Quantitative Reasoning Flashcards
1
Q
distance = ?
A
distance = speed x time
2
Q
speed = ?
A
speed = distance/time
3
Q
time = ?
A
time = distance/speed
4
Q
upstream speed = ?
A
upstream speed = distance/time upstream
5
Q
downstream speed = ?
A
downstream speed = distance/time downstream
6
Q
speed of stream = ?
A
speed of stream = downstream speed - upstream speed / 2
7
Q
speed of boat in stillwater = ?
A
speed of boat in stillwater = (downstream speed + upstream speed) / 2
8
Q
average speed of boat = ?
A
average speed of boat = downstream speed x upstream speed / (downstream speed + upstream speed / 2)
9
Q
relative speed (moving towards each other) = ?
A
relative speed (moving towards each other) = speed of object 1 + speed of object 2
10
Q
relative speed (moving in the same direction) = ?
A
relative speed (moving in the same direction) = speed of faster object - speed of slower object
11
Q
time taken = ?
A
time taken = work done/rate of work
12
Q
work done = ?
A
work done = time taken x rate of work
13
Q
rate of work = ?
A
rate of work = work done/time taken
14
Q
% = ?
A
% = (part/whole) x 100
15
Q
part = ?
A
part = (%/100) x whole
16
Q
whole = ?
A
whole = part / (%/100)
17
Q
% increase = ?
A
% increase = (new value - old value / old value) x 100
18
Q
% decrease = ?
A
% decrease = (old value - new value / old value) x 100
“decrease %: old minus new over old times 100”
19
Q
Price = ?
A
Price = (no. of items)(cost per item)
20
Q
Sale price = ?
(price at which an item is sold)
A
Sale price = Cost price (1 + Profit %/100)
21
Q
Sale price = ?
A
Sale price = Cost price (1 - Loss %/100)
22
Q
Profit = ?
A
Profit = Sale price - Cost price
23
Q
Loss = ?
A
Loss = Cost price - Sale price
24
Q
Profit % = ?
A
Profit % = (Profit/CP) x 100
25
Loss % = ?
Loss % = (Loss/CP) x 100
26
Principal = ?
Principal = (SI × 100) / RT
27
Rate of interest = ?
Rate of interest = (SI × 100) / PT
28
Time = ?
Time = (SI × 100) / PR
29
Simple interest = ?
Simple interest = PRT/100 or Total Amount (A) - Principal (P)
30
Amount after interest (A) = ?
Amount after interest (A) = Principal (P) + Simple Interest (SI)
31
Compound interest = ?
Compound interest = A - P
32
Amount after compound interest (A) = ?
Amount after compound interest (A) = P (1 + R/100) ^T
33
Compound interest (n no. of times)= ?
Compound interest = P (1 + R/n) ^n^T - P
34
Compound interest (half-yearly)= ?
Compound interest (half-yearly)= P [1 + (R/2)/100]^2T - P
35
Principal (of CI) = ?
Principal = A/(1 + R/100)^T
36
Area of square = ?
Area of square (diagonals) = ?
Perimeter of square = ?
Area of square = a^2
Area of square (diagonals) = 1/2 diagonal^2
Perimeter of square = 4a
37
Area of rectangle = ?
Perimeter of rectangle = ?
d² = ?
L = ?
Area of rectangle = LW
Perimeter of rectangle = 2(L+W)
d² = L² + W²
L = Area/W
38
Area of triangle = ?
Perimeter of triangle = ?
Area of triangle = 1/2 bh
Perimeter of triangle = sum of all sides
39
Area of parallelogram = ?
Perimeter of parallelogram = ?
Area of parallelogram = bh
Perimeter of parallelogram = 2 (base + side) | 2 (b + a)
40
Area of rhombus = ?
Perimeter of rhombus = ?
a = ?
Area of rhombus = 1/2 (diagonal1)(diagonal2)
Perimeter of rhombus = 4a
a = √(diagonal1/2)² + (diagonal2/2)²
41
Area of trapezoid = ?
Perimeter of trapezoid = ?
Area of trapezoid = 1/2 (a + b; sum of parallel sides)(h; perpendicular distance between the sides) | **1/2 (a + b)(h)**
Perimeter of trapezoid = (a + b + c + d; sum of all sides)
42
Area of circle = ?
Circumference = ?
Diameter = ?
r = ?
distance = ?
midpoint = ?
Area of circle = πr²
Circumference = 2πr
Diameter = 2r
r = Diameter/2
distance = √(x2 - x1)² + (y2 - y1)²
midpoint = (x1+x2/2 , y1+y2/2) = (h, k)
43
equation of the circle = ?
equation of the circle = (x-h)² + (y-k)² = r²
44
Cube surface area = ?
Cube volume = ?
Diagonal = ?
Cube surface area = 6a²
Cube volume = a³
Diagonal = a√3
45
Rectangle/cuboid surface area = ?
Rectangle/cuboid volume = ?
Diagonal = ?
Rectangle/cuboid surface area = 2(LW+WH+HL)
Rectangle/cuboid volume = LWH
Diagonal = √L²+W²+H²
46
Cylinder surface area = ?
Cylinder volume = ?
Cylinder surface area = 2πr² + 2πrh
Cylinder volume = πr²h
47
Sphere surface area = ?
Sphere volume = ?
Sphere surface area = 4πr²
Sphere volume = 4/3πr³
48
Cone total surface area = ?
Cone curved surface area = ?
Slant height = ?
Cone volume = ?
Cone **total** surface area = πr² + πrl
• r = radius of the base of the cone
• l = slant height of the cone
Cone **curved** surface area = πrl
Slant height (l): l² = r² + h²
Cone volume = 1/3πr²h
49
Pyramid surface area = ?
Pyramid volume = ?
Pyramid surface area =
Pyramid volume = 1/3 (base area)(h)
50
**Coordinate**
Distance = ?
Midpoint = ?
Slope (m) = ?
Slope intercept form: ?
Point-slope form: ?
Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]
Midpoint = [(x₁ + x₂)/2, (y₁ + y₂)/2]
Slope (m) = (y₂ - y₁)/(x₂ - x₁)
Slope intercept form: y = mx + b
Point-slope form: y - y₁ = m(x - x₁)
51
sin x = ?
cos x = ?
tan x = ?
sec x = ?
csc x = ?
cot x = ?
**SOH** sin x = opp/hyp
**CAH** cos x = adj/hyp
**TOA** tan x = opp/adj
**SHA** sec x = hyp/adj
**CHO** csc x = hyp/opp
**CAH** cot x = adj/hyp
52
**Pythagorean identities**
(fundamental trigonometric identities derived from the Pythagorean theorem; they relate the squares of sine, cosine, and tangent)
• ? + ? = 1
• 1 + ? = sec² x
• 1 + ? = csc² x
• sin² x + cos² x = 1
• 1 + tan² x = sec² x
• 1 + cot² x = csc² x
53
**Reciprocal identities**
sin x = 1/csc x
cos x = 1/sec x
tan x = 1/cot x
csc x = 1/sin x
sec x = 1/cos x
cot x = 1/tan x
54
**Quotient identities**
(relate the **tangent** and **cotangent** functions to sine and cosine)
• tan x = ____/cos x
• cot x = ____/sin x
• tan x = sin x/cos x
• cot x = cos x/sin x
55
**Cofunction identities**
[show the relationship between trigonometric functions of complementary angles (angles that add up to 90° or π/2)]
• sin (90° - x) = cos x, cos (90° - x) = sin x
• tan (90° - x) = cot x, cot (90° - x) = tan x
• sec (90° - x) = csc x, csc (90° - x) = sec x
56
**Sum and difference formulas**
(allow you to calculate the sine, cosine, and tangent of the sum or difference of two angles)
• sin (A+B) = ?
• sin (A-B) = ?
• cos (A+B) = ?
• cos (A-B) = ?
• tan (A+B) = ?
• tan (A-B) = ?
• sin (A+B) = sinAcosB + cosAsinB
• sin (A-B) = sinAcosB - cosAsinB
• cos (A+B) = cosAcosB - sinAsinB
• cos (A-B) = cosAcosB + sinAsinB
• tan (A+B) = (tanA + tanB)/(1 - tanAtanB)
• tan (A-B) = (tanA - tanB)/(1 + tanAtanB)
57
**Double angle formulas**
• sin (2x) = 2sinx____
• cos (2x) = cos²x - _____
• cos (2x) = ______ - 1
• cos (2x) = 1 - ______
• tan (2x) = 2tanx/(1 - _____)
• sin (2x) = 2sinxcosx
• cos (2x) = cos²x - sin²x
• cos (2x) = 2cos²x - 1
• cos (2x) = 1 - 2sin²x
• tan (2x) = 2tanx/(1 - tan²x)
58
**Half angle formulas**
• sin (x/2) = ±√[(1 - ____) / 2]
• cos (x/2) = ±√[(1 + ____) / 2]
• tan (x/2) = ±√[(1 - ____) / (1 + ____)]
• tan (x/2) = ____ / (1 + cosx)
• tan (x/2) = (1 - ____) / ____
• sin (x/2) = ±√[(1 - cosx) / 2]
• cos (x/2) = ±√[(1 + cosx) / 2]
• tan (x/2) = ±√[(1 - cosx) / (1 + cosx)]
• tan (x/2) = sinx / (1 + cosx)
• tan (x/2) = (1 - cosx) / sinx
*The ± sign depends on the quadrant in which x/2 lies: **ASTC rule (All Students Take Calculus)** to determine the sign of each function based on the quadrant.
A (All): In QI, all trigonometric functions (sin, cos, tan, and their reciprocals) are **positive**.
S (Sine): In QII, only **sin** and its reciprocal **csc** are positive. All others are **negative**.
T (Tangent): In QIII, only **tan** and its reciprocal **cot** are positive. All others are negative.
C (Cosine): In QIV, only **cos** and its reciprocal **sec** are positive. All others are negative.
**Diagram Representation**
1st Quadrant (0° to 90°): All positive
2nd Quadrant (90° to 180°): Sine positive
3rd Quadrant (180° to 270°): Tangent positive
4th Quadrant (270° to 360°): Cosine positive
59
**Product to sum formulas**
(used to rewrite the product of sine and cosine functions as a sum or difference of trigonometric functions)
**Product of sines and cosines**
sinAcosB = ?
**Product of cosines and sines**
cosAsinB = ?
**Product of cosines**
cosAcosB = ?
**Product of sines**
sinAsinB = ?
**Product of sines and cosines**
sinAcosB = (1/2) [ sin(A + B) + sin(A - B) ]
**Product of cosines and sines**
cosAsinB = (1/2) [ sin(A + B) - sin(A - B) ]
**Product of cosines**
cosAcosB = (1/2) [ cos(A + B) + cos(A - B) ]
**Product of sines**
sinAsinB = (1/2) [ cos(A - B) - cos(A + B) ]
60
**Sum to product formulas**
(used to rewrite the sum or difference of trigonometric functions as a product)
• sin(A) + sin(B) = ?
• sin(A) - sin(B) = ?
• cos(A) + cos(B) = ?
• cos(A) - cos(B) = ?
• sin(A) + sin(B) = 2sin[(A + B)/2] cos[(A - B)/2]
• sin(A) - sin(B) = 2cos[(A + B)/2] sin[(A - B)/2]
*pos: sin-cos, neg: cos-sin
• cos(A) + cos(B) = 2cos[(A + B)/2] cos[(A - B)/2]
• cos(A) - cos(B) = -2sin[(A + B)/2] sin[(A - B)/2]
*pos: cos-cos, neg: sin-sin
61
maximum value
sin (x) = ?
cos (x) = ?
minimum value
sin (x) = ?
cos (x) = ?
maximum value
sin (x) = 1
cos (x) = 1
minimum value
sin (x) = -1
cos (x) = -1
62
**Maximum and mnimum values**
R = ?
R = √A² + B²
63
1 kg = ? g
1000 g
64
1 mi = ? km
1.609344 km
1️⃣▪️6️⃣0️⃣9️⃣3️⃣4️⃣4️⃣
65
1 L = ? gallons
0.2641720524 gallons
0-264-172-0524
66
1 in = ? cm
2.54 cm
67
1 ft = ? in
12 in
68
1 m = ? cm
100 cm
69
1 km/h = ? m/s
5/18 m/s
70
m/s to km/h?
18/5 km/h
71
Radians = ?
Radians = Degrees × π/180
