Lesson 1: Basic Mathematics Answers
1.1 Mathematical Operators
Examples:
1.
2.
3.
4.
5.
12+3x5
= 27
(2 + 3) x 5
= 11
10/5 + 2
= 4
9/(5 - 2)
= 3
(3 x 5 + 5) + 40/8 -15
= 10
Examples:
1. 5 – 4 x2
2. 5 - 20 /2
3. (-5) x (-4) + 10
4. (-5) + (-20) /(-10)
= -3
= -5
= 30
= -3
Fractions
Examples:
P1: Express 1/8 as a decimal and as a percentage.
As a Decimal: 0.125
As a Percentage: 12.5%
P2: Express 0.0625 as a percentage and as a fraction.
As a Percentage: 6.25%
As a Fraction: 1/16
P3: Express 12.5% as a decimal and as a fraction.
As a Decimal: 0.125
As a Percentage: 1/8
1.4 Percentages
Examples:
1) Profit for 2011 is $100 Million, Calculate the expected profit for 2012 if the profit is expected to
(a) rise by 20%
1000000000+1000000000*(20/100) = 120000000
(b) fall by 10%
1000000000-1000000000*(10/100) = 90000000
2) Revenue of WTC is comprised of following components.
Product
Revenue $ ‘Mil
Five Roses
200
4 Horses
350
Kingfisher
250
Total
800
Express the revenue contribution from each product as a percentage.
As Percentages:
Five Roses = 25%
4 Horses = 43.75%
Kingfisher = 31.25%
Examples:
1) Selling price = $125 , Cost = $ 25 Calculate both markup and margin
Markup = (100/125)*100 = 80%
Margin = (100/25)*100 = 40%
2) Selling price = $150, Margin = 20%, Find the profit
Profit = $30
3) Selling price = $250, Margin = 25%, Find the cost
Cost = 250-62.5 = $187.5
4) Selling price = $120. Markup = 20%, Find the profit
Profit = $20
5) Profit = $ 20 , Mark up =10%, Find the selling price
Selling Price = $220
1.5 Ratios
Examples:
1) Split $100 between A and B in the ratio 2 : 3
A = $40
B = $60
2) Split $70 between B and C in the ratio 3 : 4
A = $30
B = $40
3) An amount is split between A & B in the ratio of 5 : 3 , A received $200, how much
does B receive?
B received = $180
4) An amount is split between A & B in the ratio of 10 : 2 , A received $240, how much does B receive?
B received = $48
5) A,B and C are three partners of a business, the profit for the year is $180,000. They
have decided to share the profits in the ratio of 2 :3: 1. Calculate each ones share.
A = $60,000
B = $90,000
C = $30,000
1.6 Powers & Roots
Examples:
Simplify the below expressions using the arithmetic Rules.
a)
X6.X-2
= X4
b) X6.X1/2 =X5/2
c)
X7/X4
=X3
d) X19/X-9 =X28
e)
(𝑋 5 )2
f)
𝑥 7 . √𝑥
g)
𝑥 9 . √𝑥
=X15/2
3
=X19/3
3
=X17/3
h) 𝑥 6 / √𝑥
i)
=X10
j)
𝑋7
𝑦7
.
𝑦 −3
=x12.y-10
𝑥 −5
(𝑥 2 )5 𝑦 4
𝑦2
𝑦 −3
. (𝑦4)−3
𝑥 −5
=x15.y11
1.7 Formulae & Equations
Examples:
1) Find the value of the function f(x) when x = 1, x=3 , x = -2 : f(x) = 2x +3
When x= 1: 5
When x = 3: 9
When x = -2: -1
2) Find the value of the function f(y) when y = 1, y=2 , y = -2 : f(y) = y2 +3
When y= 1: 4
When y = 2: 7
When y = -2: 7
3) Find the value of the function f(x) when x = 1, x=3 , x = -2 : f(x) = x2 + 2x +3
When x= 1: 6
When x = 3: 18
When x = -2: 3
4) Find the value of the function f(x) when x = 1, x=2 , x = -1 : f(x) = 3x2 + 2x +3
When x= 1: 8
When x = 2: 17
When x = -1: 4
5) If f(x) = x3 + 2x2 + 4 , find f(2) and f(-3)
f(2) = 20
f(-3) = -11
6) Y = axb : Find y when a= 10 , x=4 and b= -0.152
Y = 8.100034736
7) Y = log 10 (11x + 1) : Find Y when x = 9, 1, 0
When x= 9, Y = 2
When x = 1, Y = 1.079181246
When x = 0, Y = 0
8) If g(x)= 2x/(x2+4) : Find g(1) , g(-2)
g(1) = 0.4
g(-2) = -0.5
Factor each sum. Pick out the common factor.
a) 4x + 6y
=2(2x+3y
b) 6x − 6
=6(x-1)
c) 8x + 12y − 16z
=4(2x+3y-4z)
d) 12x + 3
=3(4x+1)
e) 18x − 30
=6(3x-5)
f) 2x + ax
=x(2+a)
g) x² + 4x
=x(x+4)
h) 8x² − 4x =4x(x-1)
Examples:
1) Y = 3x + 2 ; x
X = (Y-2)/3
2) S = ut + 0.5at2 ; a
a = (S-ut)/0.5t2
3) x = √𝟓𝒚 ; y
y = x2/5
4) 2x =3. √𝟐𝒚 + 1
Y = (2x-1)2/18
5) y = 49x2 ; x
x =√𝑦/7
6) DQ = PC2/ E ; C
𝐷𝑄𝐸
C=√
𝑃
7) A = (1-t) / (1+t)
t=
1−𝐴
1+𝐴
8) (1 –2x )/ (1 + 5x ) = y ; x
X = (1 − 𝑦)⁄(5𝑦 + 2)
9)
𝑽
𝑹 = 𝑸𝟐 + 𝑳 ; v
V = Q2(R-L)
10)
𝑽
𝑹 = 𝑸𝟐 + 𝑳 ; Q
𝑉
Q=√
𝑅−𝐿
11)
𝟐𝑷𝑪
𝑹 = 𝑩(𝑵+𝟏) + 𝑪 ; N
N=
12)
2𝑃𝐶
𝐵(𝑅−𝐶)
𝑭 = 𝑷(𝟏 + 𝒓)𝒏 ; P
P = F/(1+r)n
13)
𝑭 = 𝑷(𝟏 + 𝒓)𝒏 ; r
𝑛
r= √
14)
𝐹
-1
𝑃
𝑭 = 𝑷(𝟏 + 𝒓)𝒏 ; n
𝐹
n = ln (𝑃) /ln(1 + 𝑟)
15)
𝑷=𝒂 [
W =
√
𝟒𝒘𝟐 −𝟑𝒍𝟑
𝟐𝒎−𝟑𝒘𝟐
]
𝟏⁄
𝟐
2𝑚𝑝2 +3𝑎2 𝑙 3
4𝑎2 +3𝑝2
;w
1.8 Linear Equations
Example – Solve following equations
1) 7x + 5 = 19
X=2
2) 5(x-1) – 2x = 1
X=2
3) (x – 1)/ (x + 1) = 0.5
X=3
4) 4(x+2) + 8 = 6x+ 2(x -2)
X=5
5)
𝐱+𝟐
𝟑
+
𝐱−𝟑
𝟒
X=5
6)
𝟏
𝐱
𝟐
+𝐱=6
X = 0.5
7)
𝟑𝟎𝟎+𝟐𝐱
𝟑𝟎+𝟓𝐱
X = 30
8)
𝐱+𝟓
𝐱−𝟓
𝟓
=𝟑
X = 20
9)
X = 10
=2
𝟓
=2𝟔
10) Solve
X = -0.183673469
11) Solve
X = 2.805883701
Examples:
1)
John has $20 in his bank. How much money does he need to buy a game that costs $60?
Answer = $40
2)
Alice spends 20% of her income to buy cosmetics. In July she spent $50 on cosmetics. What was her income in
July?
Answer = $250
3)
Alice’s age is two times of her brother Bob. Five years before her age was 3 times of her brother’s age. How older is
Alice now?
Answer = Years 20 Old Now.
4)
Peter spent 30% of his income on food in June. In July he spent 10% more than what he spent in July on food. His
total food bill for the two months totaled $630. What is his monthly income?
Answer = $900
5)
Paul spent 20% of his income on food in June. In July he spent 10% more than what he spent in July on food. His
income for July was increased by $500 from June’s amount. His total food bill for the two months totaled $840.
What is his monthly income now?
Answer = $680 +$500 = $1180
Graphs of Linear Functions
Exercises
Draw graphs of following functions.
1.
Y = 2x , Y= 2x + 1, y = 2x -3
2.
Y=3x -1 , y= 2x +2
3.
Y= x + 2 , y = -x -3
4.
Y= -2x + 1, y= x+ 4 , y=4x
5.
2x = y -1 , x = -y + 4
1.9 Quadratic Equations
Examples:
Solve following equations
a)
X2-5x+6 = 0
X= 3 , X= 2
b) 2X2-3x-6 = 0
X = 2.63746 , X = -1.13746
c)
-2X2-7x+12 = 0
X = 1.26 , X = -4.76
d) 15p2+17p-4=0
P = 0.2 , p = -4/3
e) X2-4x + 4 = 0
X = 2,
f)
X2-4x + 5 = 0
X = 2+i , X = 2 –I (Complex Roots)
Graphs of Quadratic Functions
Exercises:
Draw graphs of following functions and solve them graphically.
a)
X2-5x+6 = 0
X = 2 , X =3
b) 2X2-3x-6 = 0
X = 2.637, X = -1.1375
c)
-2X2-7x+12 = 0
X = 1.2604 , X = -4.7604
d) 15p2+17p-4=0
p = 0.2 , p = -(4/3)
e)
X2-4x + 4 = 0
X=2
f)
X2-4x + 5 = 0
X = 2+i , X = 2-i
(Complex Roots)
1.10 Simultaneous Linear Equations
Examples
a)
2X + Y = 5, 4X + Y = 9
X = 2, Y = 1
b) 3X + 2Y = 11, 4X – 4Y = -12
X = 1, Y = 4
c)
5X + 6Y = 28, 10X + 5Y = 35
X = 2, Y = 3
d) -4x + Y = 6, -2x + 2Y = 18
X =1, Y = 10
e)
3X = 5Y -5, 7Y =4X + 10
X = 15, Y = 10
Graphical solution to Simultaneous Linear Equations
a)
2X + Y = 5, 4X + Y = 9
X = 2, Y = 1
b) 3X + 2Y = 11, 4X – 4Y = -12
X = 1, Y = 4
c)
5X + 6Y = 28, 10X + 5Y = 35
X = 2, Y = 3
d) -4x + Y = 6, -2x + 2Y = 18
X =1, Y = 10
e)
3X = 5Y -5, 7Y =4X + 10
X = 15, Y = 10
More questions
1) Three apples and four oranges cost. 2.65$. One apple and five oranges cost 2.35$. Find the
price of an apple and the price of an orange.
An Apple = $0.65769
An Orange = $0.33846
2) Two mugs and four cups cost 20$. At the same price three mugs and five cups would
28$. Find the price of a mug and the price of a cup.
cost
A Mug = $6
A Cup = $2
3) The length of a rectangle is twice its width. The perimeter is 30. Find its dimensions
Width = 5
Length = 10
4) The difference of two numbers is 3, and the sum of three times the larger one and
twice the smaller one is 19. Find the two numbers.
Larger Number = 5
Small Number = 2
5) The sum of four times the first number and three times the second number is 15. The differ
ence of three times the first number and twice the second number is 7. Find the numbers.
First Number = 3
Second Number = 1
6) 9 pens and five pencils cost $3.2, and 7pens and 8 pencils cost $2.9. Find the unit price
for each pen and pencil.
A Pen = $0.3
A Pencil = $0.1
7) A solution containing 12% alcohol is to be mixed with a solution containing 4% alcohol to
make 20 gallons of solution containing 9% alcohol. How much of each solution should be
used?
12% - 12.5 gallons
4% - 7.5 gallons
8) If sum of two numbers be 45 and their difference being 15, find the numbers.
First Number = 30
Second Number = 15
9) If twice the son’s age in years is added to the father’s age, the sum is 70. But if the
father’s age is added to the son’s age, the sum is 95. Find the ages of father and son.
????????
10) 2 tables and 3 chairs together cost 2000 dollars whereas 3 tables and 2 chairs together
cost 2500 dollars. Find the cost of a table and a chair.
A Table = 700
A Chair = 200
11) Two runners start from the same point at the same time. They will be 4 miles apart at the
end of two hours if running in the same direction, and they will be 16 miles apart at the end
of one hour if running in opposite directions. Find their speeds.
1st Person`s Speed = 9 mph
2nd Person`s Speed = 7 mph
12) 3 bags and 4 pens together cost 257 dollars whereas 4 bags and 3 pens together cost
324 dollars. Find the cost of a bag and 10 pens
A Bag = $75
10 Pens = $80
13) The sum of the numerator and denominator of fractions is 12. If the denominator is increas
ed by 3, the fraction becomes 1/2. Find the fraction
Fraction =
5
7
14) There is a two digit number; the sum of the two digits is 5. When you reverse the order of
two digits, a new number is created. The difference of values of the new number and the old
number is 9. Find the two digit number (old one)
Old Number = 23
1.11 Inequalities
(a)
Exercise
5x+6 > 2x+18
(b)
Answer = x>4
(c)
5x-2 > (1-5x)/-3
Answer = ½<x
-2x+3>7x/-4
Answer = x<12
(d)
(4x/12)-5<3-(2x/3)
Answer = x<8