NTS NAT-IM Quantitative Reasoning Syllabus

Define cost price (CP), selling price (SP), profit, and loss.

Cost price (CP): price at which an item is bought. Selling price (SP): price at which it is sold. Profit = SP − CP (when SP > CP). Loss = CP − SP (when CP > SP).

What are the formulas for profit percent and loss percent?

Profit% = (Profit ÷ CP) × 100. Loss% = (Loss ÷ CP) × 100. Both percentages are always calculated on the cost price.

How do you find selling price from cost price and profit percent?

SP = CP × (100 + Profit%)/100. For a loss, SP = CP × (100 − Loss%)/100.

What is discount, and on what amount is it calculated?

Discount is a reduction from the marked price (list price). It is calculated as a percentage of the marked price: Discount = Marked Price × Discount%/100, and SP = Marked Price − Discount.

How do you calculate marked price when an item is sold at a discount with a desired profit?

First find SP = CP × (100 + Profit%)/100, then Marked Price = SP ÷ (100 − Discount%)/100, i.e., MP = SP × 100/(100 − Discount%).

What is the formula for simple interest?

Simple Interest (SI) = (P × R × T) ÷ 100, where P = principal, R = rate per annum, T = time in years. Amount = P + SI.

What is the formula for compound interest amount, and how is CI found?

Amount A = P(1 + R/100)ⁿ, where n is the number of compounding periods. Compound Interest = A − P.

What is the key difference between simple interest and compound interest?

Simple interest is calculated only on the original principal each period. Compound interest is calculated on the principal plus accumulated interest, so it grows faster than simple interest.

For compound interest compounded half-yearly, how do you adjust rate and time?

Halve the annual rate and double the number of periods: rate becomes R/2 per period and time becomes 2T periods, so A = P(1 + R/200)^(2T).

For the same P, R, and T = 2 years, what is the difference between CI and SI?

CI − SI for 2 years = P × (R/100)². The extra amount equals the interest on the first year's interest.

What is the difference between a term, a coefficient, and a constant in an algebraic expression?

A term is a single number/variable or their product (e.g., 3x²). A coefficient is the numerical factor of a term (3 in 3x²). A constant is a term with no variable (a fixed number).

What are like terms, and why do they matter in simplification?

Like terms have the same variables raised to the same powers (e.g., 4x and 7x). Only like terms can be combined (added or subtracted) when simplifying expressions.

State the algebraic identities for (a+b)² and (a−b)².

(a+b)² = a² + 2ab + b². (a−b)² = a² − 2ab + b².

State the identity for (a+b)(a−b) and for a³+b³, a³−b³.

(a+b)(a−b) = a² − b². a³ + b³ = (a+b)(a² − ab + b²). a³ − b³ = (a−b)(a² + ab + b²).

Expand (a+b)³ and (a−b)³.

(a+b)³ = a³ + 3a²b + 3ab² + b³. (a−b)³ = a³ − 3a²b + 3ab² − b³.

What is a linear equation in one variable, and how is it solved?

A linear equation in one variable has the form ax + b = 0 (highest power of x is 1). Solve by isolating x: x = −b/a, performing the same operations on both sides.

What does the graph of a linear equation in two variables represent, and what is slope-intercept form?

It represents a straight line. Slope-intercept form is y = mx + c, where m is the slope (rate of change) and c is the y-intercept (where the line crosses the y-axis).

What are the methods to solve a pair of simultaneous linear equations?

Substitution, elimination, cross-multiplication, and graphical methods. Substitution solves one equation for a variable; elimination adds/subtracts equations to remove a variable.

How many solutions does a pair of linear equations have based on their coefficients a₁/a₂, b₁/b₂, c₁/c₂?

Unique solution if a₁/a₂ ≠ b₁/b₂ (intersecting lines). Infinitely many if a₁/a₂ = b₁/b₂ = c₁/c₂ (coincident). No solution if a₁/a₂ = b₁/b₂ ≠ c₁/c₂ (parallel).

What is the standard form of a quadratic equation?

ax² + bx + c = 0, where a, b, c are constants and a ≠ 0. The highest power of the variable is 2.

State the quadratic formula for solving ax² + bx + c = 0.

x = [−b ± √(b² − 4ac)] ÷ (2a). The roots are the two values of x that satisfy the equation.

What is the discriminant of a quadratic, and what does it reveal about the roots?

Discriminant D = b² − 4ac. If D > 0: two distinct real roots. If D = 0: two equal real roots. If D < 0: no real roots (complex roots).

For ax² + bx + c = 0, what are the sum and product of the roots?

Sum of roots = −b/a. Product of roots = c/a. (For roots α and β: α+β = −b/a, αβ = c/a.)

State the product, quotient, and power rules of exponents.

Product: aᵐ × aⁿ = aᵐ⁺ⁿ. Quotient: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. Power of a power: (aᵐ)ⁿ = aᵐⁿ.

What do a negative exponent and a fractional exponent mean?

Negative exponent: a⁻ⁿ = 1/aⁿ. Fractional exponent: a^(m/n) = ⁿ√(aᵐ), the nth root of a raised to the m power.

How do you simplify or rationalize a radical such as 1/√a?

Multiply numerator and denominator by √a: 1/√a = √a/a. Rationalizing removes the radical from the denominator. Also, √a × √b = √(ab).

What happens to the inequality sign when both sides are multiplied or divided by a negative number?

The inequality sign reverses (flips) direction. E.g., if −2x < 6, dividing by −2 gives x > −3.

What is the difference between the symbols <, ≤, and the open vs closed circle on a number line?

< (strictly less than) and > use an open circle (endpoint excluded). ≤ and ≥ use a closed/filled circle (endpoint included). Open means the boundary value is not part of the solution.

What is the relationship between complementary, supplementary, and vertically opposite angles?

Complementary angles sum to 90°. Supplementary angles sum to 180°. Vertically opposite angles (formed by two intersecting lines) are always equal.

When a transversal cuts two parallel lines, what is the relationship between corresponding, alternate, and co-interior angles?

Corresponding angles are equal. Alternate (interior and exterior) angles are equal. Co-interior (allied/same-side interior) angles are supplementary (sum to 180°).