive integers is 155. What is the largest?
(a) 34 (b) 33 (c) 29 (d) 31
- Let the first be x: 5x + 10 = 155 → x = 29.
- Largest = 29 + 4 = 33.
Answer: (b) 33
- Plug options in for age and number problems.
- Use a² − b² for products near a round number.
- Age difference is constant – use it to cross-check.
- Adding years to only one person's age.
- Sign errors when moving terms across the equals sign.
- Forgetting the negative root of a quadratic when the context allows it.
Practice set
Attempt all 12 questions (about 90 seconds each), then check the worked solutions below.
Q1. If a + b = 9 and ab = 20, what is a² + b²?
(a) 121 (b) 61 (c) 41 (d) 81
Q2. The sum of two numbers is 112 and their difference is 4. What is the larger number?
(a) 59 (b) 54 (c) 56 (d) 58
Q3. The sum of two numbers is 112 and their difference is 22. What is the larger number?
(a) 68 (b) 67 (c) 56 (d) 45
Q4. If a + b = 10 and ab = 15, what is a³ + b³?
(a) 550 (b) 1000 (c) 560 (d) 1450
Q5. The sum of 5 consecutive integers is 185. What is the largest?
(a) 39 (b) 40 (c) 37 (d) 35
Q6. If a + b = 10 and ab = 25, what is a³ + b³?
(a) 260 (b) 1750 (c) 250 (d) 1000
Q7. The sum of two numbers is 36 and their difference is 2. What is the larger number?
(a) 20 (b) 18 (c) 19 (d) 17
Q8. If a + b = 6 and ab = 11, what is a³ + b³?
(a) 216 (b) 18 (c) 414 (d) 24
Q9. What are the roots of x² + 14x + 48 = 0?
(a) 6 and 8 (b) -8 and -5 (c) 14 and 48 (d) -8 and -6
Q10. The sum of 5 consecutive integers is 245. What is the largest?
(a) 49 (b) 47 (c) 52 (d) 51
Q11. The sum of 4 consecutive integers is 46. What is the largest?
(a) 13 (b) 10 (c) 14 (d) 11
Q12. If a + b = 14 and ab = 9, what is a² + b²?
(a) 196 (b) 178 (c) 187 (d) 214
Q1 – (c) 41. a² + b² = (a + b)² − 2ab = 81 − 40 = 41.
Q2 – (d) 58. Larger = (sum + difference) ÷ 2 = (112 + 4) ÷ 2 = 58.
Q3 – (b) 67. Larger = (sum + difference) ÷ 2 = (112 + 22) ÷ 2 = 67.
Q4 – (a) 550. a³ + b³ = (a + b)³ − 3ab(a + b) = 1000 − 450 = 550.
Q5 – (a) 39. Let the first be x: 5x + 10 = 185 → x = 35. Largest = 35 + 4 = 39.
Q6 – (c) 250. a³ + b³ = (a + b)³ − 3ab(a + b) = 1000 − 750 = 250.
Q7 – (c) 19. Larger = (sum + difference) ÷ 2 = (36 + 2) ÷ 2 = 19.
Q8 – (b) 18. a³ + b³ = (a + b)³ − 3ab(a + b) = 216 − 198 = 18.
Q9 – (d) -8 and -6. Find two numbers with sum -14 and product 48: -6 and -8. (x + 6)(x + 8) = 0, so x = -8 or -6.
Q10 – (d) 51. Let the first be x: 5x + 10 = 245 → x = 47. Largest = 47 + 4 = 51.
Q11 – (a) 13. Let the first be x: 4x + 6 = 46 → x = 10. Largest = 10 + 3 = 13.
Q12 – (b) 178. a² + b² = (a + b)² − 2ab = 196 − 18 = 178.
- Words → equations, carefully
- Ages: add/subtract t for everyone
- Sum & product → quadratic
- Plug in options
