logb^c = clogb

64 = 2 x 2 x 2 x 2 x 2 x 2 = 2⁶

6log(x + 4) = log64

6log(x + 4) = log2⁶
6log(x + 4) = 6log2

Compare the left side and the right side.

x + 4 = 2
x = 2 - 4
x = -2

Even numbers are number divisible by 2 without a remainder.

Set of even integers between 11 and 97 = {12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,80,92,94,96}

The total number of elements in the set = 43

Alternate Solution

Find the interval between 97 and 11 and divide by 2

97 - 11 = 86

Even integers between 11 and 97 = $\frac{86}{2}$ = 43

The sequence is linear.

Common difference = $\frac{5}{8}$ - $\frac{1}{4}$ = 1 - $\frac{5}{8}$ = 1$\frac{3}{8}$ - 1 = $\frac{3}{8}$

U_n = a + (n - 1)d

Where n = n^th term,
a = first term
d = common difference

a = $\frac{1}{4}$

d = $\frac{3}{8}$

U₁₁ = $\frac{1}{4}$ + (11 - 1) x $\frac{3}{8}$

U₁₁ = $\frac{1}{4}$ + 10 x $\frac{3}{8}$

U₁₁ = $\frac{1}{4}$ + 5 x $\frac{3}{4}$

U₁₁ = $\frac{1}{4}$ + $\frac{15}{4}$

U₁₁ = $\frac{\mathrm{1\; +\; 15}}{4}$

U₁₁ = $\frac{16}{4}$

U₁₁ = 4

Ib = 1⁄2 (a+b)h

multiply through by 2

2 x Ib = 1⁄2(a + b)h

2b = 1(a + b)h

2b =(a + b)h

2b =ah + bh

Group like terms

factorize b out

Gives b(2 - h) = ah

Divide through by 2 - b to make b the subject

b(2 - h)⁄2-h = ah⁄2-h

b = ah⁄2-h

selling price = N6,900.00

Profit percent = 15%

percentage profit if he had sold if for N6,600.00

let C.P represent cost price

S.P represnt selling Price

% represent Profit percent

C.P = S.P - P

but put P = %P X C.P

put P = 15% x C.P

P = 0.15 C.P

Gives = C.P = S.P - ( 0.15 - C.P )

but S.P = 6,900.00

C.P = 6,900 - (0.15 C.P )

Group like terms

C.P + 0.15 C.P = 6,900.00

1.5 C.P⁄1.5 = 6,900.00⁄1.5

therefore cost price of the article is 6,000

If the owner of the article had sold it N6,600.000

Profit = Selling price - Cost Price

= 6,600 - 6000

N600.00

since profit is know

percent profit = P⁄C.P x 100

= 600⁄6000 x 100

10%

p = x - $\frac{1}{\mathrm{x}}$

Every number is divisible by 1

x = $\frac{\mathrm{x}}{1}$

p = $\frac{\mathrm{x}}{1}$ - $\frac{1}{\mathrm{x}}$

p = $\frac{\mathrm{x\; x\; x\; -\; 1}}{\mathrm{x}}$

p = $\frac{{\mathrm{x}}^2\mathrm{-\; 1}}{\mathrm{x}}$

Square both sides of the equation.

p² = ($\frac{{\mathrm{x}}^2\mathrm{-\; 1}}{\mathrm{x}}$)²

(x² - 1)² = (x² - 1) x (x² - 1)
(x² - 1)² = x²(x² - 1) - 1(x² - 1)
(x² - 1)² = x² x x² - 1 x x² - 1 xx² - 1 x -1
(x² - 1)² = x⁴ - x² - x² + 1
(x² - 1)² = x⁴ - 2x² + 1

p² = $\frac{{\mathrm{x}}^4\mathrm{-\; 2}{\mathrm{x}}^2\mathrm{+\; 1}}{{\mathrm{x}}^2}$

Divide each of the expressions at the right by the denominator (x²).

p² = $\frac{{\mathrm{x}}^4}{{\mathrm{x}}^2}$ - $\frac{2{\mathrm{x}}^2}{{\mathrm{x}}^2}$ + $\frac{1}{{\mathrm{x}}^2}$

p² = x² - 2 + $\frac{1}{{\mathrm{x}}^2}$

p² = x² + $\frac{1}{{\mathrm{x}}^2}$ - 2

But q = x² + $\frac{1}{{\mathrm{x}}^2}$

p² = q - 2

p² + 2 = q

diameter = 7cm

Area of cylinder = 165cm² ϖ = 22⁄7

since radius = 35

165 = 8(22⁄7)(3.5)(h) + 2(22⁄7)(3.5)²

165 = 22h + 44⁄7 (12.25)

165 =22h + 77

f=group like terms

165 - 77 = 22h

88 =22h

divide through by 22

h=4

Applying log_a^bc = c x log_a^b = clog_a^b

log₁₀^2y = y x log₁₀²

But log₁₀^2y = 1.8060

⇒ y x log₁₀² = 1.8060

But log₁₀² = 0.3010

Hence y x 0.3010 = 1.8060

Dividing both sides by 0.3010

⇒ y = 1.8060/0.3010 = 6

Geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, none-one number called the common ratio

Rules

For the sequence a₁, a₂, a₃, a₄ ...

Where a₁ = first term

Where a₂ = second term

Where a₃ = third term

Where a₄ = forth term

r = a₂ / a₁ = a₃ / a₂ = a₄ / a₃

Where r = common ratio

Thus the common ratio (r) is the same for all calculations

Hence 4/x = 16/4

4/x = 4

Multiplying both sides by x

x x 4/ x = 4 x x

4x = 4

Dividing both sides by 4

x = 1

16/4 = y/16

4 = y/16

Multiplying both sides by 16

16 x 4 = y/16 x 16

y = 64

⇒ x:y = 1:64

The angle adjacent to 110° is (180 - 110)° (Angles on a straight line sum up to 180°)

(x + 4y)° = The angle adjacent to 110° (Alternate angles are the same)

(x + 4y)° = (180 - 110)°
x + 4y = 70 ------ (equation 1)

(x + 4y)° = (2x + y)°(Vertically opposite angles are the same).

Since x + 4y = 70, 2x + y is also 70

2x + y = 70 ------ (equation 2)

Solve equation 1 and 2 simultaneously.

Get rid of x by multiplying equation 1 by 2 and subtracting equation 2 from equation 3.

x x 2 + 4y x 2 = 70 x 2
2x + 8y = 140 ------ (equation 3)

Equation 3 - Equation 2

7y = 70

Divide both sides by 7

y = $\frac{70}{7}$

y = 10

Subtitute y = 10 into equation 1

x + 4 x 10 = 70
x + 40 = 70
x = 70 - 40
x = 30

x + y = 30 + 10
x + y = 40

Since 3 can go into 6 and 9, split -6pqr into -3pqr and -3pqr

⇒ p²q² - 6pqr + 9r² = p²q² -3pqr -3pqr + 9r²

Now factorize

p²q² -3pqr -3pqr + 9r²

pq(pq - 3r) -3r(pq-3r)

(pq-3r)(qp-3r)

since pq-3r = pq-3r, the multiplication makes it square

(pq-3r)²

The equation above is similar to the general factorization rule of (a-b)²

(a-b)² = a² - 2ab + b²

Thus the first term square then the sign (either + or -) and 2 times the first and second terms + the second term square

Similarly (a+b)² = a² +2ab + b²

The first term for the above equation is pq and the second term is 3r and the sign is -

You can apply the (a-b)² = a²-2ab+b² to expand the (pq-3r)², you will realize you will get the expression you were asked to factorize

Prime number: A number that is divisible only by itself and 1

NB: 1 is not a prime number hence the option is not A

Integer: A whole number (not a fraction) that can be positive, negative or zero. e.g ...,-5,-4,-3,-2,-1,0,1,2,3...

P is a set of positive (No negative) integers and less than 7, hence the correct answer is B

Rational Number: A number such as 5/8 that can be expressed as quotient or fraction p/q of two integers.

Real Number: Any positive or negative number. This includes all integers and all rational and irrational numbers.

Equation of a line y = mx + c

If point (1,-5) is given and gradient 3⁄4

y = -5

x = 1

y = mx + c

Gives : -5 = 3⁄4 (1) + c

-5 = 3⁄4 + c

Group like terms

-5-3⁄4 = c

-5⁄1 -3⁄4 = c

-20-3⁄4 = c

-23⁄4

y = 3⁄4x - 23⁄4

multipy through by 4

4 x y = 4 X 3⁄4x - 23⁄4 x 4

4y =3x - 23

0 =3x - 23 - 4y = 0

3x - 4y - 23 = 0

3p = 4q ...... 1eqn

9p = 8p - 12 .......2eqn

multiply equation 1 by 3

3 x 3p = 3 x 4q

9p = 12q ..... 3eqn

put 3eqn into 2eqn

12q = 8q - 12

12q - 8q = -12

4q⁄4 =-12⁄4

q = 3

put q = -3 into 1

3p = 4(-3)

3P⁄3 = -12⁄3

Pq = (-4)(-3)

12

A = $\frac{1}{2}$b(a + b)

Get rid of the fraction by multiplying both sides by 2

2 x A = 1 x b(a + b)
2A = b(a + b)

Divide both sides by b

$\frac{\mathrm{2A}}{\mathrm{b}}$ = a + b

$\frac{\mathrm{2A}}{\mathrm{b}}$ - b = a

Significant figure rules

* Non-zero digits are always significant (1,2,3,4,5,6,7,8,9)

* Any zeros between two significant digits are significant

* A final zero or trailing zeros in the decimal portion ONLY are significant e.g .500 or .63200 the zeros are significant

.006 or .000968 the zeros are NOT significant

Applying the above rules to 9453 x ^-6

Change it to fraction

Since the exponent is negative, you have to move to the left 6 times and place the decimal point there with leading zeros before the digits 9453

9453 x ^-6 = 0.009453

The significant figures are 9,4,5 and 3. Since it is 3 significant figures, it ends on 5

Thus 0.00945 is the answer

NB: The leading zeros before 9453 are not significant as explained by the above rule. Hence they were not counted as part of the significant digits.

let represent initial term d represent difference

a = 6

d = common diffence

To find common diffence is the second term minus the first time that's current minus intial term

d = p - 6 = 14 -p

p - 6 = 14 - p

Group like terms

p + p = 14 + 6

2p⁄2 = 20⁄2

p = 10

∆OST = OTS (Isocicoles)

OTS = 50°

considering ∆OST

50 + 50 +∆SOT = 180

100 + ∆SOT =180

Group like terms

∆SOT = 180

∆SOT = 80°

∆SOT = ∆ROP = 80° (vertical opposite angles are equal

80 + OPQ = 180

OPQ =180 - 80

OPQ = 100

Perimeter is the summation of all the dimension of the boundaries of a shape.

Let w = the width of the rectangular lawn.

Length is 3 cm longer than the width → Length = w + 3

Perimeter = 2(w + 3) + 2w

But perimeter of the rectangular lawn = 42

2(w + 3) + 2w = 42
2 x w + 2 x 3 + 2w = 42
2w + 6 + 2w = 42
4w = 42 - 6
4w = 36

Divide both sides by 4

w = $\frac{36}{4}$

w = 9 cm

C

NB: Change the mixed fractions to improper fractions

Explanation:

Multiply the whole number(A) by the denominator(c) and add the numerator(b) to the result(Axc). Divide everything by the denominator(c).

⇒ 2¹⁄₂ = ^{(2x2 + 1)}⁄₂ = ⁵⁄₂
⇒ 1¹⁄₂ = ^{(1x2 + 1)}⁄₂ = ³⁄₂

Concepts

a = a/₁
b = b/₁

Explanation:

Change the ÷ to x (multiplication). To do so:
Reciprocate the fraction at the right side of the ÷
Reciprocate means the top goes to the down and the down comes to the top

Explanation:

The numerators multiply each other over the denominators multiplying each other.

Thus the products of the numerators over the products of the denominators

Product means multiplication

Applying all the above concepts

Answer is A

3 x 9 ^{1 + x} = 27^-x

Concept

A^b x A^c = A^b+c

Thus if the bases are the same, we can add the indexes (superscripts)

3 can go into the other numbers(9,27) hence we change then to have base 3

9 = 3¹x 3¹ = 3^{1 + 1}=3²

27 = 3¹x 3¹ x 3¹ =3¹⁺¹⁺¹ = 3³

⇒ 3 x 3^{2(1 + x)} = 3^3(-x)

But 3 = 3¹, thus every number is raised to the power 1
⇒ 3¹ x 3^{2(1 + x)} = 3^3(-x)

Applying A^b x A^c = A^{b + c}

3 ¹ x 3^{2(1+ x)} = 3^{1 + 2(1 + x )}

3^{1+2(1+ x)} = 3^3(-x)

Concept
If A^b = A^c ⇒ b = c

Thus if the base at the left side of the equation, is the same as the right side of the equation, then the powers are the same

∴ 1 + 2(1 + x ) = 3(-x)

Concept

a(b + c ) = a x b + a x c

a(b - c ) = a x b - a x c

Thus the number outside the bracket multiplies each of the number in the bracket and the sign ( + or - ) is(are) maintained

-x = -1 x x

3(-x) = 3 x (-1) x x = -3x

1 + 2(1 + x ) = 1 + 2 x 1 + 2 x x

                     = 1 + 2 + 2x

                     = 3 + 2x

1 + 2(1+x) = 3(-x)

⇒ 3 + 2x = -3x

Grouping the x at the left and the rest at the right

Concept
If the sign is positive(+) and crosses to the other side of the equation the sign changes to negative ( - ) and the vice versa

2x + 3x = -3

5x = -3

Concept
Divide both sides by the number multiplying the number or variable (x)

^5x⁄₅ = ^-3⁄₅

x = -3⁄5

∅ = 52

opposite ∣YZ∣ = 32

Adjacement ∣XZ∣ = ?

since opposite is known and we are ask to find adjacent (∣XZ∣)

tan∅ formular will be appiead.

which says tan∅ = opposite⁄Adjacement ∣YZ∣⁄∣XZ∣

tan(32) = 32⁄∣XZ∣

cross multiply

tan32⁄1 = 32⁄∣XZ∣

∣XZ⁄ tan(52) =32

On your scientific calculator tan(52) = 0.52

∣XZ (∅.28) = 32

making ∣XZ∣ the subject divide through by 0.53

∣XZ⁄0.53 = 32⁄1.28

∣XZ = 32⁄1.28

∣XZ∣ = 25cm

The sum of the ratio represents the entire money shared.

Sum of ratios = 3 + 2
Sum of ratios = 5

Rose's ratio is 2.

If 5 = GH₵ 875.08
2 = $\frac{\mathrm{GH₵\; 875.08\; x\; 2}}{5}$ = GH₵ 350.032

Rose's share = GH₵ 350.032

Sum of Efua's and Ama's ratios = 1 + 2
Sum of Efua's and Ama's ratios = 3

Ama's ratio = 2

The sum of ratios represents the entire money both shared.

If 3 = GH₵ 350.032
2 = $\frac{\mathrm{GH₵\; 350.032\; x\; 2}}{3}$ = GH₵ 233.3546666666667

Ama's share = GH₵ 233.35

Method I

4x + 6y = 5 --- eqn (1)
2x + 4y = 3 --- eqn (2)

Multiply equation 2 by 2 to eliminate x

2x x 2 + 4y x 2 = 3 x 2
4x + 8y = 6 --- eqn (3)

Equation 3 - Equation 1

2y = 1

Divide both sides by 2

y = $\frac{1}{2}$

Subtitute y = $\frac{1}{2}$ in equation 1

4x + 6 x $\frac{1}{2}$ = 5

4x + 3 = 5
4x = 5 - 3
4x = 2

Divide both sides by 4

x = $\frac{2}{4}$

x = $\frac{1}{2}$

x + 2y = $\frac{1}{2}$ + 2 x $\frac{1}{2}$

x + 2y = $\frac{\mathrm{1\; +\; 2}}{2}$

x + 2y = $\frac{3}{2}$

x + 2y = 1$\frac{1}{2}$

Method II

Express 2x + 4y = 3 to be in the form (x + 2y) and make (x + 2y) the subject

2x + 4y = 3

Factorize 2 out.

2(x + 2y) = 3

Divide both sides by 2 to make (x + 2y) the subject.

(x + 2y) = $\frac{3}{2}$

Change the improper fraction to mixed fraction.

(x + 2y) = 1$\frac{1}{2}$

(3 + x)(1 - x) > 9 - x²

Expand the bracket

3(1 - x) + x(1 - x) > 9 - x²

3 - 3x + x - x² > 9 - x²

3 - 2x - x² > 9 - x²

- x² cancels each other at both sides of the equation.

3 - 2x > 9
- 2x > 9 - 3
- 2x > 6

Divide both sides by - 2. When dividing by a negative, the sign changes direction.

x < $\frac{6}{-2}$

x < -3

length of an arc = ∅⁄360 X 2Πr

Angle (Π) = 72

Π = 22⁄7

length of arc = 72;⁄360 x 2 22;⁄7 x 3.5

= 4.4

diffence of two squards

(27.63 - 12.37)(27.63 + 12.37)

610.4

Run it to three dicimal place

since 4 is less than 5 you maintian it

Substitute y = x² + 4x - 6 into x² + 4x - 7 = 0

-7 = -6 - 1

x² + 4x - 7 = 0
x² + 4x - 6 - 1 = 0

But x² + 4x - 6 = y

y - 1 = 0
y = 0 + 1
y = 1

Concept

Sum of exterior angles of any polygon = 360°

61° + 76° + (x + 10)° + (2x + 45)° = 360°
61 + 76 + x + 10 + 2x + 45 = 360
3x + 192 = 360
3x = 360 - 192
3x = 168

Divide both sides by 3

x = $\frac{168}{3}$

x = 56

$\frac{\mathrm{x\; +\; y}}{5}$ = $\frac{\mathrm{x\; -\; y}}{3}$

Cross multiply. The numerator at the left side multiplies the denominator at the right then equal to then the numerator at the right also multiplies the denominator at the left.

3 x (x + y) = 5 x (x - y)
3x + 3y = 5x - 5y
3y + 5y = 5x - 3x
8y = 2x

Divide both sides by y

8 = $\frac{\mathrm{2x}}{\mathrm{y}}$

Divide both sides by 2

$\frac{8}{2}$ = $\frac{\mathrm{x}}{\mathrm{y}}$

Division is :

8:2 = x:y

2 can divide itself once and 8, 4 times.

4:1 = x:y

Inscribed angles subtended by the same are equal.

∠PSQ = ∠PRQ = 61^o

Sum of angles in are triangle is 180^o

∠QPS + ∠PSQ + ∠PQS = 180^o

∠QPS + 61 + 32 = 180

∠QPS + 93 = 180

∠QPS = 180 - 93

∠QPS = 87^o

(25⁄100)^y = 32

(1⁄4)^y 32

log(1⁄4)^y = log32

ylog 0.25 = log32

y = log32⁄log0.25

y = -2.5

y = -25⁄10

y = -5⁄2

Change the mixed fraction to an improper fraction.

Explanation:
Multiply the whole number(A) by the denominator(c) and add the numerator(b). Divide everything by the denominator(c).

⇒ 1¹⁄₁₅ = ^(1x15+1)⁄₁₅ = ¹⁶⁄₁₅

⇒ t - ⁹⁄₅ = -¹⁶⁄₁₅

Get ride of the denominators by multiplying both sides by the least common multiple(LCM)
The denominators are 5 and 15,5 can go into 15, hence the LCM is 15

tx15 -15x⁹⁄₅ = 15x-¹⁶⁄₁₅
15t - 27 = -16
15t = -16+27
15t = 11
^15t⁄₁₅ = ¹¹⁄₁₅
t = ¹¹⁄₁₅

Since the angle of the polygon are five

It means the polygon is pentagon

interior angles of polygon = n-2x180⁄n

number of side (n) = 5

Interior angles of pentagon = (5-2)X180⁄5

=3X180⁄5

=540⁄5

108

since 180 is the interior

3x which is the smalllest angle

3x⁄3 = 108⁄3

x =60°

the smallest angle is 60°

Number of Boys = 5

Number of Girls = 4

Total pupils = 8 + 4

To get the probability of boys out of lift is the number of boys over the total number of pupils

P(a boys steps out of the lift) = 8⁄12

2⁄3

y ∝ ¹⁄_x²

NB: ∝ is the symbol for proportional
inversely proportional is 1 over the expression
directly proportional is just the expression (It would had been y ∝ x²)

a constant(=k) is introduced in place of the ∝
y = kx¹⁄_x²
y = ^k⁄_x²

Use the known values to calculate the constant(k) to get the general expression

y = 100 and x = 3
⇒ 100 = ^k⁄_3²
100 = ^k⁄₉
100x9 = k
⇒ k = 900
∴ y = ⁹⁰⁰⁄_x²

when y = 25
⇒ 25 = ⁹⁰⁰⁄_x²
25 x x² = 900
x² = ⁹⁰⁰⁄₂₅
x² = 36
√x² = √36
x = 6

NB: The square root of a squared number or variable is equal to the number or variable

Diagram Illustration

Area = 49 cm²

a = 6cm, b = 7cm

⇒ 49 cm² = 1/2 x (6cm + 7 cm) x h

49cm² = 1/2 x (13cm) x h

Multiply both sides by 2

2 x 49cm² = 1 x (13cm) x h

Divide both sides by 13cm

98 cm² / 13 cm = 1 x h

⇒ h = 98/13 = 7.53846 = 7.5 (One decimal place)

Let label the part A,B and C

to find the ladder we use pythagories theorem

which says ∣AB∣²+∣BC∣² = ∣AC∣²

∣AB∣ =6m,

∣B∣=8m,

∣AB∣²+∣BC∣² = ∣AC∣²

6²+8² = ∣AC∣²

36 + 64 = ∣AC∣²

100 ∣AC∣²

Root both side

√100 = √∣AC∣²

√100 = AC

10 = AC

The ladder is 10 m long

$\frac{\mathrm{9\; -}{\mathrm{x}}^2}{\mathrm{3x\; -}{\mathrm{x}}^2}$

9 - x² is a difference of two square.

a² - b² = (a+b)(a-b)

9 = 3 x 3 = 3²

9 - x² = 3² - x²

3² - x² = (3+x)(3-x)

$\frac{\mathrm{9\; -}{\mathrm{x}}^2}{\mathrm{3x\; -}{\mathrm{x}}^2}$ = $\frac{\mathrm{(3+x)(3-x)}}{\mathrm{3x\; -}{\mathrm{x}}^2}$

Factorize x out in the denominator.

$\frac{\mathrm{9\; -}{\mathrm{x}}^2}{\mathrm{3x\; -}{\mathrm{x}}^2}$ = $\frac{\mathrm{(3+x)(3-x)}}{\mathrm{x(3-x)}}$

(3-x) cancels each other

$\frac{\mathrm{9\; -}{\mathrm{x}}^2}{\mathrm{3x\; -}{\mathrm{x}}^2}$ = $\frac{\mathrm{3+x}}{\mathrm{x}}$

The area of the trapezium is the summation of the area of the triangle and the rectangle.

Area of triangle = $\frac{1}{2}$ x base x height

Area of a rectangle = Length x Breadth

Base of the triangle = 24 - 16 = 8

Height of the triangle = x
Breadth of rectangle = x

300 = $\frac{1}{2}$ x 8 x x + 16 x x

2 divides itself once and 8, 4 times.

300 = 4x + 16x
300 = 20x

Divide both sides by 20

x = $\frac{300}{20}$ = 15

P ∝ $\frac{{\mathrm{v}}^2}{\mathrm{x}}$

Replace x by vt

P ∝ $\frac{{\mathrm{v}}^2}{\mathrm{vt}}$

v² = v x v

P ∝ $\frac{\mathrm{v\; x\; v}}{\mathrm{vt}}$

The v cancel each other.

P ∝ $\frac{\mathrm{v}}{\mathrm{t}}$

Change the mixed fraction to improper fraction

Rule

If α and β are the two roots of a quadratic equation, then the formula to construct the quadratic equation is:

x² - (α + β)x + αβ = 0

Thus, x² - (sum of roots)x + product of roots = 0

Since

and

are the roots

⇒ x² - (-1/2 + 3/2)x + (-1/2 x 3/2) = 0

-1/2 + 3/2 = (-1+3)/2 = 2/2 = 1

Rule

-1/2 x 3/2 = (-1x3)/(2x2) = -3/4

⇒ x² - (1)x + (-3/4) = 0

x² - x - 3/4 = 0

NB: + - = - ⇒ + - 3/4 = -3/4

Multiply both sides by 4

4x - 4x - 3 = 0

Length of an arc = 22 cm

raduis =

π = 22⁄7

angle (∅)= ?

formular for length of an arc = ∅⁄360 x 2πr

since length of an arc =22 cm

r = 15 and π = 22⁄7

22 = π⁄360 X 2(22)⁄7 X 15

22 = π⁄360 X 44⁄7

22 = π⁄360 X 94.286

22 = 94.2860⁄360

22⁄0.262 = 0.2620⁄0.262

π =22⁄0.262

π = 84

The angle is 84 °

Change the mixed fractions to improper fractions

Rule

Rules

Thus, reciprocate/switch the numerator and denominator at the right and change the division to multiplication

Thus, the numerators multiply each other and the denominators multiply each other

⇒

Multiply both sides by 96

96 x 9/16 = x

16 goes into 96, 6 times

6 x 9 = x

x = 54

y ∝ 1 / x

NB: Inversely = 1 / the relation

Varies directly = Relation 1 ∝ Relation 2

Find the constant of the relation

Let k = the constant

y = k x 1/x

When y = 6, x is 3

⇒ 6 = k/3

Multiplying both sides by 3

⇒ k = 6 x 3 = 18

Hence y = 18/x

When x = 9

⇒ y = 18 / 9 = 2

Answer is D

Use a venn diagram to illustrate

In a class of 39 student ⇒ n(u)=39
let F represent students who offer Fante ⇒ n(F)= 25
Let T represent students who offer Twi ⇒ n(T)= 19
Five student do not offer any of the two languages ⇒ 5 are outside the sets F and T
Let n represent number of student who offer both languages

NB: You need to substract the number of the students who offer both languaes from each of the languages to get the students who offer only that languages.

When we add all the parts in the venn diagram we are to get the total number of students n(u) = 39

(25 - n )+ n + (19 - n) + 5 =39
25 - n + n + 19 -n + 5 = 39
-n + 49 = 39
-n = 39 - 49
-n = -10
^-n⁄_-1 = ^-10⁄ _-1
n = 10
Only Twi = 19-n
⇒ Only Twi = 19 - 10 = 9

4b² – ab + (a + 9b)² – a²

step 1: Expand (a + 9b)² to give (a + 9b) (a + 9b)

4b² – ab + (a + 9b) (a + 9b) – a²

step 2: Expand the bracket:

=4b² – ab + a² + 18ab + 81b² – a²

step 3: Group like terms:

= 4b² + 81b² – ab + 18ab + a² - a²

= 4b² + 81b² – ab + 18ab

= 85b² + 17ab

step 4: Factorize 17b out:

= 17b(5b + a)

Since Z = 8, x = 4, y = 1/8, substitute them in place of the variables.

Notes

Arithmetic Progression

Is a sequence of numbers such that the difference between the consecutive terms is constant.

E.g The sequence 1,3,5,7,9.. is an arithmetic progression with a common difference of 2

a_n = a₁ + (n - 1)d

a_n = The value of the sequence number of on the n^th position.

a₁ = the first number in the sequence

d = the common difference

d = a_{n + 1} - a_n (The difference between two consecutive sequence numbers)

d is the same for all consecutive numbers used for the calculator.

Thus a_{n + 2} - a_{n + 1} = a_{n + 1} - a_n

For instance, the common difference for the sequence a₁, a₂, a₃, a₄, a₅, a₆ .... will be calculated as:
a₂ - a₁ = a₃ - a₂ = a₅ - a₄ = a₆ - a₅

NB: The common difference (d) is the higher sequence number value - lower previos sequence number.

For the sequence 1,3,5,7,9

⇒ d = 3 - 1 = 5 - 3 = 7 - 5 = 9 - 7 = 2

Summation of Arithmetic Progression

S_n = ⁿ/₂ (a₁ + a_n)

S_n = ⁿ/₂ [2a₁ + (n - 1)d]

If the last number is know the summation can also be writen as

S_n = ⁿ/₂ [a₁ + l]

where l = last term

From the above diagram, the length of the arc becomes the circumference of the base of the cone.

radius of the circle = 22cm
θ = 60^o

NB: Unit of area = unit square (e.g cm²,m²) and unit of volume = unit cubic (e.g cm³, m³)

class="remain-inline"> the subject.

Cross multiple:

5(5y – x) = 1(8y + 3X)

Expand the bracket:

25Y – 5X = 8Y + 3X

Group like terms and add:

25Y- 8Y = 3X + 5X

17Y = 8X

Divide through by 8.

Divide through by Y to make X the numerator