Fundamentals
Simplify: 3/4 + 2/5
23/20 = 1 3/20
15/20 + 8/20
Solve for x: 2x - 5 = 11
x = 8
2x = 16; 16/2 = 8
What is the mean of 3, 7, 5, 9
6
sum = 24, n = 4; 24/4 = 6
In a triangle, two angles measure 40∘ and 65∘. What is the measure of the third angle?
75 degrees
Sum of angles = 180; 180−(40+65)=75∘
If sinθ= 3/5, find cosθ
cos = 4/5
Opp=3, Hyp=5 → Adj=4 → cos=4/5
Evaluate: (23)2 X 2-4
4
Solution: (23)2=26; 26 X 2-4 = 22= 4
Find the discriminant of the quadratic equation: 3x2 - 7x + 4 = 0
1
Solution: Discriminant = b2 - 4ac
(-7)2 - 4(3)(4) = 49; 49 - 48 = 1
Find the median of 15, 10, 8, 12, 20, 18
13.5
Solution: sort - 8, 10, 12, 15, 18, 20
(12+15)/ 2 = 13.5
A circle has a radius of 7 cm. Find its circumference.
14π
C=2πr=2 x π x 7= 14π
If tanθ=7/ 24, find sinθ (leave it as a fraction)
Sin = 7/25
Opp=7, Adj=24 → Hyp=25 → sin=7/ 25
Simplify: 5(x + 2) - 3x
2x + 10
Solution: 5x + 10 - 3 = 2x + 10
Solve: square root of x + 3 = 5
22
Solution: square both sides: x + 3 = 25
The mean of 6 numbers is 12. If one of the number is removed, the mean of the remaining is 10. Find the number removed.
22
Total sum = 6 × 12 = 72; Remaining sum = 5 × 10 = 50; Removed number = 72 - 50 = 22.
In a right triangle, the sides are 6 cm and 8 cm. Find the hypotenuse
10
Solution: C2=62+82 =36+64=100⇒h=10
In a right triangle, one acute angle measures 40∘ and the hypotenuse is 10 units long.
Find the length of the side opposite the 40∘ angle.
(Round to the nearest tenth.)
6.4
sin(40∘)=opp/ hyp
sin(40∘)=x/10\s
x = 10 x sin (40) ≈6.4
5/6 divided by 10/9
3/4
Solution: Multiply the reciprocal- 5/6 X 9/10 = 45/60 = 3/4
Solve: 3(x -2) + 4 = 2x
2
3x - 6 + 4 = 2x ; 3x - 2 = 2x ; x = 2
The average score of 10 students is 75. The top 4 students have an average of 90. Find the average of the other 6 students.
65
Total sum = 10 × 75 = 750; Top 4 sum = 4 × 90 = 360; Remaining 6 sum = 750 - 360 = 390; Average = 390 ÷ 6 = 65.
In a circle, a central angle measures 72∘ and the radius is 5 cm. Find the length of the intercepted arc.
2π = 6.28 cm
Arc length = θ/360 × 2πr (circumference) =72/ 360× (2 x π x 5) =1/5×10π= 2π = 6.28 cm
Find θ, if sinθ=0.6
θ≈36.87 degrees (inverse sine)
Evaluate: 8 - [32 - (4 / 2)3]
7
Solution: Inside parentheses: 4÷2=2;23=8 ; 9−8=1 → Final: 8−1=7
Solve 3x2 + 5x - 2 = 0 using the quadratic formula
x = 3, -2
A bag contains 4 green marbles and 6 yellow marbles. You draw one marble, record its color, replace it, then draw a second marble. What is the probability both marbles are green? (leave it as a fraction)
hint: multiply the first probability with the second probability pick.
4/25
First pick: 4/10 = 2/5
Second pick: 4/10 = 2/5
2/5 x 2/5 = 4/25
A right triangle has one leg measuring 9cm and a hypotenuse measuring 15 cm. The triangle is inscribed in a circle with the hypotenuse as the diameter.
Find the area of the circle.
56.25π = 176.71
Solution:
Since the hypotenuse is the diameter:
diameter=15 cm
radius=7.5 cm
Area: π r2 = π (7.5)2 = 56.25π
In right triangle, adjacent = 9, hypotenuse = 15. Find tanθ (Leave it as a fraction)
tan=12/9=4/3
cosθ=9/15 =3/5 → Opp = 12 → tan=12/9=4/3