RRB NTPC (Non-Technical Popular Categories) Mathematics (Quantitative Aptitude) Syllabus

How do you find the overall gain when two successive discounts of x percent and y percent are given?

Equivalent single discount = (x + y - xy/100) percent. For example, 20 percent and 10 percent give 20 + 10 - 2 = 28 percent.

What is the formula for Simple Interest?

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

What is the formula for Compound Interest (compounded annually)?

Amount A = P(1 + R/100)^T, and Compound Interest = A - P. When compounded n times a year, A = P(1 + R/(100n))^(nT).

For 2 years, what is the difference between CI and SI on a principal P at rate R?

Difference (CI - SI) for 2 years = P(R/100)^2 = P x R^2 / 10000.

What is the formula for the average of n numbers, and the effect of adding/removing a value?

Average = (Sum of all observations)/(Number of observations). If a new value equal to the current average is added, the average is unchanged. Sum = Average x Number of observations.

What is the average of the first n natural numbers?

Average of first n natural numbers = (n + 1)/2. Their sum = n(n+1)/2.

State the alligation rule for mixing two ingredients.

(Quantity of cheaper)/(Quantity of dearer) = (Price of dearer - Mean price)/(Mean price - Price of cheaper). The mean price lies between the two prices.

If a vessel has milk and water and some mixture is removed and replaced with water repeatedly, what is the milk left formula?

After n operations, milk remaining = Initial quantity x (1 - amount removed each time / total)^n. For volume V, removing x each of n times: V(1 - x/V)^n.

If A can do a work in x days and B in y days, how long do they take together?

Together they take (xy)/(x + y) days. Their combined one-day work = 1/x + 1/y.

What is the relationship between number of workers, days, hours, and work done?

(M1 x D1 x H1)/W1 = (M2 x D2 x H2)/W2, where M = men, D = days, H = hours per day, W = work. This is the chain rule for work problems.

What are the key unit conversions for speed: km/hr to m/s and back?

To convert km/hr to m/s, multiply by 5/18. To convert m/s to km/hr, multiply by 18/5. Speed = Distance/Time.

What is the average speed for equal distances travelled at speeds x and y?

Average speed = 2xy/(x + y) (the harmonic mean), not the simple average, because the distances are equal but times differ.

How long does a train of length L take to cross a stationary platform of length P?

Time = (L + P)/Speed of train. To cross a stationary pole or man, time = L/Speed (only the train's own length matters).

How do you find the time for two trains to cross each other moving in the same and opposite directions?

Sum of their lengths divided by relative speed. Opposite directions: relative speed = sum of speeds. Same direction: relative speed = difference of speeds.

State the boats and streams formulas for downstream and upstream speeds.

Downstream speed = (boat speed + stream speed); Upstream speed = (boat speed - stream speed). Boat speed in still water = (down + up)/2; Stream speed = (down - up)/2.

If the present ages of A and B are in ratio a:b, how do you handle ages t years ago or hence?

Write present ages as ax and bx. For t years ago subtract t (ax - t, bx - t); for t years hence add t (ax + t, bx + t). Form an equation from the second given ratio and solve for x.

State the algebraic identities for (a + b)^2, (a - b)^2, and a^2 - b^2.

(a + b)^2 = a^2 + 2ab + b^2; (a - b)^2 = a^2 - 2ab + b^2; a^2 - b^2 = (a + b)(a - b).

State the identities for (a + b)^3, (a - b)^3, and a^3 + b^3, a^3 - b^3.

(a+b)^3 = a^3 + b^3 + 3ab(a+b); (a-b)^3 = a^3 - b^3 - 3ab(a-b); a^3 + b^3 = (a+b)(a^2 - ab + b^2); a^3 - b^3 = (a-b)(a^2 + ab + b^2).

What is the identity for a^2 + b^2 + c^2 + 2(ab + bc + ca)?

a^2 + b^2 + c^2 + 2(ab + bc + ca) = (a + b + c)^2. Also a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca).

What is the quadratic formula for ax^2 + bx + c = 0?

x = [-b plus or minus root(b^2 - 4ac)] / (2a), where b^2 - 4ac is the discriminant.

What does the discriminant of a quadratic equation tell you about its roots?

If D = b^2 - 4ac > 0, two distinct real roots; if D = 0, two equal real roots; if D < 0, no real roots (complex roots).

For a quadratic ax^2 + bx + c = 0, what are the sum and product of the roots?

Sum of roots = -b/a; Product of roots = c/a.

What is the difference between mean, median, and mode?

Mean is the arithmetic average. Median is the middle value when data is ordered (average of the two middle values if even count). Mode is the most frequently occurring value.

What is the formula for range and what does it measure?

Range = Highest value - Lowest value. It is the simplest measure of dispersion (spread) of a data set.

In a pie chart, how do you convert a percentage of data into degrees and vice versa?

Degrees = (Percentage/100) x 360, or directly (Value/Total) x 360. Conversely, Percentage = (Degrees/360) x 100. The whole circle is 360 degrees representing 100 percent.

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

The sum of the interior angles of a triangle is 180 degrees. An exterior angle equals the sum of the two opposite (remote) interior angles.

State the Pythagoras theorem and the angle relationships for parallel lines cut by a transversal.

Pythagoras: in a right triangle, hypotenuse^2 = base^2 + perpendicular^2. For parallel lines with a transversal: corresponding angles are equal, alternate angles are equal, and co-interior (allied) angles sum to 180 degrees.

Give the area and perimeter formulas for a rectangle, square, and triangle.

Rectangle: area = l x b, perimeter = 2(l + b). Square: area = side^2, perimeter = 4 x side. Triangle: area = (1/2) x base x height.

What are the area and circumference formulas for a circle, and area of a sector?

Circle area = pi r^2; circumference = 2 pi r. Sector area = (theta/360) x pi r^2; arc length = (theta/360) x 2 pi r, where theta is the central angle in degrees.

State Heron's formula for the area of a triangle and the area of an equilateral triangle.

Heron's formula: area = root[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2. Equilateral triangle of side a: area = (root 3 / 4) a^2.

Give the volume and total surface area formulas for a cube and a cuboid.

Cube (side a): volume = a^3, total surface area = 6a^2, diagonal = a root 3. Cuboid (l, b, h): volume = lbh, total surface area = 2(lb + bh + hl), diagonal = root(l^2 + b^2 + h^2).

Give the volume and surface area formulas for a cylinder, cone, and sphere.

Cylinder: volume = pi r^2 h, curved surface = 2 pi r h. Cone: volume = (1/3) pi r^2 h, curved surface = pi r l (l = slant height). Sphere: volume = (4/3) pi r^3, surface area = 4 pi r^2.

State the standard trigonometric ratios sin, cos, and tan in terms of a right triangle.

sin(theta) = Opposite/Hypotenuse; cos(theta) = Adjacent/Hypotenuse; tan(theta) = Opposite/Adjacent = sin/cos. Also cosec = 1/sin, sec = 1/cos, cot = 1/tan.

Give the values of sin and cos at 0, 30, 45, 60, and 90 degrees.

sin: 0, 1/2, 1/root2, root3/2, 1. cos: 1, root3/2, 1/root2, 1/2, 0 (for 0, 30, 45, 60, 90 degrees respectively). tan: 0, 1/root3, 1, root3, undefined.

State the fundamental trigonometric identities used in heights and distances.

sin^2 theta + cos^2 theta = 1; 1 + tan^2 theta = sec^2 theta; 1 + cot^2 theta = cosec^2 theta. In heights/distances, tan(angle of elevation) = height/horizontal distance.