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Civil Service Numerical and Analytical Methods

Essential formulas, quantitative shortcuts, number sequence patterns, and analytical reasoning principles for the Philippine Civil Service Examination.

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What is the formula to calculate percentage increase between an original value V1 and a new value V2?
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1
What is the formula to calculate percentage increase between an original value V1 and a new value V2?
Percentage increase = ((V2 - V1) / V1) * 100%. The difference is always divided by the original base value.
2
How do you calculate the simple interest earned on a principal P at annual rate r over t years?
Interest = P r t. The total accumulated amount is A = P + I = P(1 + rt).
3
What is the combined work rate formula when person A finishes a job in a hours and person B finishes it in b hours?
Combined rate = 1/a + 1/b. The time required working together is T = (a * b) / (a + b).
4
In a distance-rate-time problem, what is the formula relating distance (d), speed (r), and time (t)?
Distance = Rate Time (d = r t); consequently, Rate = d / t and Time = d / r.
5
How do you compute the average speed for a round trip covering equal distances at speeds v1 and v2?
Harmonic mean: Average Speed = (2 v1 v2) / (v1 + v2), not the arithmetic average of the two speeds.
6
What is the formula for the nth term of an arithmetic sequence with first term a1 and common difference d?
an = a1 + (n - 1) * d.
7
What is the formula for the sum Sn of the first n terms of an arithmetic progression?
Sn = (n / 2) (2 a1 + (n - 1) d) = (n / 2) (a1 + an).
8
What is the formula for the nth term of a geometric sequence with first term a1 and common ratio r?
an = a1 * r^(n - 1).
9
How do you find the ratio of parts when dividing an amount T into a ratio a : b?
Sum the ratio parts (a + b). The first portion is T (a / (a + b)) and the second is T (b / (a + b)).
10
What is the weighted average formula for groups with values x1, x2, ... and frequencies w1, w2, ...?
Weighted Average = (w1x1 + w2x2 + ... + wn*xn) / (w1 + w2 + ... + wn).
11
How do you find the perimeter and area of a rectangle with length L and width W?
Perimeter = 2(L + W); Area = L W.
12
What is the Pythagorean theorem relating legs a and b to hypotenuse c in a right triangle?
a^2 + b^2 = c^2, so c = sqrt(a^2 + b^2). Common triples include 3-4-5, 5-12-13, and 8-15-17.
13
How do you find the circumference and area of a circle with radius r?
Circumference = 2 pi r; Area = pi * r^2.
14
What is the volume of a right circular cylinder with base radius r and height h?
Volume = pi r^2 h.
15
In number series questions, what is a second-order arithmetic sequence?
A sequence where the differences between consecutive terms are not constant, but the differences of those differences are constant.
16
What is an alternating number sequence?
A series consisting of two interleaved sub-sequences (e.g., odd positions follow one pattern, even positions follow another).
17
What is the rule of contrapositive in formal logic?
The statement 'If P, then Q' is logically equivalent to its contrapositive: 'If not Q, then not P'.
18
What is the fallacy of affirming the consequent?
Assuming that because Q is true, P must be true in the conditional 'If P, then Q'. (Invalid deduction).
19
What is the fallacy of denying the antecedent?
Assuming that because P is false, Q must be false in the conditional 'If P, then Q'. (Invalid deduction).
20
In categorical syllogisms, what conclusion follows from 'All A are B' and 'All B are C'?
All A are C (transitive property of universal inclusion).
21
What does a Venn diagram intersection A intersect B represent in data interpretation?
The subset of elements that belong to both category A and category B simultaneously.
22
In word association analogies, what relationship exists between 'SPARK' and 'CONFLAGRATION'?
Degree or intensity: a spark is an incipient or minimal fire, while a conflagration is an extensive, destructive fire.
23
In probability, what is the formula for the probability of the union of two events P(A or B)?
P(A or B) = P(A) + P(B) - P(A and B).
24
How do you calculate the discount price given marked price M and discount rate d%?
Discounted Price = M (1 - d/100). The discount amount is M (d/100).
25
How do you calculate consecutive percentage discounts of d1% followed by d2%?
Net multiplier = (1 - d1/100) (1 - d2/100). Effective single discount = (d1 + d2 - (d1d2)/100)%.
26
In clock angle problems, what is the formula for the angle between the hour and minute hands at H hours and M minutes?
Angle = |30H - 5.5M| degrees (or 360 - angle if taking the smaller interior angle).

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