NTS NAT-IGS Quantitative Reasoning Syllabus

State the alligation rule for mixing two ingredients.

(Quantity of cheaper)/(Quantity of dearer) = (Price of dearer − Mean price)/(Mean price − Price of cheaper).

A vessel has milk and water in ratio a:b. What does alligation/replacement help find?

It finds the proportion to mix two solutions to reach a target concentration, or the amount of pure liquid left after repeated replacement using milk_left = initial × (1 − replaced/total)^n.

What is the average speed for a journey covering equal distances at speeds x and y?

Average speed = 2xy/(x+y) (the harmonic mean), not the simple arithmetic average.

Define cost price, selling price, profit and loss.

Cost Price (CP) is the buying price; Selling Price (SP) is the selling price. If SP > CP, Profit = SP − CP; if SP < CP, Loss = CP − SP.

State the formulas for profit percent and loss percent.

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

How is selling price found from cost price and a profit/loss percentage?

SP = CP × (100 + Profit%)/100, or SP = CP × (100 − Loss%)/100.

What is discount and on what value is it calculated?

Discount is a reduction on the Marked Price (list price): Discount = MP − SP, and Discount% = (Discount/MP)×100. SP = MP × (100 − Discount%)/100.

A shopkeeper allows two successive discounts of x% and y%. What single equivalent discount results?

Equivalent single discount = (x + y − xy/100)%.

State the simple interest formula and the amount.

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

State the compound interest amount formula for annual compounding.

A = P(1 + R/100)^T, and Compound Interest = A − P.

How is the compound interest formula adjusted for half-yearly and quarterly compounding?

Half-yearly: A = P(1 + R/200)^(2T). Quarterly: A = P(1 + R/400)^(4T) — the rate is divided and the period multiplied by the number of compounding intervals per year.

For the same P, R and T, how do simple and compound interest compare, and what is the 2-year difference?

Compound interest is greater than simple interest (equal only in the first period). For 2 years, CI − SI = P(R/100)².

Expand the identity (a + b)² and (a − b)².

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

State the identity for a² − b² and for (a + b)(a − b).

a² − b² = (a + b)(a − b); equivalently (a + b)(a − b) = a² − b².

Expand (a + b)³ and (a − b)³.

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

State the identities for a³ + b³ and a³ − b³.

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

Expand (x + a)(x + b).

(x + a)(x + b) = x² + (a + b)x + ab.

What is the identity for a² + b² + c² when a + b + c and ab+bc+ca are known?

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca), so a² + b² + c² = (a+b+c)² − 2(ab+bc+ca).

How do you solve a single linear equation in one variable?

Isolate the variable by performing the same operations on both sides (collect variable terms on one side, constants on the other) until the variable is alone.

Name the three algebraic methods for solving a pair of linear equations in two variables.

Substitution, elimination (addition/subtraction), and cross-multiplication.

For two lines a1x+b1y+c1=0 and a2x+b2y+c2=0, what conditions give one, none, or infinitely many solutions?

Unique solution: a1/a2 ≠ b1/b2 (intersecting). No solution: a1/a2 = b1/b2 ≠ c1/c2 (parallel). Infinite solutions: a1/a2 = b1/b2 = c1/c2 (coincident).

State the standard form of a quadratic equation and the quadratic formula.

Standard form: ax² + bx + c = 0 (a ≠ 0). Roots: x = [−b ± √(b² − 4ac)]/(2a).

What does the discriminant D = b² − 4ac tell you about the roots of a quadratic?

D > 0: two distinct real roots; D = 0: two equal (real) roots; D < 0: no real roots (two complex conjugate roots).

State the sum and product of the roots of ax² + bx + c = 0.

Sum of roots = −b/a; product of roots = c/a.

How do you build a quadratic equation from a given sum S and product P of its roots?

x² − Sx + P = 0, i.e. x² − (sum)x + (product) = 0.

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

The direction of the inequality sign reverses (e.g. −2x < 6 becomes x > −3).

How do you solve the absolute-value inequality |x| < a (a > 0) and |x| > a?

|x| < a means −a < x < a; |x| > a means x < −a or x > a.

State the product rule, quotient rule, and power rule for exponents.

a^m · a^n = a^(m+n); a^m / a^n = a^(m−n); (a^m)^n = a^(mn).

What are a^0, a^(−n), and a^(1/n) equal to?

a^0 = 1 (a ≠ 0); a^(−n) = 1/a^n; a^(1/n) = the n-th root of a (ⁿ√a).

How do you rationalize a denominator of the form 1/(√a + √b)?

Multiply numerator and denominator by the conjugate (√a − √b), giving (√a − √b)/(a − b).

State the angle sum of a triangle and the exterior angle theorem.

The three interior angles of a triangle sum to 180°. The exterior angle equals the sum of the two non-adjacent (remote) interior angles.

Define 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 equal.

When a transversal cuts two parallel lines, which angle pairs are equal and which are supplementary?

Corresponding angles and alternate (interior/exterior) angles are equal; co-interior (allied) angles are supplementary, summing to 180°.