CELE Engineering Mathematics — Algebra and FundamentalsCheat Sheet
Algebra and Fundamentals cheat sheet for CELE aspirants. If you could only take one sheet of paper into your review session, this is what it would look like. Professional Regulation Commission (PRC) — Board of Civil Engineering's most-tested concepts, all in one place.
Exam context
On the CELE 2026, the Engineering Mathematics subtest carries a "Core" weight in Professional Regulation Commission (PRC) — Board of Civil Engineering's pattern. Algebra and Fundamentals lands at position 1st out of 10 in the standard review order. Target score is 70% weighted average, no sub-test below 50%, and roughly a meaningful share of items come from Engineering Mathematics on a typical CELE paper.
Algebra and Fundamentals - Cheat Sheet
Your last-minute revision companion for algebra essentials: exponents, radicals, logarithms, quadratics, progressions, and word problems. Every formula, every pitfall, every exam trigger condensed into one rapid-fire reference.
Sections
Formulas
Formula
a^m · a^n = a^(m+n)
Meaning
a = base; m, n = exponents
Watch Out
Do NOT add bases: a^m + a^n ≠ a^(m+n). Only works for multiplication.
When To Use
When multiplying powers with the same base
Formula
(a^m)^n = a^(mn)
Meaning
Nested exponents; a = base; m, n = exponents
Watch Out
a^(m^n) ≠ (a^m)^n — parentheses matter; always compute m×n first.
When To Use
Power raised to another power
Formula
(ab)^n = a^n · b^n
Meaning
Distribute exponent to each factor
Watch Out
(a+b)^n ≠ a^n + b^n — this rule does NOT apply to sums.
When To Use
Exponent applied to a product
Formula
a^(-n) = 1/a^n
Meaning
Negative exponent means reciprocal
Watch Out
a^(-n) ≠ -a^n. Negative exponent flips; negative sign is different.
When To Use
Simplifying expressions with negative powers
Formula
a^0 = 1 (a ≠ 0)
Meaning
Any nonzero base to the power 0 equals 1
Watch Out
0^0 is indeterminate — never appears in exam if you're careful. a must be nonzero.
When To Use
Simplification; elimination of terms
Formula
a^(m/n) = ⁿ√(a^m) = (ⁿ√a)^m
Meaning
Fractional exponent = nth root; m = power, n = root index
Watch Out
Order matters: a^(m/n) ≠ a^(n/m). Always reduce m/n first if possible.
When To Use
Converting between radical and exponential form
Formula
ⁿ√a = a^(1/n)
Meaning
nth root equals exponent 1/n
Watch Out
ⁿ√(a+b) ≠ ⁿ√a + ⁿ√b — radicals do NOT distribute over sums.
When To Use
Converting radicals to exponent form for simplification
Formula
ⁿ√(ab) = ⁿ√a · ⁿ√b
Meaning
nth root of product = product of nth roots
Watch Out
Only works for products. ⁿ√(a/b) = ⁿ√a / ⁿ√b (same rule for division).
When To Use
Simplifying radicals by factoring
Formula
ⁿ√(a^m) = a^(m/n)
Meaning
Radical with power exponent
Watch Out
Simplify m/n before converting: e.g., ⁶√(a⁴) = a^(4/6) = a^(2/3) = ³√(a²).
When To Use
Simplifying nested radicals and fractional exponents
Common Values
Value
≈ 1.414
Symbol
√2
Quantity
√2
Value
≈ 1.732
Symbol
√3
Quantity
√3
Value
≈ 2.236
Symbol
√5
Quantity
√5
Value
√2 ≈ 1.414
Symbol
2^0.5
Quantity
2^(1/2)
Value
≈ 1.260
Symbol
³√2
Quantity
Cube root of 2
Section Title
Exponents and Radicals
Important Facts
- a^m / a^n = a^(m-n) (quotient rule: subtract exponents)
- √a · √b = √(ab) for a, b ≥ 0 (product rule for square roots)
- √a / √b = √(a/b) for b ≠ 0 (quotient rule for radicals)
- Rationalize denominators: multiply by the conjugate (e.g., 1/√2 × √2/√2 = √2/2)
- ⁿ√(a^n) = |a| for even n; = a for odd n (absolute value needed for even roots of negatives)
- Laws hold for any real base and exponent (when defined); negative bases with fractional exponents can yield complex numbers
Key Definitions
Term
Exponent (Power)
Example
In 2³ = 8, the exponent 3 means 2 × 2 × 2.
Definition
Number of times a base is multiplied by itself; in a^n, n is the exponent.
Term
Radical
Example
√16 = 4; ³√8 = 2.
Definition
Expression written as ⁿ√a; the inverse of raising to a power; ⁿ√a = a^(1/n).
Term
Rational Exponent
Example
x^(3/2) = (√x)³ = x√x.
Definition
Exponent expressed as a fraction m/n, where a^(m/n) = (ⁿ√a)^m.
Diagrams To Know
- Law of exponents table (multiplication → add, division → subtract, power → multiply)
- Radical simplification tree (nested roots collapse via addition of exponent fractions)
Formulas
Formula
log_b(x) = y ⟺ b^y = x
Meaning
b = base, x = argument, y = logarithm; logarithm is the exponent
Watch Out
b > 0, b ≠ 1, and x > 0 (logarithm undefined for x ≤ 0 or invalid base).
When To Use
Converting between logarithmic and exponential form
Formula
log(MN) = log(M) + log(N)
Meaning
Logarithm of a product = sum of logarithms
Watch Out
log(M) + log(N) ≠ log(M + N). Product rule, NOT sum rule.
When To Use
Expanding log of products; solving equations
Formula
log(M/N) = log(M) − log(N)
Meaning
Logarithm of a quotient = difference of logarithms
Watch Out
Applies to division ONLY. log(M − N) ≠ log(M) − log(N).
When To Use
Expanding log of fractions
Formula
log(M^p) = p·log(M)
Meaning
Power moves out front; p can be any real number
Watch Out
p·log(M) ≠ log(p) + log(M). The power becomes a coefficient, not a separate term.
When To Use
Simplifying log of powers or extracting exponents
Formula
log_b(b) = 1
Meaning
Logarithm of the base in its own system equals 1
Watch Out
log_10(10) = 1, ln(e) = 1. Each base has its own identity.
When To Use
Simplification; recognizing identity
Formula
log_b(1) = 0
Meaning
Logarithm of 1 in any base is 0 (since b^0 = 1)
Watch Out
Always true regardless of base (as long as base is valid).
When To Use
Simplification; solving log equations
Formula
log_b(x) = ln(x) / ln(b) = log(x) / log(b)
Meaning
Change of base formula; convert any logarithm to natural or common log
Watch Out
The NEW base (denominator) must be what you're converting TO. Never flip numerator/denominator.
When To Use
Evaluating logarithms on calculators; switching bases
Formula
b^(log_b(x)) = x
Meaning
Exponential and logarithm are inverses; cancellation rule
Watch Out
Only valid when base b and argument x are compatible (b > 0, b ≠ 1, x > 0).
When To Use
Solving log and exponential equations
Formula
log_b(b^x) = x
Meaning
Logarithm of base to a power = the exponent
Watch Out
Does NOT apply to log_b(a^x) where a ≠ b.
When To Use
Inverse property; simplification
Common Values
Value
≈ 0.301
Symbol
log(2)
Quantity
log₁₀(2)
Value
≈ 0.693
Symbol
ln(2)
Quantity
ln(2)
Value
= 1
Symbol
ln(e)
Quantity
ln(e)
Value
= 1
Symbol
log(10)
Quantity
log₁₀(10)
Value
≈ 2.718
Symbol
e
Quantity
e (Euler's number)
Section Title
Logarithms
Important Facts
- Logarithmic function is the inverse of exponential function: if y = b^x, then x = log_b(y)
- Domain of log_b(x): x > 0; range: all real numbers
- Base must satisfy b > 0 and b ≠ 1 for logarithm to be defined
- log(0) is undefined; log of negative numbers yields complex results in real domain
- ln(e) = 1, log(10) = 1 (identity in respective bases)
- Logarithmic equations often require checking solutions (argument must remain positive)
- ln ≈ 0.693 (natural log of 2); log₁₀(2) ≈ 0.301
Key Definitions
Term
Logarithm
Example
log₂(8) = 3 because 2³ = 8.
Definition
The exponent y such that b^y = x, written as log_b(x) = y.
Term
Common Logarithm
Example
log(100) = 2 because 10² = 100.
Definition
Logarithm with base 10, written as log(x) or log₁₀(x).
Term
Natural Logarithm
Example
ln(e) = 1; ln(e²) = 2.
Definition
Logarithm with base e ≈ 2.718, written as ln(x) or log_e(x).
Term
Change of Base
Example
log₃(9) = ln(9) / ln(3) = 2.197 / 1.099 = 2.
Definition
Formula to convert logarithm from one base to another: log_b(x) = ln(x) / ln(b).
Diagrams To Know
- Graph of y = log_b(x) (approaches −∞ as x → 0⁺, passes through (1,0), increases without bound)
- Comparison of bases: log_b(x) for b > 1 vs 0 < b < 1 (increasing vs decreasing curves)
Reactions Or Equations
Note
Logarithm and exponential forms are equivalent; converting between them is key to solving equations.
Equation
log_b(x) = y ⟺ x = b^y
Conditions
b > 0, b ≠ 1, x > 0
Formulas
Formula
x = (−b ± √(b² − 4ac)) / (2a)
Meaning
a, b, c = coefficients of ax² + bx + c = 0; quadratic formula gives both roots
Watch Out
Discriminant (b² − 4ac) sign determines root type: >0 (real distinct), =0 (repeated), <0 (complex). Always use ± to get both roots.
When To Use
Solving any quadratic equation when factoring is difficult
Formula
Δ = b² − 4ac
Meaning
Discriminant; determines nature of quadratic roots
Watch Out
Δ < 0 means NO real roots (only complex conjugates). Δ = 0 means roots are equal/repeated.
When To Use
Analyzing root type without solving; checking if real solutions exist
Formula
Sum of roots = −b/a
Meaning
If roots are r₁ and r₂, then r₁ + r₂ = −b/a
Watch Out
Sign: SUM is −b/a, not +b/a. Watch the coefficient signs in the original equation.
When To Use
Finding one root if other is known; checking answers
Formula
Product of roots = c/a
Meaning
If roots are r₁ and r₂, then r₁ · r₂ = c/a
Watch Out
If c and a have opposite signs, product is negative (one positive, one negative root).
When To Use
Finding one root if other is known; analyzing root signs
Formula
(x − r₁)(x − r₂) = 0 where roots are r₁, r₂
Meaning
Factored form of quadratic; expansion gives ax² + bx + c = 0
Watch Out
This is the factored form only if roots are integers or simple fractions. Use quadratic formula first if roots are irrational.
When To Use
Writing quadratic when roots are known
Formula
x² − (sum)x + (product) = 0
Meaning
Quadratic in terms of sum and product of roots
Watch Out
Remember the signs: −(sum) and +(product).
When To Use
Building a quadratic when sum and product are given
Formula
For ax² + bx + c = 0 with real coefficients: if one root is p + qi, the other is p − qi
Meaning
Complex roots of polynomials with real coefficients always occur in conjugate pairs
Watch Out
True only for REAL coefficients. If coefficients are complex, this rule does not hold.
When To Use
Analyzing quadratics with negative discriminant
Section Title
Quadratic and Polynomial Equations
Important Facts
- If a = 1, quadratic is monic: x² + bx + c = 0
- Quadratic always has exactly 2 roots (real or complex, counting multiplicity)
- If roots are integers and product/sum are integers, try factoring first (faster than quadratic formula)
- Complex roots of real quadratics always come in conjugate pairs: if a + bi is a root, so is a − bi
- Vertex form: y = a(x − h)² + k; vertex at (h, k); axis of symmetry x = h
- For ax² + bx + c = 0, vertex x-coordinate: x = −b / (2a); this is also the line of symmetry
- Graph opens upward if a > 0, downward if a < 0
Key Definitions
Term
Quadratic Equation
Example
2x² − 5x + 3 = 0; roots are x = 1 and x = 3/2.
Definition
Polynomial equation of degree 2: ax² + bx + c = 0, where a ≠ 0.
Term
Discriminant
Example
For x² − 5x + 6 = 0: Δ = 25 − 24 = 1 > 0 → two real distinct roots.
Definition
Expression Δ = b² − 4ac; determines nature (real/complex) and number of roots.
Term
Vieta's Formulas
Example
For x² − 5x + 6 = 0: sum = 5, product = 6; roots satisfy these via (3+2=5, 3×2=6).
Definition
Relations between roots and coefficients: sum = −b/a, product = c/a.
Term
Repeated Root (Double Root)
Example
x² − 4x + 4 = (x − 2)² = 0 has double root x = 2.
Definition
When Δ = 0, the quadratic has one root with multiplicity 2; graph touches x-axis at one point.
Diagrams To Know
- Parabola with two real roots (Δ > 0), one real repeated root (Δ = 0), no real roots (Δ < 0)
- Vertex, axis of symmetry, focus, and directrix of parabola y = ax² + bx + c
Reactions Or Equations
Note
Factored form allows immediate identification of roots; expand to verify.
Equation
ax² + bx + c = a(x − r₁)(x − r₂)
Conditions
r₁, r₂ are the roots of the quadratic
Note
Transforms to vertex form; reveals vertex and aids in solving.
Equation
x² + px + q = (x + p/2)² − (p²/4 − q)
Conditions
Completing the square; general technique
Formulas
Formula
a_n = a₁ + (n − 1)d
Meaning
a_n = nth term; a₁ = first term; d = common difference; n = term number
Watch Out
n starts at 1, not 0. Off-by-one error is common: double-check which term you need.
When To Use
Finding a specific term in an arithmetic sequence
Formula
d = a_n − a_(n−1) = a₂ − a₁
Meaning
Common difference; constant difference between consecutive terms
Watch Out
d can be positive (increasing), negative (decreasing), or zero (constant sequence).
When To Use
Identifying or verifying an AP; computing d from given terms
Formula
S_n = (n/2)[2a₁ + (n − 1)d]
Meaning
Sum of first n terms; n = number of terms; a₁ = first term; d = common difference
Watch Out
Common error: forgetting the (n−1)d term or using wrong parentheses. Test with n=1 (should equal a₁).
When To Use
Computing sum of first n terms of an AP
Formula
S_n = (n/2)(a₁ + a_n)
Meaning
Alternative sum formula using first and last terms
Watch Out
Only works if you know the last term a_n. Verify n is correct.
When To Use
When both a₁ and a_n are known (often simpler than other form)
Formula
a_n = a_m + (n − m)d
Meaning
Relating any two terms; a_m and a_n at positions m and n
Watch Out
Must identify both m and n correctly from problem statement.
When To Use
Finding term at position n given term at position m
Formula
2a_n = a_(n−1) + a_(n+1)
Meaning
Each term is the arithmetic mean of its neighbors
Watch Out
This is the AP characterization; if ANY term fails this, sequence is not arithmetic.
When To Use
Verifying AP property; finding unknown term in sequence
Section Title
Arithmetic Progression (AP)
Important Facts
- An AP is uniquely determined by any two distinct terms (can solve for a₁ and d)
- Sum S_n grows quadratically with n for d ≠ 0; linearly for d = 0
- If d > 0, AP increases; d < 0, AP decreases; d = 0, all terms equal
- For finite AP, the sum can also be computed as S_n = n × (median term), where median = (a₁ + a_n)/2
- The last term in sum formula is a_n = a₁ + (n−1)d, not a_(n+1)
- Inserting m arithmetic means between two numbers a and b creates AP with a+1 terms total and d = (b−a)/(m+1)
Key Definitions
Term
Arithmetic Progression (AP)
Example
3, 7, 11, 15, ... has a₁ = 3, d = 4; the 5th term is 3 + 4(4) = 19.
Definition
Sequence where consecutive terms have a constant difference d; a_n = a₁ + (n−1)d.
Term
Common Difference
Example
In 10, 7, 4, 1, ..., d = −3 (decreasing AP).
Definition
Constant d such that each term is d more (or less) than the previous term.
Term
Arithmetic Mean
Example
In 2, 5, 8, the mean of 2 and 8 is (2+8)/2 = 5.
Definition
Middle term of three consecutive AP terms; a_n = (a_(n−1) + a_(n+1)) / 2.
Diagrams To Know
- AP sequence on a number line showing equal spacing
- Linear graph of term vs. term number (a_n vs n) — should be a straight line
Formulas
Formula
a_n = a₁ · r^(n−1)
Meaning
a_n = nth term; a₁ = first term; r = common ratio; n = term number
Watch Out
n starts at 1. If r = 1, all terms equal a₁ (degenerate case). If r = 0, only a₁ ≠ 0.
When To Use
Finding a specific term in a geometric sequence
Formula
r = a_n / a_(n−1) = a₂ / a₁
Meaning
Common ratio; ratio of any term to the previous term
Watch Out
r can be positive, negative, or even complex. If |r| > 1, terms grow in magnitude; |r| < 1, terms decay.
When To Use
Identifying or verifying a GP; computing r from given terms
Formula
S_n = a₁(r^n − 1) / (r − 1) for r ≠ 1
Meaning
Sum of first n terms; a₁ = first term; r = common ratio; n = number of terms
Watch Out
Only valid for r ≠ 1. If r = 1, use S_n = n·a₁. Sign errors are common: (r^n − 1) not (1 − r^n) in numerator.
When To Use
Computing sum of first n terms of a GP
Formula
S_n = a₁(1 − r^n) / (1 − r) for r ≠ 1
Meaning
Alternative form of GP sum (algebraically equivalent to above)
Watch Out
Equivalent to standard form; use whichever feels natural. Both are correct.
When To Use
When you prefer factor out (1 − r) in denominator; often clearer for |r| < 1
Formula
S_∞ = a₁ / (1 − r) for |r| < 1
Meaning
Sum of infinite GP; only converges if |r| < 1
Watch Out
CRITICAL: If |r| ≥ 1, series diverges (sum is not finite). Must check |r| < 1 first.
When To Use
Finding sum of infinite geometric series with |r| < 1
Formula
a_n = a_m · r^(n−m)
Meaning
Relating any two terms; a_m and a_n at positions m and n
Watch Out
If m > n, exponent is negative; handle carefully.
When To Use
Finding term at position n given term at position m
Formula
a_n² = a_(n−1) · a_(n+1)
Meaning
Each term is the geometric mean of its neighbors
Watch Out
This is the GP characterization (for positive terms). If sequence has mixed signs, be careful with square roots.
When To Use
Verifying GP property; finding unknown term in sequence
Section Title
Geometric Progression (GP)
Important Facts
- A GP is uniquely determined by any two distinct terms (can solve for a₁ and r via simultaneous equations)
- For |r| > 1, terms grow without bound; for |r| < 1, terms approach 0; for r = 1, all terms equal a₁
- Infinite sum S_∞ exists ONLY if |r| < 1; otherwise series diverges
- If r < 0, terms alternate in sign (e.g., 1, −2, 4, −8, ...)
- GP with r = 0 is degenerate: a₁, 0, 0, 0, ... (only first term matters)
- For finding n terms to reach a sum close to S_∞, check convergence: as n → ∞, r^n → 0 when |r| < 1
Key Definitions
Term
Geometric Progression (GP)
Example
2, 6, 18, 54, ... has a₁ = 2, r = 3; the 5th term is 2·3⁴ = 162.
Definition
Sequence where consecutive terms have a constant ratio r; a_n = a₁·r^(n−1).
Term
Common Ratio
Example
In 100, 50, 25, 12.5, ..., r = 0.5 (decreasing GP).
Definition
Constant r such that each term is r times the previous term.
Term
Geometric Mean
Example
In 1, 3, 9, the geometric mean of 1 and 9 is √(1×9) = 3.
Definition
Middle term of three consecutive GP terms; a_n = √(a_(n−1) · a_(n+1)).
Term
Convergent Series
Example
1/2 + 1/4 + 1/8 + ... = (1/2)/(1 − 1/2) = 1.
Definition
Infinite GP with |r| < 1; sum converges to finite value S_∞ = a₁/(1−r).
Diagrams To Know
- GP sequence on logarithmic scale (appears linear if plotted as log(a_n) vs n for r > 0)
- Exponential graph for r > 1 (growth), exponential decay for 0 < r < 1
Formulas
Formula
(a + b)^n = Σ(k=0 to n) C(n,k) · a^(n−k) · b^k
Meaning
Sum of all binomial terms; C(n,k) = n!/(k!(n−k)!); k ranges 0 to n
Watch Out
The exponents of a and b must sum to n at each term. Off-by-one errors in indexing are common.
When To Use
Expanding (a+b)^n for any positive integer n
Formula
C(n,k) = n! / (k!(n−k)!)
Meaning
Binomial coefficient; number of ways to choose k items from n
Watch Out
C(n,k) = C(n, n−k) (symmetry). 0! = 1, not 0.
When To Use
Computing the coefficient of the kth term in binomial expansion
Formula
Term (r+1) = C(n,r) · a^(n−r) · b^r
Meaning
The (r+1)th term; r = 0, 1, 2, ..., n; note: r+1 because we start counting from term 1
Watch Out
The rth term (counting from 1) uses C(n, r−1), not C(n, r). Always clarify: 'find the kth term' vs 'find the term with r'.
When To Use
Finding a specific term without expanding the entire binomial
Formula
Coefficient of a^p in (a+b)^n is C(n, n−p) · b^p
Meaning
Extracting coefficient of a specific power of a
Watch Out
The exponent sum (n−p) + p = n must hold. Verify power of a is achievable.
When To Use
Finding coefficient of a desired term when a and b are specified
Formula
(1 + x)^n = 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + ...
Meaning
Special case a=1, b=x; series form useful for fractional or negative n
Watch Out
For |x| small and n not a positive integer, this is an infinite series, NOT finite. Convergence depends on |x|.
When To Use
Expanding (1+x)^n or deriving series approximations
Common Values
Value
= 1
Symbol
Always 1
Quantity
C(n, 0)
Value
= n
Symbol
Linear in n
Quantity
C(n, 1)
Value
= 1
Symbol
Always 1
Quantity
C(n, n)
Value
= n(n−1)/2
Symbol
Triangular numbers
Quantity
C(n, 2)
Section Title
Binomial Theorem
Important Facts
- Binomial expansion has n+1 terms (from k=0 to k=n)
- Sum of all binomial coefficients in row n: C(n,0) + C(n,1) + ... + C(n,n) = 2^n
- The expansion is symmetric: coefficient of a^k equals coefficient of a^(n−k)
- For (a−b)^n, alternate signs: odd k terms are negative
- Factorial: n! = n(n−1)(n−2)...1; 0! = 1 by definition
- For large n, middle terms (k near n/2) have the largest coefficients
- Binomial theorem for negative or fractional n yields infinite series (use (1+x)^n form)
Key Definitions
Term
Binomial Theorem
Example
(x+2)³ = C(3,0)x³ + C(3,1)x²(2) + C(3,2)x(4) + C(3,3)(8) = x³ + 6x² + 12x + 8.
Definition
Formula for expanding (a+b)^n into a sum of n+1 terms, each with binomial coefficient C(n,k).
Term
Binomial Coefficient
Example
C(5,2) = 5!/(2!3!) = 10.
Definition
Symbol C(n,k) or ⁿC_k; equals n!/(k!(n−k)!); represents coefficient in binomial expansion.
Term
Pascal's Triangle
Example
Row 3: 1, 3, 3, 1 → coefficients for (a+b)³.
Definition
Triangular array of binomial coefficients; each row n gives coefficients for (a+b)^n.
Diagrams To Know
- Pascal's Triangle (rows 0–10) showing binomial coefficients
- Graph of C(n,k) vs k for fixed n (bell-shaped for large n)
Reactions Or Equations
Note
Difference eliminates even-power terms; useful for extracting odd terms.
Equation
(a+b)^n − (a−b)^n = 2[b-terms with odd powers]
Conditions
Subtracting expansions with + and − in middle
Note
Sum eliminates odd-power terms; useful for extracting even terms.
Equation
(a+b)^n + (a−b)^n = 2[terms with even powers of b]
Conditions
Adding expansions with + and −
Formulas
Formula
Rate of work = 1 / (time to complete job) [jobs/time]
Meaning
Fraction of job completed per unit time
Watch Out
ADD rates, NOT times. If A takes 6 days, B takes 4 days: combined rate = 1/6 + 1/4 = 5/12 jobs/day, not 1/(6+4).
When To Use
Work-rate problems; multiple workers
Formula
Combined rate × time = 1 (job); t = 1 / (combined rate)
Meaning
Time to complete job together = 1 / (sum of individual rates)
Watch Out
Only valid if working simultaneously on the SAME job. If they do different parts, use different equations.
When To Use
Finding time for multiple workers to finish together
Formula
Amount = Concentration × Volume (or Quantity = Concentration × Amount of Solution)
Meaning
For mixtures: mass of solute = (concentration) × (total mass/volume)
Watch Out
Keep units consistent. Concentration can be %, ratio, molarity, etc. — define before solving.
When To Use
Mixture problems; tracking amounts of different components
Formula
Initial amount + Added − Removed = Final amount
Meaning
Conservation equation for mixture problems
Watch Out
Sign convention: added is +, removed is −. Be careful with rates (e.g., drain removes per unit time).
When To Use
Tracking total quantity when adding/removing components
Formula
Distance = Rate × Time; d = rt
Meaning
Fundamental motion equation
Watch Out
Units must match: if d in km and t in hours, r must be km/h. Common error: mixing units.
When To Use
Any motion problem; uniform or average velocity
Formula
Age problem: current age + years elapsed = future age
Meaning
Linear relationship for ages; set up equations comparing ages at different times
Watch Out
All parties age at the same rate (1 year per year). Ratio of ages changes, but differences may stay constant.
When To Use
Age relationship problems; comparing ages now vs. past/future
Section Title
Word Problems – Work, Mixture, Age, and Motion
Important Facts
- In work problems: total work = (combined rate) × (time); for multiple workers: 1 = (r₁ + r₂ + ...)t
- If a drain empties while pipes fill, subtract drain rate from total fill rate
- In mixture problems: (concentration₁)(volume₁) + (concentration₂)(volume₂) = (final concentration)(final volume)
- Average speed ≠ average of speeds; use total distance / total time
- In age problems, differences between ages remain constant over time, but ratios change
- For motion: if two objects move toward each other, add speeds; if same direction, subtract
- Relative velocity: if object A moves at v_A and B at v_B, A's velocity relative to B is v_A − v_B
Key Definitions
Term
Work Rate
Example
If a job takes 5 hours, the rate is 1/5 job/hour.
Definition
Fraction of a job completed per unit time; inverse of time to complete the full job.
Term
Mixture Problem
Example
Mix 30% salt solution with 10% salt solution to get 20 L of 20% solution.
Definition
Problem involving combining solutions of different concentrations or materials of different values/types.
Term
Age Problem
Example
Father is 3 times son's age now; in 10 years, he will be twice the son's age. Find current ages.
Definition
Problem comparing ages of two or more people at different times (now, past, or future).
Diagrams To Know
- Work problem timeline (parallel work vs. sequential work)
- Mixture diagram (two inputs, different concentrations, one output)
- Motion diagram (distance-time or position-time for multiple objects)
Must Remember
Quadratic formula: x = (−b ± √(b²−4ac))/(2a); discriminant Δ = b²−4ac determines root type (>0 real, =0 double, <0 complex).
Priority
1
AP sum: S_n = (n/2)[2a₁+(n−1)d] or (n/2)(a₁+a_n); GP sum: S_n = a₁(r^n−1)/(r−1) [r≠1]; infinite GP: S_∞ = a₁/(1−r) ONLY if |r|<1.
Priority
2
Laws of exponents: a^m · a^n = a^(m+n), (a^m)^n = a^(mn), a^(−n) = 1/a^n, a^0 = 1. Radicals: ⁿ√(ab) = ⁿ√a · ⁿ√b (products only).
Priority
3
Logarithm rules: log(MN) = log(M)+log(N), log(M/N) = log(M)−log(N), log(M^p) = p·log(M). Change base: log_b(x) = ln(x)/ln(b).
Priority
4
Sum and product of quadratic roots: sum = −b/a, product = c/a (Vieta's formulas). Use to verify answers or build quadratic from roots.
Priority
5
Binomial coefficient: C(n,k) = n!/(k!(n−k)!); (r+1)th term of (a+b)^n is C(n,r)·a^(n−r)·b^r. Sum of row: Σ C(n,k) = 2^n.
Priority
6
Work problems: ADD RATES, NOT TIMES. If A finishes in t₁ and B in t₂, combined rate = 1/t₁ + 1/t₂; time together = 1/(rate sum).
Priority
7
GP convergence: S_∞ exists ONLY if |r|<1. If r=1, sum = n·a₁; if |r|>1 or |r|=1 (r≠1), series diverges.
Priority
8
Inverse: log_b(x) = y ⟺ b^y = x. Check domains: b>0, b≠1, x>0 for real logarithms. b^(log_b(x)) = x and log_b(b^x) = x (inverse properties).
Priority
9
Common mistakes: (a+b)^n ≠ a^n+b^n (power does NOT distribute), log(M+N) ≠ log(M)+log(N) (log is NOT linear), ⁿ√(a+b) ≠ ⁿ√a + ⁿ√b.
Priority
10
Last Minute Tips
Tip
Quadratic formula is your safety net: if factoring fails or seems slow, use x = (−b±√(b²−4ac))/(2a) immediately. Check discriminant sign FIRST to know if real roots exist.
Rationale
Exam time pressure makes factoring risky; formula is mechanical and reliable. Discriminant check saves wasted work on complex-root problems.
Tip
In work problems, ALWAYS ask: 'Am I adding rates or times?' If workers work simultaneously, add rates (1/t₁ + 1/t₂); if sequentially, add times. Exam loves this confusion.
Rationale
This is the #1 algebraic mistake on word-problem questions. One sentence in your mind prevents the wrong setup.
Tip
For infinite GP sum: FIRST check |r|<1. If you skip this check, you'll get a wrong 'answer' from S_∞ = a₁/(1−r) on a divergent series. Mark it DIVERGES if |r|≥1.
Rationale
Exam graders love this trap. One mental check ('Is |r| less than 1?') prevents automatic point loss.
Tip
When expanding (a+b)^n with binomial theorem: write out the PATTERN of exponents first (a^n, a^(n−1)b, a^(n−2)b², ..., b^n) before computing coefficients. This avoids off-by-one errors.
Rationale
Most errors come from coefficient confusion, not formula. Visual template reduces careless mistakes under time pressure.
Tip
For logarithm problems: convert to exponential form (log_b(x)=y → b^y=x) if stuck. Often reveals the answer or simplifies algebra dramatically.
Rationale
Logarithmic form can feel abstract; exponent form is concrete and more intuitive. Conversion is a powerful troubleshooting tool.
Comparison Tables
Rows
Values
- (ab)^n
- Not applicable
- = a^n · b^n
Property
Power of a product
Values
- a^m · a^n
- Not applicable
- = a^(m+n)
Property
Product of powers
Values
- (a^m)^n
- Not applicable
- = a^(mn)
Property
Power of a power
Values
- a^(m/n)
- ⁿ√(a^m)
- equivalent forms
Property
Radical to exponent
Values
- a^(1/n) · b^(1/n)
- ⁿ√a · ⁿ√b
- = ⁿ√(ab)
Property
Product of radicals
Values
- a^(1/n) / b^(1/n)
- ⁿ√a / ⁿ√b
- = ⁿ√(a/b)
Property
Quotient of radicals
Columns
- Operation
- Exponent Form
- Radical Form
- Rule/Property
Table Title
Exponents vs. Radicals: Quick Reference
Rows
Values
- a_n = a₁ + (n−1)d
- a_n = a₁ · r^(n−1)
Property
General term
Values
- Common difference d (addition)
- Common ratio r (multiplication)
Property
Constant between consecutive terms
Values
- S_n = (n/2)[2a₁ + (n−1)d] = (n/2)(a₁ + a_n)
- S_n = a₁(r^n − 1)/(r − 1) [r ≠ 1]
Property
Sum of first n terms
Values
- Diverges unless d = 0 (converges to a₁)
- S_∞ = a₁/(1−r) only if |r| < 1
Property
Infinite sum
Values
- 2a_n = a_(n−1) + a_(n+1) (arithmetic mean)
- a_n² = a_(n−1) · a_(n+1) (geometric mean)
Property
Middle term relationship
Values
- If a₁ and d known: a_m + a_n = a₁(2 + (m+n−2)d/1)
- If a₁ and r known: a_m · a_n = a₁² · r^(m+n−2)
Property
Sum of two given terms
Values
- Linear (a_n vs n is a straight line)
- Exponential (a_n vs n curves up/down depending on r)
Property
Graph shape
Columns
- Property
- Arithmetic Progression (AP)
- Geometric Progression (GP)
Table Title
AP vs. GP: Side-by-Side Comparison
Rows
Values
- log_b(MN) = log_b(M) + log_b(N)
- log₁₀(2×5) = log₁₀(2) + log₁₀(5) = log₁₀(10) = 1
- Does NOT apply to sums: log(M+N) ≠ log(M) + log(N)
Property
Product rule
Values
- log_b(M/N) = log_b(M) − log_b(N)
- ln(e²/e) = 2 − 1 = 1
- Only for division; log(M−N) ≠ log(M) − log(N)
Property
Quotient rule
Values
- log_b(M^p) = p · log_b(M)
- log₂(8³) = 3 log₂(8) = 3(3) = 9
- Power becomes a coefficient, not separate term
Property
Power rule
Values
- log_b(x) = log_c(x) / log_c(b)
- log₃(9) = ln(9) / ln(3) = 2.197 / 1.099 ≈ 2
- New base (denominator) is what you're converting TO
Property
Change of base
Values
- b^(log_b(x)) = x; log_b(b^x) = x
- 2^(log₂(5)) = 5; log₃(3^7) = 7
- Exponent and log must have same base
Property
Inverse property
Columns
- Rule Name
- Formula
- Example
- Watch Out
Table Title
Logarithm Rules Summary
Rows
Values
- Δ > 0
- Two distinct real roots
- 2
- x² − 5x + 6 = 0; Δ = 1; roots: 2, 3
Property
Δ = b² − 4ac
Values
- Δ = 0
- One repeated real root (double root)
- 1 (with multiplicity 2)
- x² − 4x + 4 = 0; Δ = 0; root: x = 2 (double)
Property
Δ = 0
Values
- Δ < 0
- Two complex conjugate roots
- 0 (in reals)
- x² + x + 1 = 0; Δ = −3; roots: (−1 ± i√3)/2
Property
Δ < 0
Columns
- Discriminant Δ
- Sign of Δ
- Root Type
- Number of Real Roots
- Example
Table Title
Root Types: Discriminant (Δ) Analysis
Rows
Values
- r₁ + r₂ = combined rate; t = 1/(r₁+r₂)
- r₁ = 1/t₁, r₂ = 1/t₂
- Adding TIMES (not rates) — WRONG.
Property
Work (parallel workers)
Values
- c₁V₁ + c₂V₂ = c_f(V₁+V₂)
- c = concentration, V = volume, subscript f = final
- Forgetting units; mixing % and decimal concentration
Property
Mixture (two solutions)
Values
- d = v₁t ± v₂t = (v₁ ± v₂)t
- d = distance, v = velocity, t = time
- Using average speed incorrectly; mixing units (km vs m)
Property
Motion (same direction)
Values
- Age_now + Δt = Age_future; set up ratio at past/present/future
- Assign variables to current ages; work backward/forward
- Forgetting that all parties age at 1 year/year
Property
Age (comparing times)
Columns
- Problem Type
- Key Equation(s)
- Variable Setup
- Common Pitfall
Table Title
Word Problem Strategy: Problem Type vs. Method
Ready to practise for the CELE 2026?
Super Tutor's AI review plan adapts to your weak areas and builds a weekly practice schedule around your target CELE exam date.