MATH CONCEPTS
109. (Imagine Their Shapes Also) 1. Circular Ring 🔹A? 2. Sphere 🔸Surface Area? 🔹Volume? 3. Trapezoid 🔸P? 🔹A? 4. Rectangular Box 🔸A? 🔹V? 5. Cube 🔸A? 🔹V? 6. Cyclinder 🔸A? 🔹V?
1. Circular Ring 🔹A = π(R² - r²) R->bigger radius r->smaller radius 2. Sphere 🔸Surface Area = 4πr² 🔹Volume = (4πr²)/3 3. Trapezoid 🔸P = a+b+c+d 🔹A = h[(a+b)/2] 4. Rectangular Box 🔸A = 2ab+2ac+2bc 🔹V = abc 5. Cube 🔸A = 6s² 🔹V = s³ 6. Cyclinder 🔸A = 2πr(r+h) 🔹V = πr²h
108. (Imagine Their Shapes Also) 1. Square 🔸P? 🔹A? 2. Rectangle 🔸P? 🔹A? 3. Circle 🔸P? 🔹A? 4. Triangle 🔸P? 🔹A? 5. Paralellogram 🔸P? 🔹A? 6. Right Circular Cone 🔸A? 🔹Surface Area? 🔻V?
1. Square 🔸P = 4s 🔹A = s² 2. Rectangle 🔸P = 2L + 2W 🔹A = LxW 3. Circle 🔸P = 2πr or dr 🔹A = πr² or πd 4. Triangle 🔸P = a+b+c 🔹A = ½bh 5. Paralellogram 🔸P = 2L+2W 🔹A = bh 6. Right Circular Cone 🔸A = πr² + πrs 🔹Surface Area = all√r² + h 🔻V = ⅓πr²h
68. What is the value of Golden Ratio?
1.618
83. How to solve an=k(an-1)+L Ex: an=3(an-1)+1 and a1=10, then a3=?
1st: Substitute 2 to n, to get a2. an=3(an-1)+1 a2=3(a2-1)+1 a2=3(a1)+1 a2=3(10)+1 a2=31 2nd: After substituting 2, now substitute 3 and so on to get desired value. an=3(an-1)+1 a3=3(a3-1)+1 a3=3(a2)+1 a3=3(31)+1 a3=94 a3=94
62. Simplify i²¹³
1st: Subtract the exponent with a multiple of 4 until you make it 1, 2, 3, or 4. = 213 - 212 = 1 2nd: i¹ = i i² = -1 i³ = -1 i^4 = 1 3rd: Since the answer was 1, therefore, i²¹³ is equals i¹, then it is i.
24. What is the shortcut for getting factors of quadratic equation?
2ax + b +- √(b²-4ac)
9. How to solve if there is ratio? Men to women is 2:3, total is 350, how many women?
2x + 3x =350 5x=350 x=70 Women: 3x = 3(70) = 210
106. Differentiate: •Arithmetic Progression •Geometric Progression •Harmonic Progression •Fibonacci Progression
> Arithmetic progression has a common difference while Geometric progression has a common ratio. > Also, Harmonic progression is similar to arithmetic progression but is just the reciprocal. > In Fibonacci progression, you just add two preceding numbers to know the next term.
21. How do you interpret graph to get its equation of a line
All you need to do is get m (slope) and get b 1st: To get slope, look at the two endpoints of the line and get its coordinates (x1,y1) and (x2,y2). Then solve for m. (You know the formula) 2nd: To get b, choose from the two coordinates, you choose either (x1,y1) or (x2,y2) and substitute also the value of m to y = mx+b. Then you can get b. 3rd: Substitute the value of m and value of b to the equation of a line. (y = mx+b)
1. What will you do if you are solving equation with multiple signs. 2 + 3 - 6 divided by 9(2) =
Ans: 2 Use PEMDAS! remember to add or subtract from left to right, dili sunod'2 nga unahon ang add or subtract, basta start ka sa left okay?
34. You have (x + 1) and (x - 6), what are the solutions? a) +1 and -6 b) -1 and +6
B. -1 and +6 Always remember that if naay choices, solve first the factors like (x + 1) and (x - 6) before ka MO ANSWER!
85. How many terms in an binomial expansion? Ex: How many terms in (1+2x-7y+d)^6
Formula: e=exponent k=no. of terms e+(k-1) C (k-1) 6+(4-1) C (4-1) 9C3 Then solve for 9!/(9-3)!3! Ans: 84 terms
76. What is the formula of Infinite geometric sequence Ex: What is the sum of 2+1+1/2+1/4+...
Formula: Sn= a1/(1-r) Sn=2/(1-1/2) Sn=2/(1/2) Sn=4
59. What part of x is y? Ex: What part is 10 of 100?
Formula: The number with is before goes to numerator and the other goed to denominator. = 10/100 = 1/10
60. What percentage of x is y? Ex: What percentage of 30 is 10?
Formula: The number with is before goes to numerator, the other goes to denominator. = 10/30 = 0.33 = 33%
97. Formula to know how many diagonals can be formed in a polygon? Ex: How many diagonals can be formed in nonagon?
Formula: [n(n-3)]/2 = [9(9-3)] = [9(7)] = 63
65. How to solve angle of clock problems? Ex: What is the angle between hands when time is 9:34
Formula: Φ=| 30H-11/2M | where: H= hours of given time M= minutes of given time Φ=| 30(9)-11/2(34) | Φ=| 270 - 187 | Φ=| 83 | Φ=83
7. What if the exponent of 10 is negative? Ex: 9.75 x 10^-4
If exponent of 10 is negative, go to left, if positive, go to right. Ans: 0.000975
32. There is 200 meter dash contest. Jerry ran with speed 2.4 kph. Ryan ran with speed 3 kph. How many minutes longer did it take for Jerry to complete the race?
You should get the fraction that represents their speed. 1st: RYAN *(200m)(1hr/3000m) * = 1/15 hr * = 4 minutes 2nd: JERRY *(200m)(1hr/2400) * = 1/12 hr * = 5minutes 3rd: Subtract Jerry and Ryan time: * 5 min. - 4 min. * 1 min. Ans. 1 min.
23. How do you solve √21+q=3+√q ?
You square both sides! To eliminate the radical root signs. Then, always remember; (3+√q)² is NOT (9+q) It is (3+√q)(3+√q) DON'T FORGET!
8. 2 x 10^-2 where will the decimal point go?
left
49. 3x²-8x-3/2x²+7x-4 Find: 📌y-intercept 📌x-intercept or zeroes 📌vertical asymptotes 📌horizontal asymptotes
📌y-intercept *just substitute zero to all x. Ans: 3/4 📌x-intercept or zeroes *factor first the numerator: 3x²-8x-3 = (3x+1)(x-3) 3x+1, x = -1/3 x-3, x = 3 📌vertical asymptotes *factor first the denominator: 2x²+7x-4 = (2x-1)(x+4) 2x-1, x = 1/2 x+4, x = -4 📌Horizontal Asymptote n=m 3/2
104. ABRIDGED MULTIPLICATION 🔸Sum of Square 🔸Difference of Square 🔸Difference of Two Squares 🔸Sum of Cube 🔸Difference of Cube 🔸Sum of Two Cubes 🔸Difference of Two Cubes
🔹Sum of Square (a+b)² = (a²+2ab+b²) 🔹Difference of Square (a-b)³ = (a²-2ab+b²) 🔹Difference of Two Squares (a²-b²) = (a-b)(a+b) 🔹Sum of Cube (a+b)³ = (a³+3a²b+3ab²+b³) 🔹Difference of Cube (a-b³) = (a³-3a²b+3ab²-b³) 🔹Sum of Two Cubes (a³+b³) = (a+b)(a²-ab+b²) 🔹Difference of Two Cubes (a³-b³) = (a-b)(a²+ab+b²)
16. The capacity was 40% below full capacity, then it was raised to a level of 40%. What is tge equation?
1st: *40% below full capacity *It is 60% 2nd: *It was raised to 40% *Multiply (0.40x0.60) *= 24% 3rd: *60% + 24% *=84%
102. How to solve Average of two Averages? Ex: Average of six numbers is 4, if the average of two of the six numbers is 2, what is the average of the remaining four numbers?
1st: Get two totals 🔹If the average of six numbers is 4 then total is: 6x4 = 24 🔹If average of two numbers is 2, then total is: 2x2 = 4 2nd: Subtract two totals 24-4 = 20 3rd: Divide the difference to the question, which is: "what is the average of four remaining numbers" = 20/4 = 5
81. If there are 5 students who love English and 4 who love History and 2 loves both, how many students love English or History?
= 5 + 4 - 2 = 7
29. In a remainder theorem, what will you do? Will you add or subtract?G
After you bring down first digit, multiply it to the factor then ADD to the digits.
42. Formula to convert Degrees to Radians?
Degrees x π/180
36. You are asked to solve for the measurement of each interior angles of a regular polygon. What formula will you use?
Each Interior Angle = 180°-360/n
103. Differentiate Function and Relation.
Function = {(1,b) (5,a) (8,b)} Relation = {(1,b) (5,a) (5,c)} 💠 Function: 🔸one domain per range 🔸one input has exacty one output 💠 Relation 🔸has 2 domain per range 🔸one input has many outputs
64. Roman numerals I=? V=? X=? L=? C=? D=? M=?
I=1 V=5 X=10 L=50 C=100 D=500 M=1000
45. If the lengths of the sides of a square are doubled, what is the effect on its area?
It will increase four times.
30. How to solve questions with carabaos and ducks (or any animals) given with total of their heads and total feet?
Let: x=ducks y=carabao You should create 2 equations, one represents the feet and one represents the head. 1st: For FEET *2x + 4y = total no. feet 2nd: For HEAD *x + y = total no. head 3rd: *Manipulate the equation to get x and y by using the two equations.
2. Shortcut of arranging fractions? Ex: Arrange in ascending, 1/2, 2/3, 2/7
Multiply denominator and numerator vice versa 2/7, 1/2, 2/3
89. END BEHAVIOR Ex: f(x)=ax^n a) [a+] [n even] R=? L=? b) [a+] [n odd] R=? L=? c) [a-] [n even] R=? L=? d) [a-] [n odd] R=? L=?
REMEMBER: +If a is positive, then right arrow in Q1, if a is negative, then right arrow in Q4. +If n is even, directions should be symmetrical, if n is odd, then the opposite. a) R=Q1 , L=Q2 b) R=Q1 , L=Q3 c) R=Q4 , L=Q3 d) R=Q4 , L=Q2
38. What happens when lengths of the sides of square are doubled, what happens to its area?
The area increases 4 times.
92. How to know if points lies on the graph. Ex: What points lies on y=2x-1, (2,1) or (3,5) .
You just substitute values of x and y, then if two equated values are the same, then those points lie on that graph. y=2x-1 (2,1) 1=2(2)-1 1=3 ❌ y=2x-1 (3,5) 5=2(3)-1 5=5 ✔️ Thus, points (3,5) lies on the graph y=2x-1.
69. For Proportion questions, what will you multiply? Ex: •a:b = c:d •5:10 = 4:__
You multiply what's outside and inside. ______ | | •a:b = c:d |__________| • (a)(d) = (b)(c) • 5:10 = 4:__ (5x)=(10)(4) 5x= 40 x=40/5 x=8
19. How to solve equation of a line given two points (1,1) and parallel to line y = 2x-5?
You only need to have the value of m and b! 1st: To get me, since it is parallel to y = 2x-5, so it has the same m (slope) which is 2. So, m=2 2nd: Get the value of by substituting the 2 points given (1,1) to y = mx + b. 1 = 2(1) + b 1 = 2 + b So, b=-1 Substitute m=2 and b=-1 y = mx + b y = 2x - 1
66. Mentally create an outline for solution problems
a + b = a+b x% y% = z%
61. Formula to get: 📌 Number is increased by x% 📌 Number is decreased by x% Ex: a) 432 is increased by 5%? b) 123 is decreased by 20%
a) = 432(1+0.05) = 432(1.05) = 453.60 b) = 123(1-0.20) = 123(0.80) = 98.4
101. FACTOR POLYNOMIALS a) Factor difference of two squares. 🔹Factor 4x²-9 b) Factor Sum/Difference of two squares 🔹Factor x³-8
a) Difference of Two Squares (a²-b²) = (a+b)(a-b) = 4x²-9 = (2x+3)(2x-3) b) Sum/Difference of Two Squares (a³+b³) = (a+b)(a²-ab+b²) = x³-8 = (x-2)(x²+2x+4)
90. What is the slope of the: a) — Horizontal Line b) | Vertical Line
a) — Slope = 0 b) | Slope = undefined
37. Give the needed formula: a) What is the total measurement of all interior angle of a regular polygon. b) What is the measurement of each interior angle of a regular polygon. c) How many sides does polygon have if it has sum of 720° interior angle. d) What is the measurement of one exterior angle of a regular polygon?
n = total sides a) To get total measurement of all interior angles use: *Sum of Interior Angles=180°(n-2) b) To get measurement of each interior angles, use: *Each Interior Angle=180° - 360°/n c) To get how many sides a polygon has, use: *Sum of Interior Angles = 180°(n-2) d) To get one exterior angle, use: 360°/n
22. What is equation of a line?
y = mx + b
73. Explain rule of Combined Variation Ex: Suppose x varies jointly to y and z and inversely to w. If x=30 when y=2, z=3, and w=1, what is the value of x if w=2, y=1 and z=2?
•FACT Combined Variation has both dorect/joint and inverse variations •RULE x varies directly to y and inversely to z, then x=ky/z, where k is constant 1st: Change first x=ky/z to given variables. x=ky/z x=kyz/w 2nd: Find the value of k by substituting given value of x, y and z (or any variable) to x=ky/z x=ky/z x=kyz/w (30)=k(2)(3)/(1) k=5 3rd: Substitute the arrived value of k and the number given in question. x=kyz/w x=5yz/w "what is the value of x if w=2, y=1, and z=2" x=5yz/w x=5(1)(2)/2 x=5
63. What is the value of e? And what is it called?
•e means Eulers Number •It is equivalent to 2.718
110. Recite Converting Metric Lengths
▶️ km (x1000) m (x100) cm (x10) mm ◀️ km (/1000) m (/100) cm (/10) mm
112. Recite Converting Metric Weights
▶️ mile (x1760) yard (x3) ft (x12) in ◀️ mile (/1760) yard (/3) ft (/12) in
111. Recite Converting Metric Weights
▶️ tonne (x1000) kg (x1000) g (x100) cg (x10) mg ◀️ tonne (x1000) kg (/1000) g (/100) cg (/10) mg
47. What is Domain and Range?
📌DOMAIN: *Values of x which the function can take. Ex: 2x/x-1, the domain is all real numbers such that x is not equals to 1. 📌RANGE: *All possible values of y corresponds to value of x.
107. EXPONENT RULE 🔹1. a²•a³ 🔹2. a²/a³ 🔹3. (a²)³ 🔹4. (a/b)² 🔹5. a^0 🔹6. a^-3 🔹7. a^(x/y)
🔸1. a²+³ 🔸2. a²-³ 🔸3. a²•³ 🔸4. a²/b² 🔸5. 1 🔸6. 1/a³ 🔸7. y√a^x
91. Slope Questions a) What is the slope of Line A if slope of Line B is 4 and they are parallel lines? b) What is the slope of Line A if slope of Line B is -2 and they are perpendicular lines
a) FACT: 📌Slope of parallel lines is the same •Therefore, Slope of Line A is also 4 b) FACT: 📌Given slope of one perpindicular line, slope of other perpindicular line is -1/m (m=slope) •If they are perpindicular and slope of Line B is -2, then slope of Line A is: = -1/-2 (since denominator is negative) = - -1/2 = +1/2 or = 1/2
27. In the perfect square trinomial equation 4x² - 20x + 25, what will you substitute to (ax +- b)² ?
* "a" is the square root of first coefficient and "b" is the square root of the last coefficient. 4x² - 20x +25 = (ax +- b)² = (2x +- 5)²
31. How to get speed given distance (d) and time (t)? And what if it is stated late and stated early, what will you do to time?
*To get speed, use s = d/t *If it is stated that object is late, add that hours of being late to time *If it is stated that object is early, subtract that hours ob being early to time.
5. At first route, driver unloaded 1/5 of packages in his truck. After he unloaded 9 packages at his stop, 1/2 of original number remained. How many packages in the van before at first delivery? Put in equation.
1st: *Let x be total number 2nd: Multiply sa nabilin ang given *x(unloaded 1/5) - unloaded 9 packages *x(4/5)-9 3rd: Equate sa nabilin *x(4/5)-9 = 1/2 of original number remained *x(4/5)-9 = 1/2x Ans: 30ii
17. Grades are 83, 86, 89 and 90. There is one more quiz and you want an average not lower than 85. How to solve this?
1st: *You add all and also add x to represent other grade. (83 + 86 + 89 + 90 + x)/5 = 85 2nd: (348 + x)/5 = 85 3rd: Manipulate Ans: 77
10. You sell object 25% more than selling price. Then you're given 50% tax of your income. How much will tax will you pay?
1st: *let x be original price of object 2nd: get selling price *x + (rate you increase)(original price) *x + 0.25x or *x + 1/4x 3rd: get tax *(percent tax)(amount you increase to original price) *0.50(1/4x) = 1/2(1/4x) = 1/8x
3. How to transform questions nga naay gikuha nga part'2 into equation? ex: A rope is cut into three and one-third is used. Then one-fourth of the remaining rope is cut of and used. The piece left is 6 meters long. How many meters was the rope originally?
1st: *Let x be the rope 2nd: I multiply ang mga nabilin nga parts sa x *x(one third is used)(one-fourth of rope was used) *x(2/3 nabilin)(3/4 nabilin) *x(2/3)(3/4) 3rd: equate kung pila ang nabilin nga parts *x(2/3)(3/4)=6 SOLVE ans: 12
28. How will you know if an equation is a perfect square trinomial? Ex: Is x² + 10x + 36 a perfect square trinomial?
1st: *Use (ax +- b)² *Substitute: (ax +- b)² (1x +- 6)² You should solve (x + 6)² and (x - 6)², if any of the two gives the equation x² + 10x + 36, then it is a perfect square trinomial, if none of them gives the said equation, then it is not a perfect square trinomial. 2nd: *(x + 6)² is x²+12x+36 *(x - 6)² is x²-12x+36 3rd: *None of them gives x² + 10x + 36, therefore, this equation is not a perfect square trinomial.
88. How to find inverse of a function? Ex: What is the inverse of the function f(x)=2x+3 ?
1st: Equate to x then change to y x=2x+3 x=2y+3 2nd: Solve for y x=2y+3 x-3=2y x-3/2=y Ans: f^-1(x)=x-3/2
4. What about this: Rocky had a party. Midway in the party, half of people left. A fifth of those remaining started to sing. There were 16 remaining who did not sing. How many people attended? x(1/2)(4/5)=16 Question: nganong 4/5 man ang gibutang, if a fifth remaining started to sing, diba dapat 1/5?
4/5 dapat because ang gihatag na whole number is 16 WHO DID NOT SING, so if a fifth or 1/5 STARTED TO SING, meaning, 4/5 WALA NIKANTA. So kung unsa ang gihatag na whole number, 16 who sing, so 4/5 ang ihatag kay mao man ang ni sing pod, dapat naay parallelism. Okay? Ans: 40
43. Use the fingers technique to solve trigonometric values. a) cos 30° b) sin 45° c) cos 60° d) sin 90° e) cos 0° f) sin 0°
5 values are in each fingers are: 0°, 30°, 45°, 60°, 90° a) √3/2 b) √2/2 c) 1/2 d) 1 e) 1 f) 0
67. Solve the ratio The ratio of x to y is 3/4 and ratio of y to z is 2/3. Find the ratio of x to z.
= (3/4)(2/3) = 6/12 = 1/2
82. If there are students who only love English and 4 who only love History and 2 loves both, how many students love English or History?
= 5 + 2 + 4 = 11
15. 70% were 15 years old and 63 were 15 years old or below, what is the total number of people?
Always remember to equate given percentage or fraction to its equivalent given whole number. 63 is the number for 15 years old or below so find given similar to this 70% is 15 years old above, meaning 30% is 15 years old and below. Always equate similar given, for parallelism 0.30x = 63 x = 63/0.30 x = 210
99. How to find Coterminal Angles? a) What is the positive coterminal angle of 143° b) Find at least 3 positive and negative coterminal angles of 260°
Coterminal Angles are angles that shares a common terminal side. a) Just simply add or subtract 360° 143° + 360° = 503° 143° - 360° = -217° Since we are looking for a positive coterminal angle, then positive coterminal angle of 143° is 503°. b) We'll use Φ+360°(k) and substitute -1, -2, -3, 1, 2, 3 since it looks for 3 positives and 3 negative coterminal angles. Answers are: 260° + 360°(1) = 620° 260° + 360°(2) = 980° 260° + 360°(3) = 1340° 260° + 360°(-1) = -100° 260° + 360°(-2) = -460° 260° + 360°(-3) = -820°
86. Explain formula of Binomial Theorem Ex: Solve (x+y)^4
Formula: (a+b)^n = (4C0)a^4 b^0 a=magkaubos until zero b=magkasaka until n (x+y)^4= (4C0)a^4b^0 + (4C1)a^3b^1 + (4C2)a^2b^2 + (4C3)a^1b^3 + (4C4)a^0b^4
87. How to find coeffecient of rth term? Ex: What is the coefficient of x²y² in (x+y)^4.
Formula: (exponent of binomial)C(rth term y exponent)^(rth term y coefficient)^(rth term y exponent) x²y² in (x+y)^4 exponent of binomial = 4 rth term y exponent = 2 binomial y coefficient = 1 rth term y exponent = 2 =4C2^1^2 =4C2 =16
57. How to know the discount percentage? Ex: There is a store. It sells apple cost 30 each. There is a sale, and it only cost 23 each. What is the discount percentage?
Formula: 1 - discounted price/original price = 1 - 23/30 = 1 - 0.77 = 0.23 = 23%
58. x is y% of what number? Ex: 10 is 5% of what number?
Formula: 100(x/y) x is number you have y is percentage = 100(10/5) = 100(2) = 200
74. How to form an Arithmetic Sequence Ex: What is the value of s if s+4, 10, 7s forms an arithmetic sequence?
Formula: 3rd term-2nd term=2nd term-1st term 7s-10=10-s+4 8s=16 s=2 Ans: 6, 10, 14
75. How to form a geometric Sequence? Ex: What is/are the values of k if the numbers k+6, 16, 16k form a geometric sequence?
Formula: 3rd term/2nd term=2nd term/1st term 16k/16=16/k+6 k=16/k+6 k²+6k=16 k²+6k-16=0 (k+8)(k-2)=0 k= -18 and 2 If k=-8 -2, 16, -128 If k=2 8, 16, 32
84. How to find rth term of Binomial Theorem?
Formula: eCr-1 x^(e-(r-1)) y^(r-1) e = exponent r = term you desire
96. How many triangles can be formed in polygon? Ex: How mny triangles can be formed in decagon?
Formula: n-2 =10-2 =8
95. You are given slope-intercept form, how do you look for y-intercept? Ex: What is Y-intercept of y=-3x+2?
Given slope-interceot form, its y intercept is (0,b) y=-3x+2 (0,b) (0,2) y-intercept is (0,2)
13. You have 100 pesos original price and decreased it by 25% discount which is 25 pesos, and a tax of 50% rate. How will you get tax?
Multiply tax by the decreased amount which is 75 pesos. So, multiply 50% tax rate to 75 pesos.
12. You have 100 pesos original price and increased it by 25% which is 25 pesos, and a tax of 50% rate. How will you get tax?
Multiply tax rate by the increased amount which is 25 pesos.
33. One number is 17 less than another. Their sum is 125. What is the smaller number? You already got the value of x. Is that the final answer?
NO! PLEASE NO! *If you got the value of x, please check first if the value is right. *Let's say x is 71, subtract it firat to 125, answer is 54. *So the values are 54 and 71, you are asked what the smaller number is, and that is 54. *SEE? If you answered directly 71 because that is the value of x on you solution, you are already WRONG! YOU LOSE ONE POINT! 😏
18. If you will be asked, what of the following are consecutive ODD integers, will you only substitute 1? Ex: Is this a consecutive odd integer: n, n+2, n+4 ?
NO. You should not only substitute 1 to know if it represents consecutive odd integers, also try 2. If it is still odd after substituting 2, then it represents consecutive odd integer. ans: Try: n, n+2, n+4 *For value of 1 = 1, 3, 5 ✔️ (it is consecutive odd) *BUT try value of 2 =2, 4, 6 ❌ (it is not consexutive odd) THEREFORE, n, n+2, n+4 does not represent sequence of consecutive odd integers.
93. How to solve Table of Linear Function? The table represents some values of a linear function. What is c? f(x) 3 6 9 x 10 c 18
Numbers in first coloumn (3, 6, 9) increases by the same value, which is 3 consecutively, therefore numbers below should also increase by the same value. 18-c=c-10 18+10=c+c 28=2c c=14
56. How to get percent to a percent? Ex: In a club of 100 members, 25% of its members play piano, 20% play piano and bass. How many members play both piano and bass?
Percent of a Percent means if there are x students, y% are handsome, z% are tall and handsome, then there are x(y%) who is handsome and x(y%)(z%) who are handsome and tall. =100(0.25)(0.20) =25(0.20) =5 members
41. Formula to convert Radians to Degrees
Radians x 180/π
50. How to find range of a function? Ex: x² + 1
To find range: 1st: Equate to y: y = x² + 1 2nd: Solve for x: y = x² + 1 y-1 = x² √y-1 = x x = √y-1 3rd: Form an inequality, what values will make x = √y-1 imaginary? Well, y should not be zero, if its zero, it will be √-1 which is imaginary. Therefore: y-1>0 y>1 So, the RANGE of x² + 1, is All real numbers greater than 1.
11. How do you get total amount paid if for example you are given 10% discount on original price and charged 10% tax on selling price?
Task: -Get selling price -Get tax -Add selling price and tax, you get amount paid 1st: *let x be total amount 2nd: *Get selling price on the question *original price minus discounted amount *x - 0.10x =0.9x 3rd: *Get tax: *It is said to get 10% tax on selling price *(tax rate)(discounted amount) *(0.10)(0.9x) =0.09x 4th: *Get amount paid *amount paid = selling price plus tax *0.9x + 0.09x *0.99x is amount paid
46. The length of a rectangle is twice the width and is equal to the side of a square. What happens to their area?
The area of the square is 1/2 times that of the rectangle.
51. Someone started a monthly salary of 5,600 and steadily recieved an annual salary increase of P250. How much did she earn in 11 years? How to solve this?
This is an example of arithmetic progression. Use: 5,600 x 12 = 67,200 an=a1 + (n-1)d a11=67,200 + (11-1)250 a11=67,200 + (10)250 a11=67,200 + 2500 a11=69,700 Ans: 69,700
26. What will you add to make this a perfect square trinomial? x² - 12x + ___
Use (b/√a)² (b/√a)² (-6/√1)² (-6/1)² (-6)² 36 36 is the missing number to make x² - 12x + ___ a perfect square trinomial.
25. What number will you add to make this a perfect square trinomial: 4x² - 20x + ___
Use formula (b/√a)² 4x² - 20x + ___ *b= -20/2 = -10 *a = 4 √a = 2 (-10/2)² (-5)² 25 *25 is the missing number to make 4x² - 20x + ___ a perfect square trinomial.
40. What is the value of an exterior angle of a regular polygon? a) What is the measurement of exterior angle of a regular decagon?p
Use formula 360°/n a) 360°/10 = 36°
39. How do you find area of Kite or Rhombus given 2 diagonals?
Use formula: 1/2(d1)(d2)
20. How to get slope given two points (2,-4) and (-5,7)?
Use slope formula! m = (y2-y1)/(x2-x1) m = (7+4)/(-5-2) m = 11/-7 Ans: -11/7
35. 📌How will you get the sum of interior angles of a regular polygon? 📌How will you get the total sides of a regular polygon?
Use the formula: *Sum of Interior Angles = 180°(n-2) Just substitute what is given and find or manipulate the formula to get what is asked. NOTE: You can only use this if the given or asked is the SUM of interior angles OR the number of sides.
55. Convert Percent to Fraction Ex: Convert 45.2% in fraction
Use: x/100 (x=percent) =45.2/100 =452/1000 =113/250
54. Formula for solving volume of pyramid? Ex: A pyramid has base area of 9m² and an altitude of 200cm. What is its volume?
Vp = 1/3(base area)(altitude) Ans: =1/3(9m²)(200cm/100) =1/3(9m²)(2m) =1/3(18m³) =6m³
80. Explain what are their names and their rules, IMAGINE also the venn diagram. a) AuB b) AnB c) A-B or A\B d) AΦB e) A^c A= {1, 2, 3, 4} B= {3, 4, 5}
a) A union B •includes all elements from circles A and B >AuB = {1, 2, 3, 4, 5} b) A intersect B •includes only the element found in intersection of A and B >AnB = {3, 4} c) Set Difference or set A minus set B •Elements only in A >A-B = {1, 2} d) AΦB (this is plus sign inscribed in circle) or also called disjunctive union and symmetric difference, is the set containing all elements that are either in A or in B, but not in both. > AΦB = {1, 2, 4, 5} e) A complement contains all elements not in A. > A^c = {5}
98. Reference and Coterminal Angles a) If the Angle is in quadrant 1, what is the reference angle? b) If the Angle is in quadrant 2, what is the reference angle? c) If the Angle is in quadrant 3, what is the reference angle? d) If the Angle is in quadrant 4, what is the reference angle?
a) Reference Angle is itself. b) Reference Angle is 180°-Φ c) Reference Angle is Φ-180° d) Reference Angle is 360-Φ
6. a) What if you move exponent to right, the power of 10 is? b) What if you move exponent to left, the power of 10 is? Ex: 1) 0.000875 to scientific notation 2) 875000.0 to scientific notation
a) negative b) positive 1) 8.75 x 10^-4 2) 8.75 x 10^5
94. Forms of Linear Equation: a) Point-Slope Form b) Slope-Intercept Form c) Standard Form d) General Form
a) y-y1=m(x-x1) b) y=mx+b c) ax+by=c d) ax+by+c=0
44. In sin and cos, what is y, and what is x?
sin = y cos = x
77. Simplify: x³(x-1)³ ---------- (x^5)(x-1)²
•FACT If fraction has a common factor but unequal exponent, subtract the exponents and the factor must be on where the highest exponent was. (x-1) ----- (x²)
79: How many elements are in the set? Ex: A= {2, 4, 6, 8, 10} B= {5, 10, 15}
•FACT: In a set, write only the same element once. AuB {2, 4, 5, 6, 8, 10, 15} Thus, there is only 7 elements in that set because you only write 10 once.
78. How to solve question with inclusive word? Carl reads pages 139 to 225, inclusive, in his History book. How many pages did he read?
•FACT: Inclusive means that a range includes two extreme ends. 1st: Subtract two numbers. =225-139 =86 2nd: Add one to difference. =86+1 =87N
71. Explain rule of Inverse or Indirect Variation. Ex: If j varies inversely to x and j=4 when x=1, what is the value of j if x=4?
•RULE: If x varies directly to y, then x=k/y. 1st: Change first x=k/y to given variables. x=k/y j=k/x 2nd: Find the value of k by substituting given value of x and y (or any variable) to j=k/x. j=k/x 4=k/1 k=4 3rd: Substitute the arrived value of k and the number given in question. j=k/x j=4/x "what is the value of j if x=4" j=4/4 j=1
72. Explain rule of Joint Variation. Ex: If x variest jointly to c and h and x=24 when c=2 and h=1, what is the value of x if c=3 and h=2?
•RULE: If x varies jointly to y and z, then x=kyz, where k is a constant. 1st: Change first x=kyz to given variables. x=kyz x=kch 2nd: Find the value of k by substituting given value of x and y (or any variable) to x=kch x=kch 24=k(2)(1) k=12 3rd: Substitute the arrived value of k and the number given in question. x=kch x=(12)ch "what is the value of x if c=3 and h=2? x=(12)ch x=(12)(3)(2) x=72
70. Explain rule of Direct Variation. Ex: If x varies directly to z, and x=30 when z=6, what is the value of x if z=2?
•RULE: If x varies directly to y, then x=ky. 1st: Change x=ky first according to given variables x=ky x=kz 2nd:Find the value of K by substituting given value of x and z (or any variable) to x=ky. x=kz (30)=k(6) k=5 3rd: Substitute the arrived value of k and the number given in question. x=kz x=5z "what is the value of x if z=2" x=5(2) x=10
14. Analyze tax and where to multiply tax rate.
📌 If the amount has original price and discounted it by certain percent, multiply tax rate to the discounted amount. 📌 If the amount has original price and increased it by certain percent, multiply tax rate to the amount you add to the original price.
53. 📌General formula for geometric progression? 📌Formula for sum of Geometric progression? Ex: What is the sum of 2, 4, 8, ...,256
📌An = a1r^(n-1) 📌Sn = (a1-an^r)/(1-r) or a1(1-r^n)/(1-r) 1st: Find n to know how many terms there are. an=a1r^(n-1) 256=2(2)^(n-1) 2^8=(2^1)(2^n-1) exponents: 1+n-1 = n 2^8=2^n n=8 2nd: Solve using geometric sum formula. Sn=(a1-anr)/(1-r) S8=(2-256(2))/1-2 S8=(-510)/(-1) S8=510
48. How do you look for the Horizontal Asymptote?
📌Horizontal Asymptote: *Let "n" be the exponent of numerator, and "m" is the exponent of denominator: •If n<m, hor. asy. is y=0 •If n=m, hor. asy. is a/b (a is coffecient of the highest exponent in numerator, and b is the coefficient of the gighest exponent of the denominator. •If n>m, there is no hor. asy.
52. 📌General formula for arithmetic progression? 📌Formula for sum of arithmetic progression? Ex: What is the sum of 2, 4, 6, ..., 44
📌an = a1 + (n-1)d 📌Sn = n/2(a1 + an) or Sn = n/2(2a+(n-1)d) 1st: Find n first, n means how many terms there are. an=a1+(n-1)d 44=2+(n-1)2 44=2+(2n-2) 44-2+2=2n 44=2n n=22 2nd: Substitute to sum of arithmetic formula. Sn=n/2(a1+an) S22=22/2(2+44) S22=11(46) S22=506
100. What is reference angle of: 🔶69° 🔶5π/6 🔶110° 🔶Determine which quadrant 10π/3 is located.
🔶69° is in Q1, so its reference angle is also 69° 🔶5π/6 > make it degrees first =(5π/6)(180/π) =150° Since 150° is in quadrant 2, so its reference angle is 180-Φ. =180°-Φ =180°-150° =30° 🔶We need to know where is -110° located at. = -110°+360° = 250° Since 250° is in quadrant 3, so its reference angle is Φ-180°. = Φ-180° = 250°-180° = 70° 🔶10π/3 > first know where it is located. = (10π/3)(180°/π) = 600° Express 600° as 360° + 240° We know 240° is in quadrant 3, hence, 10π/3 is located at quadrant 3.
