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

For the equation to be undefined, the denominator must be 0.

2x² - 13x + 15 = 0

Solve for the value of x

2 x 15 = 30

Two numbers when multiplied the result will be 30 and when added the result will be - 13.

The numbers are -10 and - 3.

Express - 13x as -10x - 3x

2x² - 10x - 3x + 15 = 0

Factorize out

2x(x - 5)-3(x-5) = 0
(x - 5)(2x-3) = 0

(x - 5) = 0 or (2x-3) = 0

x - 5 = 0
x = 0 + 5
x = 5

2x-3 = 0
2x = 3

Divide both sides of the equation by 2

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

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

Bearing of Q from P = 90^o + 90^o + 65^o

Bearing of Q from P = 245^o

Length of arc = $\frac{\mathrm{\theta }}{360}$ x 2πr

Where r = radius of the circle.

Length of the arc = 50 cm
θ = 75

π = $\frac{22}{7}$

$\frac{75}{360}$ x 2 x $\frac{22}{7}$ x r = 50

$\frac{\mathrm{75\; x\; 2\; x\; 22}}{\mathrm{360\; x\; 7}}$ x r = 50

$\frac{3300}{2520}$ x r = 50

Get rid of the fraction by multiplying both sides of the equation by the denominator (2520)

3300 x r = 50 x 2520
3300 x r = 126000

Divide both sides of the equation by 3300

r = $\frac{126000}{3300}$

r = 38.1818 ≈ 38.2 cm (three significant figures)

(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

Probability = $\frac{\mathrm{Number\; of\; possible\; selection}}{\mathrm{Number\; of\; samples}}$

Probability of is + Probability of not = 1

You can find the probability of picking a red ball and subtract it from 1 to get the not or find the total number of balls which are not red and divide by the total balls.

Total balls = 32 + 28 + 20
Total balls = 80

Method I

Number of red balls = 20

Probability of picking red ball = $\frac{20}{80}$

Probability of picking red ball = $\frac{1}{4}$

Probability of not red ball = 1 - $\frac{1}{4}$

Every number is being divided by 1.

1 = $\frac{1}{1}$

Probability of not red ball = $\frac{1}{1}$ - $\frac{1}{4}$

Probability of not red ball = $\frac{\mathrm{1x\; 4\; -\; 1}}{4}$

Probability of not red ball = $\frac{\mathrm{4\; -\; 1}}{4}$

Probability of not red ball = $\frac{3}{4}$

Method II

Balls which are not red = 32 + 28
Balls which are not red = 60

Probability of not red ball = $\frac{60}{80}$

Probability of not red ball = $\frac{3}{4}$

$\frac{1}{2}$(x-1)-$\frac{1}{3}$($\frac{1}{2}$x-1)

Expand the brackets

$\frac{1}{2}$(x-1)-$\frac{1}{3}$($\frac{1}{2}$x-1) = $\frac{\mathrm{x}}{2}$ - $\frac{1}{2}$ - $\frac{\mathrm{x}}{6}$ + $\frac{1}{3}$

Note: - x - = +

-$\frac{1}{3}$ x - 1 = $\frac{1}{3}$

The Least Common Multiples (L.C.M) of the denominators (2,3 and 6) is 6

$\frac{1}{2}$(x-1)-$\frac{1}{3}$($\frac{1}{2}$x-1) = $\frac{\mathrm{3\; x\; x\; -\; 3\; x\; 1\; -\; 1\; x\; x\; +\; 2\; x\; 1}}{6}$

$\frac{1}{2}$(x-1)-$\frac{1}{3}$($\frac{1}{2}$x-1) = $\frac{\mathrm{3x\; -\; 3\; -\; x\; +\; 2}}{6}$

$\frac{1}{2}$(x-1)-$\frac{1}{3}$($\frac{1}{2}$x-1) = $\frac{\mathrm{3x\; -\; x\; -\; 3\; +\; 2}}{6}$

$\frac{1}{2}$(x-1)-$\frac{1}{3}$($\frac{1}{2}$x-1) = $\frac{\mathrm{2x\; -\; 1}}{6}$

$\frac{1}{2}$(x-1)-$\frac{1}{3}$($\frac{1}{2}$x-1) = $\frac{1}{6}$(2x - 1)

Average = $\frac{\mathrm{Sum\; of\; items}}{\mathrm{Number\; of\; items}}$

Number of items = 3 + 9
Number of items = 12

Average cost = $\frac{\mathrm{Sum\; of\; cost}}{12}$

Sum of cost = 3 x 0.4 + 9 x 0.2
Sum of cost = 1.2 + 1.8
Sum of cost = GH₵3

Average cost = $\frac{\mathrm{GH₵3}}{12}$

Average cost = GH₵ 0.25

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$\frac{1}{2}$

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

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

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

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 ∝ $\frac{\mathrm{v}}{\mathrm{t}}$

$\frac{9}{28}$ - $\frac{2}{7}$ = $\frac{\mathrm{1\; x\; 9\; -\; 4\; x\; 2}}{28}$

$\frac{9}{28}$ - $\frac{2}{7}$ = $\frac{\mathrm{9\; -\; 8}}{28}$

$\frac{9}{28}$ - $\frac{2}{7}$ = $\frac{1}{28}$

$\frac{9}{28}$ - $\frac{2}{7}$ = 0.0357142857142857 ≈ 0.036 (Two significant figures)

Significant figures

Significant figures (or significant digits) are the number of digits important to determine the accuracy and precision of measurement, such as length, mass, or volume.

Significant digits in math convey the value of a number with accuracy. They are considered substantial figures that contribute to the precision of a number.

Rules to determine the number of significant figures in a number

1. Leading zeros are not significant
2. Non-zeros are significant
3. Zeros in between non-zero digits are significant
4. Trailing zeros to the right of the decimal point are significant

Examples

How to Round Off Significant Figures

To round off significant figures, we have to omit one or more digits from the right side of the number until we reach the number of significant digits that we want to round it off to.

First, we have to look at the digit on the right end of the number (to the right of the digit we want to round it off to).

If the digit is lower than 5, the number is rounded off to the lower number.

If the digit is greater than or equal to 5, the number is rounded up to the higher number.

If the number to be rounded off is a whole number, the remaining significant figures are replaced by 0.

Rounding off 0.0357142857142857

The significant figures are 3,5,7,1,4,2,8,5,7,1,4,2,8,5,7.

The second significant digit is 5 and the digit next to it (7) is greater than 5 hence we have to round up the 5 to 6 by simply adding 1 to the 5.

0.0357142857142857 = 0.036 (Two significant figures).

Vertically opposite angles are the same.

Sum of angles in a triangle is 180^o

3y + 2y + 3y + 12 = 180
8y = 180 - 12
8y = 168

Divide both sides by 8

y = $\frac{168}{8}$

y = 21^o

If you convert each of them to base 10, you should get the same value since they are equal.

2 x x¹ + 3 x x⁰ = 1 x 2³ + 1 x 2² + 1 x 2¹ + 1 x 2⁰

a⁰ = 1 (Every number raised to the power 0 is 1

a¹ = a (Every number to the power 1 is the same number)

2 x x + 3 x 1 = 1 x 8 + 1 x 4 + 1 x 2 + 1 x 1
2x + 3 = 8 + 4 + 2 + 1
2x = 8 + 4 + 2 + 1 - 3
2x = 12

Divide both sides by 2

x = $\frac{12}{2}$

x = 6

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

9 = 3 x 3 = 3²
27 = 3 x 3 x 3 = 3³

a^-b = $\frac{1}{{\mathrm{a}}^{\mathrm{b}}}$

$\frac{1}{27}$ = $\frac{1}{3^3}$ = 3^-3

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

Since the bases are the same, the powers are also the same.

2(y + 1) = -3(y - 2)
2y + 2 = -3y + 6

Note: - x - = +
-3 x -2 = + 6

2y + 2 = -3y + 6
2y + 3y = 6 - 2
5y = 4

Divide both sides by 5

y = $\frac{4}{5}$

Discount % = $\frac{\mathrm{Discount}}{\mathrm{Actual\; Price}}$ x 100%

Discount = Actual Price - Selling Price

Actual price for the 5 tins = GH₵ 15.00 x 5
Actual price for the 5 tins = GH₵ 75.00

Discount = GH₵ 75.00 - GH₵ 66.00
Discount = GH₵ 9.00

Discount % = $\frac{\mathrm{GH₵\; 9.00}}{\mathrm{GH₵\; 75.00}}$ x 100%

Discount % = 12%

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

Exterior angle = $\frac{360}{\mathrm{No\; of\; sides}}$

Let x = exterior angle

Interior angle = 2 x the exterior angle
Interior angle = 2 x x

Interior angle + Exterior angle = 180^o

2x + x = 180^o
3x = 180^o

Divide both sides by 3

x = $\frac{180}{3}$

x = 60^o

The exterior angle = 60^o

60 = $\frac{360}{\mathrm{No\; of\; sides}}$

60 x No of sides = 360

Divide both sides by 60

No of sides = $\frac{360}{60}$

No of sides = 6

Let x = Fafa's present age.

Metuh is nine times as old as Fafa → Metuh's present age = 9 x Fafa's present age.

Metuh's present age = 9 x x

In six year's time, Metuh's age will be 9x + 6

In six year's time, Fafa's age will be x + 6

In six years time, he will be five times as old as Fafa

9x + 6 = 5 (x + 6)
9x + 6 = 5x + 30
9x - 5x = 30 - 6
4x = 24

Divide both sides by 4

x = $\frac{24}{4}$

x = 6

Fafa's present age is 6 years

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

53,000,000 = 53 x 10⁶
0.002 = 2 x 10^-3
0.0004 = 4 x 10^-4

53,000,000 x 0.002 = 53 x 10⁶ x 2 x 10^-3

Law of multiplication of indices

a^m x aⁿ = a^{m + m}

53 x 10⁶ x 2 x 10^-3 = 53 x 2 x 10^6-3
53 x 10⁶ x 2 x 10^-3 = 106 x 10³

$\frac{\mathrm{53,000,000\; x\; 0.002}}{0.0004}$ = $\frac{\mathrm{106\; x}{10}^3}{\mathrm{4\; x}{10}^{-4}}$

Law of division of indices

$\frac{{\mathrm{a}}^{\mathrm{m}}}{{\mathrm{a}}^{\mathrm{n}}}$ = a^{m - n}

$\frac{\mathrm{106\; x}{10}^3}{\mathrm{4\; x}{10}^{-4}}$ = 26.5 x 10^3-(-4)

$\frac{\mathrm{106\; x}{10}^3}{\mathrm{4\; x}{10}^{-4}}$ = 26.5 x 10³⁺⁴

$\frac{\mathrm{106\; x}{10}^3}{\mathrm{4\; x}{10}^{-4}}$ = 26.5 x 10⁷

$\frac{\mathrm{106\; x}{10}^3}{\mathrm{4\; x}{10}^{-4}}$ = 2.65 x 10¹ x 10⁷

$\frac{\mathrm{106\; x}{10}^3}{\mathrm{4\; x}{10}^{-4}}$ = 2.65 x 10¹⁺⁷

$\frac{\mathrm{106\; x}{10}^3}{\mathrm{4\; x}{10}^{-4}}$ = 2.65 x 10⁸

Note: For standard form, the decimal point must be after the first non-zero digit.

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

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

Concept

Parallel lines have the same gradient.

The general equation of a line is given as y = mx + b
m = slope/gradient
b = y - intercept

Determine the gradient/slope in the given equation.

2x + 5y = -10

Make y the subject

5y = -2x - 10

Divide both sides by 5

y = -$\frac{2}{5}$x - 2

Compare the equation with the general equation for a line (y = mx + b).

Gradient/slope = -$\frac{2}{5}$

Since the line is parallel to the line with gradient -$\frac{2}{5}$, that line has the same gradient.

Gradient = $\frac{\mathrm{Change\; in\; Y}}{\mathrm{Change\; in\; X}}$ = $\frac{{\mathrm{Y}}_2-{\mathrm{Y}}_1}{{\mathrm{X}}_2-{\mathrm{X}}_1}$

Calculate the gradient of the line using the given point (4,1) and (x,y), equate it to the known gradient and simplify.

Cross multiply.

5(y - 1) = -2(x - 4)
5y - 5 = -2x + 8
5y + 2x = 8 + 5
2x + 5y = 13

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

Modulo is a math operation that finds the remainder when one integer is divided by another.

x = 7 because 17 = 2 x 7 + 3

Tan(θ) = $\frac{\mathrm{Opposite}}{\mathrm{Adjacent}}$

Tan(θ) = $\frac{2}{0.5}$

θ = Tan^-1($\frac{2}{0.5}$)

θ = 76° (nearest degree)

Note: M ⇒ N means M implies N.

Method I

The angle for the entire circle is 360°

If 140° = 40 cm²
360° = $\frac{\mathrm{40\; x\; 360}}{140}$ = 102.86 cm²

Method II

Area of a sector = $\frac{\mathrm{\theta }}{360}$ x πr²

But πr² = area of the circle

Area of a sector = $\frac{\mathrm{\theta }}{360}$ x area of circle

Area of a sector = 40 cm²
θ = 140

40 = $\frac{140}{360}$ x area of circle

Make area of circle the subject.

Multiply both sides by 360

40 x 360 = 140 x area of circle

Divide both sides by 140

$\frac{\mathrm{40\; x\; 360}}{140}$ = area of circle

Area of circle = 102.86 cm²

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

Probability = $\frac{\mathrm{Number\; of\; possible\; selection}}{\mathrm{Number\; of\; samples}}$

Numbers in a die rolled once = {1,2,3,4,5,6}

Numbers less than 3 = {1,2}

Number of possible selection = 2
Number of samples = 6

Probability of less than 3 = $\frac{2}{6}$ = $\frac{1}{3}$

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

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

Let M represent Mathematics
B represent Biology
x represent students that take only Biology

13 + 5 + x + 8 = 30
26 + x = 30
x = 30 - 26
x = 4

Number of students that take Biology = 5 + 4 = 9

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

48 = 3 x 16
6 = 3 x 2

$\sqrt{\mathrm{ab}}$ = $\sqrt{a}$ x $\sqrt{b}$

$\sqrt{\mathrm{48}}$ = $\sqrt{\mathrm{16\; x\; 3}}$ = $\sqrt{\mathrm{16}}$ x $\sqrt{3}$

$\sqrt{\mathrm{16}}$ = 4

$\sqrt{6}$ = $\sqrt{\mathrm{3\; x\; 2}}$ = $\sqrt{3}$ x $\sqrt{2}$

$\frac{\sqrt{3}\mathrm{+}\sqrt{\mathrm{48}}}{\sqrt{6}}$ = $\frac{\sqrt{3}\mathrm{+}\sqrt{\mathrm{16}}\mathrm{x}\sqrt{3}}{\sqrt{3}\mathrm{x}\sqrt{2}}$

$\frac{\sqrt{3}\mathrm{+}\sqrt{\mathrm{48}}}{\sqrt{6}}$ = $\frac{\sqrt{3}\mathrm{+\; 4}\sqrt{3}}{\sqrt{3}\mathrm{x}\sqrt{2}}$

$\frac{\sqrt{3}\mathrm{+}\sqrt{\mathrm{48}}}{\sqrt{6}}$ = $\frac{5\sqrt{3}}{\sqrt{3}\mathrm{x}\sqrt{2}}$

$\sqrt{3}$ cancels each other.

$\frac{\sqrt{3}\mathrm{+}\sqrt{\mathrm{48}}}{\sqrt{6}}$ = $\frac{5}{\sqrt{2}}$

$\sqrt{2}$ = 2^½

$\frac{1}{{\mathrm{a}}^{\mathrm{b}}}$ = a^-b

$\frac{\sqrt{3}\mathrm{+}\sqrt{\mathrm{48}}}{\sqrt{6}}$ = 5 x 2^-½

Law of multiplication of indices

a^m x aⁿ = a^{m + m}

5 x 2^-½ = 5 x 2^-1 x 2^½

5 x 2^-½ = $\frac{5}{2}$ x 2^½

2^½ = $\sqrt{2}$

5 x 2^-½ = $\frac{5}{2}$$\sqrt{2}$

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

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

Speed = $\frac{\mathrm{Distance}}{\mathrm{Time}}$

Speed x Time = Distance

Time = $\frac{\mathrm{Distance}}{\mathrm{Speed}}$

30kmh^-1 = 30000m^-1

Time = $\frac{200}{30000}$

Note: 200 divides itself once and 30000, 150 times.

Time = $\frac{1}{150}$ hour

1 hour = 3600seconds
$\frac{1}{150}$ hour = $\frac{1}{150}$ x 3600 seconds

Time = 24 seconds

y ∝ x²

Remove the proportionality by introducing a constant.

y = kx²

Subtitute the known values to solve for the constant k

y = 3 when x = $\sqrt{2}$

3 = k x ${\left(\sqrt{2}\right)2}^{}$

When you square a square root, the square cancels out the square root.

3 = k x 2

Divide both sides by 2

k = $\frac{3}{2}$

Subtitute the value of k into the equation

y = $\frac{3}{2}$ x x²

When x = 2

y = $\frac{3}{2}$ x 2²

2² = 2 x 2

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

The 2 dividing 3 cancels out one of the 2s

y = 3 x 1 x 2

y = 6

Angles subtended by the diameter (or semi-circle) is 90^o.

∠PQR = 90^o

∠PQS + ∠RQS = ∠PQR = 90^o

∠RQS = 23^o

∠PQS + 23^o = 90^o

∠PQS = 90^o - 23^o

∠PQS = 67^o

Since triangle PQS is isosceles triangle, the base angles are the same.

∠PQS = ∠PSQ = 67^o

Sum of angles in a triangle is 180^o

x + ∠PQS + ∠PSQ = 180^o

x + 67^o + 67^o = 180^o

x + 134^o = 180^o

x = 180^o - 134^o

x = 46^o

Range = Largest Value - Smallest Value

Range = 83 - 47

Range = 36

Mean = $\frac{\mathrm{Sum\; of\; numbers}}{\mathrm{Number\; of\; numbers}}$

Mean = $\frac{\mathrm{83\; +\; 47\; +\; 62\; +\; 49\; +\; 55\; +\; 72\; +\; 58\; +\; 62}}{8}$

Mean = $\frac{488}{8}$ = 61

Tan 30^o = $\frac{1}{\sqrt{3}}$

sin 60^o = $\frac{\sqrt{3}}{2}$

tan 30^o - 2sin 60^o = $\frac{1}{\sqrt{3}}$ - 2 x $\frac{\sqrt{3}}{2}$

Note: The 2 dividing $\sqrt{3}$ cancels the 2 multiply.

tan 30^o - 2sin 60^o = $\frac{1}{\sqrt{3}}$ - $\sqrt{3}$

Every number is being divided by 1.

$\sqrt{3}$ = $\frac{\sqrt{3}}{1}$

tan 30^o - 2sin 60^o = $\frac{1}{\sqrt{3}}$ - $\frac{\sqrt{3}}{1}$

Simplify the fraction.

tan 30^o - 2sin 60^o = $\frac{\mathrm{1\; -}\sqrt{3}x\sqrt{3}}{\sqrt{3}}$

From indices, $\sqrt{a}$ x $\sqrt{a}$ = a

Hence $\sqrt{3}$ x $\sqrt{3}$ = 3

tan 30^o - 2sin 60^o = $\frac{\mathrm{1\; -}3}{\sqrt{3}}$

tan 30^o - 2sin 60^o = $\frac{-2}{\sqrt{3}}$

Multiply both the numerator and the denominator by $\sqrt{3}$

tan 30^o - 2sin 60^o = $\frac{-2x\sqrt{3}}{\sqrt{3}x\sqrt{3}}$

$\sqrt{3}$ x $\sqrt{3}$ = 3

tan 30^o - 2sin 60^o = $\frac{-2\sqrt{3}}{3}$

8$\frac{1}{6}$ - $(11\frac{2}{3}÷2\frac{2}{3})$ = 8$\frac{1}{6}$ - $(11\frac{2}{3}÷2\frac{2}{3})$

Change the mixed fractions to improper fractions

8$\frac{1}{6}$ = $\frac{\mathrm{6\; x\; 8\; +\; 1}}{6}$ = $\frac{49}{6}$

11$\frac{2}{3}$ = $\frac{\mathrm{3\; x\; 11\; +\; 2}}{3}$ = $\frac{35}{3}$

2$\frac{2}{3}$ = $\frac{\mathrm{3\; x\; 2\; +\; 2}}{3}$ = $\frac{8}{3}$

8$\frac{1}{6}$ - $(11\frac{2}{3}÷2\frac{2}{3})$ = $\frac{49}{6}$ - $(\frac{35}{3}÷\frac{8}{3})$

Reciprocate the fraction at the right of ÷ and change the ÷ to multiplication (x)

Note: reciprocate means the numerator becomes the denominator and the denominator, numerator.

8$\frac{1}{6}$ - $(11\frac{2}{3}÷2\frac{2}{3})$ = $\frac{49}{6}$ - $(\frac{35}{3}x\frac{3}{8})$

Note: 3 cancel out each other.

8$\frac{1}{6}$ - $(11\frac{2}{3}÷2\frac{2}{3})$ = $\frac{49}{6}$ - $\left(\frac{35}{8}\right)$

Remove the bracket.

8$\frac{1}{6}$ - $(11\frac{2}{3}÷2\frac{2}{3})$ = $\frac{49}{6}$ - $\frac{35}{8}$

The L.C.M of 6 and 8 is 24.

8$\frac{1}{6}$ - $(11\frac{2}{3}÷2\frac{2}{3})$ = $\frac{\mathrm{49\; x\; 4\; -\; 35\; x\; 3}}{24}$

8$\frac{1}{6}$ - $(11\frac{2}{3}÷2\frac{2}{3})$ = $\frac{\mathrm{196\; -\; 105}}{24}$

8$\frac{1}{6}$ - $(11\frac{2}{3}÷2\frac{2}{3})$ = $\frac{91}{24}$

8$\frac{1}{6}$ - $(11\frac{2}{3}÷2\frac{2}{3})$ = 3$\frac{19}{24}$

P(X) = $\frac{1}{2}$

P(Y) = $\frac{2}{3}$

P(Z) = $\frac{1}{4}$

P(only X and Z) = P(X) x P(Z) x P(Y not occurring)

Probability of not occurring = 1 - Probability of occurring

P(Y not occurring) = 1 - $\frac{2}{3}$

Every number is being divided by 1

P(Y not occurring) = $\frac{1}{1}$ - $\frac{2}{3}$

P(Y not occurring) = $\frac{\mathrm{3\; x\; 1\; -\; 1\; x\; 2}}{3}$

P(Y not occurring) = $\frac{\mathrm{3\; -\; 2}}{3}$

P(Y not occurring) = $\frac{1}{3}$

P(only X and Z) = $\frac{1}{2}$ x $\frac{1}{4}$ x $\frac{1}{3}$

P(only X and Z) = $\frac{\mathrm{1\; x\; 1\; x\; 1}}{\mathrm{2\; x\; 4\; x\; 3}}$

P(only X and Z) = $\frac{1}{24}$

Let the length of the rectangle = 8 cm and b cm be the breadth.

From Pythagorean theorem,

The hypothenuse (c) is the longest side. The side facing the right-angle (∟)

The 10 cm is the hypothenuse, since it's facing the right-angle and also the longest side.

10² = 8² + b²

100 = 64 + b²

100 - 64 = b²

36 = b²

Take square root of both sides

$\sqrt{\mathrm{36}}$ = b

6 = b

Area of a rectangle = Length x Breadth

Length = 8 cm
Breadth = 6 cm

Area = 8 cm x 6 cm

Area = 48 cm²

Central angle is twice any inscribed angle subtended by the same arc.

∠ EPG = 2 x ∠ EFG

∠ EPG = 2 x 142^o

∠ EPG = 284^o

∠ EPG + x = 360^o (Sum of angles of a circle)

284^o + x = 360^o

x = 360^o - 284^o

x = 76^o

Sum of interior angles of a quadrilateral is 360^o

Note: sum of interior angles = (n - 2) x 180^o, where n is the number of sides.

63^o + 142^o + ∠ FEP + x = 360^o

63^o + 142^o + ∠ FEP + 76^o = 360^o

∠ FEP + 281^o = 360^o

∠ FEP = 360^o - 281^o

∠ FEP = 79^o

Angles subtended by the diameter (or semi-circle) is 90^o.

∠PRQ = 90^o

∠PRQ = ∠QRS + ∠PRS

∠PRS = ∠PRQ - ∠QRS

∠PRS = 90^o - ∠QRS

36^o + 64^o + ∠QRS = 180^o (sum of angles in a triangle)

100^o + ∠QRS = 180^o

∠QRS = 180^o - 100^o

∠QRS = 80^o

∠PRS = 90^o - 80^o

∠PRS = 10^o

x : y = 2 : 7

Change the ratios to frations.

$\frac{\mathrm{x}}{\mathrm{y}}$ = $\frac{2}{7}$

Cross multiply.

7 x x = 2 x y

7x = 2y

Divide both sides by 7

x = $\frac{\mathrm{2y}}{7}$ ----- eqn 1

Sum of co-interior angles of parallel lines is 180°.

x + y = 180° ------ eqn 2

Subtitute x = $\frac{\mathrm{2y}}{7}$ into ----- eqn 2

$\frac{\mathrm{2y}}{7}$ + y = 180

Get rid of the fraction by multiplying each side by the denominator 7

2y + 7y = 180 x 7

9y = 1260

Divide both sides by 9

y = $\frac{1260}{9}$ = 140

y + (7m - 16) = 180 (Sum of angles on a straight line)

But y = 140

140 + 7m - 16 = 180

7m = 180 + 16 - 140

7m = 56

Divide both sides by 7

m = $\frac{56}{7}$ = 8