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

-$\frac{2}{5}$ = $\frac{\mathrm{4\; -\; 6}}{\mathrm{x\; -\; 3}}$

-$\frac{2}{5}$ = $\frac{-2}{\mathrm{x\; -\; 3}}$

The numerator at the left equals the numerator at the right and the denominator at the left equals the denominator at the right.

x - 3 = 5
x = 5 + 3
x = 8

Volume of a cylinder = πr²h

Where r = radius of the cylinder
h = depth of the water

Volume of a cylinder = 792 cm³
r = 6cm

π = $\frac{22}{7}$

r² = 6 x 6

792 = $\frac{22}{7}$ x 6 x 6 x h

Get rid of the fraction by multiplying both sides of the equation by 7

792 x 7 = 7 x $\frac{22}{7}$ x 6 x 6 x h

792 x 7 = 22 x 6 x 6 x h

Divide both sides by 22 x 6 x 6

h = $\frac{\mathrm{792\; x\; 7}}{\mathrm{22\; x\; 6\; x\; 6}}$

22 divides 792, 36 times.

h = $\frac{\mathrm{36\; x\; 7}}{\mathrm{6\; x\; 6}}$

6 x 6 = 36

h = $\frac{\mathrm{36\; x\; 7}}{36}$

h = 7

Depth of kerosene = 7 cm.

Women = 60

Children = $\frac{10}{100}$ x 200

Note:
Zeros (0) at the end of 100 and 200 cancels each other

Children = 10 x 2
Children = 20

Total people in the hall = 200

Men = 200 - (20 + 60)
Men = 200 - 80
Men = 120

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

Probability(Men) = $\frac{120}{200}$

Note:
1. Zeros (0) at the end of 120 and 200 cancels each other
2. 4 divides 12, 3 times and 20, 5 times

Probability(Men) = $\frac{3}{5}$

Let x = Juliet's age

Badu is four times as old as Juliet.

Badu's age = 4 x x
Badu's age = 4x

In 10 years,
Badu's age = 4x + 10
Juliet's age = x + 10

10 years Badu will be twice as old as Juliet

4x + 10 = 2 x (x + 10)
4x + 10 = 2 x x + 2 x 10
4x + 10 = 2x + 20
4x - 2x = 20 - 10
2x = 10

Divide both sides by 2

x = $\frac{10}{2}$

x = 5

Juliet's age = 5 years

64 = 4 x 4 x 4
64 = 4³

4^x = $\frac{1}{4^3}$

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

$\frac{1}{4^3}$ = 4^-3

4^x = 4^-3

Since the bases are the same, the powers should be the same.

x = -3

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

Distance = 15 km
Time = 30 minutes

Speed = $\frac{\mathrm{15\; km}}{\mathrm{30\; minutes}}$

1 hour = 60 minutes

1 hour 30 minutes = 60 minutes + 30 minutes.

1 hour 30 minutes = 90 minutes

Speed for distance in 1 hour 30 minutes is the same.

$\frac{\mathrm{15\; km}}{\mathrm{30\; minutes}}$ = $\frac{\mathrm{Distance}}{\mathrm{90\; minutes}}$

Cross multiply.

15 km x 90 minutes = Distance x 30 minutes

Divide both sides by 30 minutes

Distance = $\frac{\mathrm{15\; km\; x\; 90\; minutes}}{\mathrm{30\; minutes}}$

Note:
30 divides 90, 3 times

Distance = 15 km x 3
Distance = 45 km

3u + 2v = 3$\left(\begin{array}{c}2\\ 1\end{array}\right)$ + 2$\left(\begin{array}{c}-1\\ 3\end{array}\right)$

3u + 2v = $\left(\begin{array}{c}\mathrm{3\; x\; 2}\\ \mathrm{3\; x\; 1}\end{array}\right)$ + $\left(\begin{array}{c}\mathrm{2\; x\; -1}\\ \mathrm{2\; x\; 3}\end{array}\right)$

3u + 2v = $\left(\begin{array}{c}6\\ 3\end{array}\right)$ + $\left(\begin{array}{c}-2\\ 6\end{array}\right)$

3u + 2v = $\left(\begin{array}{c}\mathrm{6\; +\; -2}\\ \mathrm{6\; +\; 3}\end{array}\right)$

3u + 2v = $\left(\begin{array}{c}\mathrm{6\; -\; 2}\\ \mathrm{6\; +\; 3}\end{array}\right)$

3u + 2v = $\left(\begin{array}{c}4\\ 9\end{array}\right)$

The ratio for the shorter boy is the lesser ratio, 4 and that of the taller boy is 5

If 4 = 80 cm

5 = $\frac{\mathrm{80\; cm\; x\; 5}}{4}$

Note
4 divides itself, 1 and 80, 20 times

Taller boy's height = 20 cm x 5

Taller boy's height = 100 cm

Interior angle = $\frac{\mathrm{(n\; -\; 2)\; x\; 180}}{\mathrm{n}}$

Where n = number of sides of the polygon.

Interior angle = 120^o.

$\frac{\mathrm{(n\; -\; 2)\; x\; 180}}{\mathrm{n}}$ = 120

Get rid of the fraction by multiplying both sides by n.

(n - 2) x 180 = 120 x n

n x 180 - 2 x 180 = 120 x n

180n - 360 = 120 x n
180n - 120 x n = 360
60n = 360

Divide both sides by 60

n = $\frac{360}{60}$

Note:
1. The zero (0) at the end of 360 and 60 cancels each other
2. 6 divides itself 1 and 36, 6 times

n = 6

Union (∪) is simply the listing all the elements in both sets once in ascending order.

x : y is 9 : 5

$\frac{\mathrm{x}}{\mathrm{y}}$ = $\frac{9}{5}$

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

But $\frac{\mathrm{x}}{\mathrm{y}}$ = $\frac{9}{5}$

$\frac{\mathrm{2x}}{\mathrm{y}}$ = 2 x $\frac{9}{5}$

$\frac{\mathrm{2x}}{\mathrm{y}}$ = $\frac{\mathrm{2\; x\; 9}}{5}$

$\frac{\mathrm{2x}}{\mathrm{y}}$ = $\frac{18}{5}$

Perimeter is the sum of all the sides of a shape
The triangle is an isosceles triangle (two sides are equal)
Let the two equal sides length = x = AB = AC
Perimeter = x + x + 4
Perimeter = 2x + 4
But the perimeter = 22
22 = 2x + 4
22 - 4 = 2x
18 = 2x
Divide both sides by 2

x = $\frac{18}{2}$

x = 9

∴ AB = 9

Sum of angles in a triangle is 180°

x + y + 112 = 180

Substitute the value of x (3y) in the above equation.

3y + y + 112 = 180

4y + 112 = 180

4y = 180 - 112

4y = 68

Divide both sides by 4.

y = $\frac{68}{4}$

y = 17

Change the decimal to whole number x 10^-n.

Where n is the number of times you have to move the decimal point to the right to obtain a whole number.

0.28 = 0.⌒2⌒8

0.28 = 28 x 10^-2

0.8 = 0.⌒8

0.8 = 8 x 10^-1

$\frac{0.28}{0.8}$ = $\frac{\mathrm{28\; x}{10}^{-2}}{\mathrm{8\; x}{10}^{-1}}$

$\frac{0.28}{0.8}$ = $\frac{\mathrm{7\; x}{10}^{\mathrm{-2-(-1)}}}{2}$

Note:
1. 4 divides 28, 7 times and 8, 2 times
2. From the law of indices the power dividing can be subtracted from the power at the top since the bases are the same

$\frac{0.28}{0.8}$ = $\frac{\mathrm{7\; x}{10}^{\mathrm{-2+1}}}{2}$

Note: -- = +

$\frac{0.28}{0.8}$ = $\frac{\mathrm{7\; x}{10}^{-1}}{2}$

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

10^-1 = $\frac{1}{{10}^1}$

10^-1 = $\frac{1}{10}$

$\frac{0.28}{0.8}$ = $\frac{7}{2}$ x $\frac{1}{10}$

$\frac{0.28}{0.8}$ = $\frac{\mathrm{7\; x\; 1}}{\mathrm{2\; x\; 10}}$

$\frac{0.28}{0.8}$ = $\frac{7}{20}$

The more the workers, the lesser the time.

Hence time (t) is inversely proportional to the number of workers (w).

t ∝ $\frac{1}{\mathrm{w}}$

Remove the proportionality by introducing a constant (k).

t = k x $\frac{1}{\mathrm{w}}$

Use the known information to solve for k

15 workers used 8 days.

8 = k x $\frac{1}{15}$

k = 8 x 15

k = 120

The general equation is:

t = 120 x $\frac{1}{\mathrm{w}}$

t = $\frac{120}{\mathrm{w}}$

Calculate how many workers would be needed for 6 days.

t = 6

6 = $\frac{120}{\mathrm{w}}$

Multiply each side of the equation by the denominator w

6 x w = w x $\frac{120}{\mathrm{w}}$

6w = 120

Divide both sides by 6

w = $\frac{120}{6}$

w = 20

Note: 6 divides itself 1 and 12, 2 times then join the zero (0)

∴ It will take 20 workers to complete in 6 days.

More workers needed = 20 workers - 15 workers
More workers needed = 5 workers

y = $\frac{\mathrm{5x\; +\; 9}}{\mathrm{x}}$

Get rid of the denominator (x) by multiplying both sides of the equation by the denominator (x).

x x y = x x $\frac{\mathrm{5x\; +\; 9}}{\mathrm{x}}$

xy = 5x + 9

xy - 5x = 9

Factorize x out.

x(y - 5) = 9

Divide both sides by (y - 5)

x = $\frac{9}{\mathrm{y\; -\; 5}}$

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

Where θ = angle subtends at the centre
r = radius of the circle

Length of arc = 22 cm
Radius (r) = 15 cm

π = $\frac{22}{7}$

$\frac{\mathrm{\theta }}{360}$ x 2 x $\frac{22}{7}$ x 15 = 22

Get rid of the fractions by multiplying both sides by 360 x 7

360 x 7 x $\frac{\mathrm{\theta }}{360}$ x 2 x $\frac{22}{7}$ x 15 = 22 x 360 x 7

θ x 2 x 22 x 15 = 22 x 360 x 7

Divide both sides by 2 x 22 x 15

θ = $\frac{\mathrm{22\; x\; 360\; x\; 7}}{\mathrm{2\; x\; 22\; x\; 15}}$

Note:
1. 22 at the numerator and denominator cancels each other
2. 15 divides itself 1 and 360, 24 times
3. 2 divides itself 1 and 24, 12 times

θ = 12 x 7
θ = 84

(2x + 2y)(x - y) + (2x - 2y)(x + y) = (2x + 2y)(x - y) + (2x - 2y)(x + y)
Factorize the 2s out
(2x + 2y)(x - y) + (2x - 2y)(x + y) = 2(x + y)(x - y) + 2(x - y)(x + y)

The order of multiplication doesn't matter.

(x + y)(x - y) is the same as (x - y)(x + y) so you can simply add them.

(2x + 2y)(x - y) + (2x - 2y)(x + y) = 4(x - y)(x + y)

Initial length = 15 cm
Initial width = x cm

Double means 2 x initial

New length = 2 x 15 cm
New length = 30 cm
New width = 2 x x cm
New width = 2x cm

Area of a rectangle = Length x Width
New area = 30 cm x 2x cm
New area = 60x cm²

From pythagoras theorem

Let l = Length of the ladder

l² = 8² + 6²
l² = 64 + 36
l² = 100

100 = 10 x 10
100 = 10²

l² = 10²

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

l = 10

Change the decimal and percentage to fraction.

0.45 = 0.⌒4⌒5 = $\frac{45}{100}$ = $\frac{9}{20}$

% means over 100.

25% = $\frac{25}{100}$ = $\frac{1}{4}$

0.45, $\frac{3}{4}$, 25% = $\frac{9}{20}$, $\frac{3}{4}$, $\frac{1}{4}$

Before you can order fractions, the denominators must be the same. Follow the steps below to ensure the denominators are the same.

1. Find the L.C.M of the denominators

The L.C.M of 20 and 4 is 20.

2. Multiply both the numerator and the denominator by the number which when multiplied by the denominator, would result the L.C.M

For the denominator 4, 4 x 5 = 20, hence both the numerator and the denominators are multiplied by 5

For the denominator 20, 20 x 1 = 20, hence both the numerator and the denominator are multiplied by 1. But 1 times any number is the same so there is no need to multiply.

$\frac{3}{4}$ = $\frac{\mathrm{3\; x\; 5}}{\mathrm{4\; x\; 5}}$ = $\frac{15}{20}$

$\frac{1}{4}$ = $\frac{\mathrm{1\; x\; 5}}{\mathrm{4\; x\; 5}}$ = $\frac{5}{20}$

Once the denominators are the same, you can order the fractions by the numerators.

Ascending means from the lowest to the highest.

5 < 9 < 15

$\frac{9}{20}$, $\frac{15}{20}$, $\frac{5}{20}$ = $\frac{5}{20}$, $\frac{9}{20}$, $\frac{15}{20}$ ascending

Substitute the equivalent fractions by their actual fraction/decimal/percentage.

0.45, $\frac{3}{4}$, 25% = 25%,0.45,$\frac{3}{4}$ ascending

Mode is the score which appeared the most.

13 appeared more than the rest of the distribution. Hence the mode is 13.

Median is the middle score of the ordered scores. If there are two scores in the middle, add the two scores and divide by 2.

The ordered distribution is 8,8,10,11,13,13,13,14,14,15,16,17,18,18

13 and 14 are at the middle, hence add them and divide by 2

Median = $\frac{\mathrm{13\; +\; 14}}{2}$

Median = $\frac{27}{2}$

Median = 13.5

$\frac{\frac{1}{4}x\; 2\frac{1}{2}}{12\; ÷\; 1\frac{1}{2}}$

Change the mixed fractions to improper fractions.

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

1$\frac{1}{2}$ = $\frac{\mathrm{1\; x\; 2\; +\; 1}}{2}$ = $\frac{\mathrm{2\; +\; 1}}{2}$ = $\frac{3}{2}$

$\frac{1}{4}x\; 2\frac{1}{2}$ = $\frac{1}{4}$ x $\frac{5}{2}$

$\frac{1}{4}x\; 2\frac{1}{2}$ = $\frac{\mathrm{1\; x\; 5}}{\mathrm{4\; x\; 2}}$

$\frac{1}{4}x\; 2\frac{1}{2}$ = $\frac{5}{8}$

$12\; ÷\; 1\frac{1}{2}$ = 12 ÷ $\frac{3}{2}$

Reciprocate the fraction (numerator becomes denominator and denominator becomes numerator) at the right and change the division (÷) to multiplication (x) sign.

$12\; ÷\; 1\frac{1}{2}$ = 12 x $\frac{2}{3}$

Note: 3 divides itself 1 and 12, 4 times.

$12\; ÷\; 1\frac{1}{2}$ = $\frac{\mathrm{4\; x\; 2}}{1}$

$12\; ÷\; 1\frac{1}{2}$ = $\frac{8}{1}$

$\frac{\frac{1}{4}x\; 2\frac{1}{2}}{12\; ÷\; 1\frac{1}{2}}$ = $\frac{5}{8}$ ÷ $\frac{8}{1}$

Reciprocate the fraction (numerator becomes denominator and denominator becomes numerator) at the right and change the division (÷) to multiplication (x) sign.

$\frac{\frac{1}{4}x\; 2\frac{1}{2}}{12\; ÷\; 1\frac{1}{2}}$ = $\frac{5}{8}$ x $\frac{1}{8}$

$\frac{\frac{1}{4}x\; 2\frac{1}{2}}{12\; ÷\; 1\frac{1}{2}}$ = $\frac{\mathrm{5\; x\; 1}}{\mathrm{8\; x\; 8}}$

$\frac{\frac{1}{4}x\; 2\frac{1}{2}}{12\; ÷\; 1\frac{1}{2}}$ = $\frac{5}{64}$

Draw a Venn diagram to illustrate.

Let x = number of students who offer both Chemistry and Biology.

U represent the total students, C represent Chemistry and B represent Biology.

28 - x + x + 25 - x = 45
53 - x = 45
53 - 45 = x
8 = x

Students who studied only Chemistry = 28 - x

But x = 8

Students who studied only Chemistry = 28 - 8
Students who studied only Chemistry = 20

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

P(Chemistry only) = $\frac{20}{45}$ = $\frac{4}{9}$

Note: 5 divides 20, 4 times and 45, 9 times.

Expand the brackets. The negative (-) affect each letter in the bracket.

3x - (p - x) - (r - p) = 3x - (p - x) - (r - p)
3x - (p - x) - (r - p) = 3x - p + x - r + p

Note: -- = +

3x - (p - x) - (r - p) = 3x + x + p - p - r
3x - (p - x) - (r - p) = 4x - r

$\frac{1}{\mathrm{5x}}$ + $\frac{1}{\mathrm{x}}$ = 3

Get rid of the fractions by multiplying both sides of the equation by the L.C.M of the denominators.

L.C.M of 5x and x is 5x

5x x $\frac{1}{\mathrm{5x}}$ + 5x x $\frac{1}{\mathrm{x}}$ = 5x x 3

1 + 5 = 15x
6 = 15x

Divide both sides by 15

x = $\frac{6}{15}$

Note: 3 divides 6, 2 times and 15, 5 times.

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

Perimeter is the sum of all the sides of a shape.

A rectangle has 2 equal lenths and 2 equal widths.

2 x Length + 2 x Width = Perimeter.

But the perimeter is 54 cm

2 x Length + 2 x Width = 54

Divide both sides by 2

Length + Width = $\frac{54}{2}$

Length + Width = 27 cm

Ratio 5 represents the length and ratio 4 represents the width.

The sum of the length and width (Length + Width) represents the sum of the ratios.

Sum of ratios = 5 + 4
Sum of ratios = 9

If 9 = 27

5 = $\frac{\mathrm{5\; x\; 27}}{9}$

Note: 9 divides itself 1 and 27, 3 times.

Length = 5 x 3 cm
Length = 15 cm

Median is the middle of the ordered distribution. If there are two numbers in the middle, add the two numbers and divide by 2.

Mean is the sum of all the numbers divided by how many numbers in the distribution.

x - 2, x + 2, 4, 2x + 18

There are four ordered items/numbers

The middle are x + 2 and 4, hence add them and divide by 2

Median = $\frac{\mathrm{x\; +\; 2\; +\; 4}}{2}$

Median = $\frac{\mathrm{x\; +\; 6}}{2}$

Mean = $\frac{\mathrm{x\; -\; 2\; +\; x\; +\; 2\; +\; 4\; +\; 2x\; +\; 18}}{4}$

Mean = $\frac{\mathrm{x\; +\; x\; +\; 2x\; -\; 2\; +\; 2\; +\; 4\; +\; 18}}{4}$

Mean = $\frac{\mathrm{4x\; +\; 22}}{4}$

But median = mean

$\frac{\mathrm{x\; +\; 6}}{2}$ = $\frac{\mathrm{4x\; +\; 22}}{4}$

Cross multiply

4 x (x + 6) = 2 x (4x + 22)
4x + 24 = 8x + 44
24 - 44 = 8x - 4x
-20 = 4x

Divide both sides by 4

x = $\frac{-20}{4}$

x = -5

-3x + y = 1

The general equation of a line is:

y = mx + c

Where m = gradient of the line
c = y - intercept

Make y the subject in the given equation and compare it with the general equation to determine the gradient.

-3x + y = 1
y = 3x + 1

Comparing y = 3x + 1 to the general equation y = mx + c, m = 3, hence the gradient is 3.

Prime numbers are numbers with only two factors, 1 and itself. Thus numbers which are not divisible by any other number except 1 and itself. Note: 1 is not a prime number.

P = {prime numbers less than 20}
P = {2,3,5,7,11,13,17,19}

Odd numbers are numbers which are not divisible by 2.

Intersection (∩) means elements that can be found in both sets.

P ∩ Q = {3,5,7}

Method I

Using the factors and selecting the highest common in all.

Factors of 48 = {1, 2, 3, 4, 6, 8, 12, 16, 24, 48}

Factors of 30 = {1, 2, 3, 5, 6, 10, 15, 30}

Factors of 18 = {1, 2, 3, 6, 9, 18}

From the list, 6 is the highest common in all

Method II

Using prime factors

Prime factors of 48

$$\begin{array}{cc}\mathrm{}& 48\\ 2& 24\\ 2& 12\\ 2& 6\\ 2& 3\\ 3& 1\end{array}$$

Prime factors of 48 = 2 x 2 x 2 x 2 x 3

Prime factors of 48 = 2⁴ x 3

Prime factors of 30

$$\begin{array}{cc}\mathrm{}& 30\\ 2& 15\\ 3& 5\\ 5& 1\end{array}$$

Prime factors of 30 = 2 x 3 x 5

Prime factors of 18

$$\begin{array}{cc}\mathrm{}& 18\\ 2& 9\\ 3& 3\\ 3& 1\end{array}$$

Prime factors of 18 = 2 x 3 x 3

Prime factors of 18 = 2 x 3²

The H.C.F is the prime factor common in all in their lowest power

H.C.F = 2 x 3
H.C.F = 6

A standard form should be in the form a.bcde x 10ⁿ

The decimal point must be after the first non-zero digit.

The n is the number of times the decimal point has to be moved to be right after the first non-zero digit.

If it is moved to the left, the n is positive and if to the right, n is negative.

In 356.07, the decimal point has to be after the 3. Hence we have to move the decimal point two times to the left. Since we are moving to the left, our n (2) would be positive.

3.⌒5⌒6.07

356.07 = 3.5607 x 10² in standard form.

$(1\frac{1}{2}+\frac{1}{4})$ ÷   $(1\frac{1}{2}-\frac{1}{4})$

Change the mixed fractions to improper fractions.

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

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

1$\frac{1}{2}$ = $\frac{3}{2}$

$(1\frac{1}{2}+\frac{1}{4})$ = $\frac{3}{2}$ + $\frac{1}{4}$

$(1\frac{1}{2}+\frac{1}{4})$ = $\frac{\mathrm{2\; x\; 3\; +\; 1\; x\; 1}}{4}$

$(1\frac{1}{2}+\frac{1}{4})$ = $\frac{\mathrm{6\; +\; 1}}{4}$

$(1\frac{1}{2}+\frac{1}{4})$ = $\frac{7}{4}$

$(1\frac{1}{2}-\frac{1}{4})$ = $\frac{3}{2}$ - $\frac{1}{4}$

$(1\frac{1}{2}-\frac{1}{4})$ = $\frac{\mathrm{2\; x\; 3\; -\; 1\; x\; 1}}{4}$

$(1\frac{1}{2}-\frac{1}{4})$ = $\frac{\mathrm{6\; -\; 1}}{4}$

$(1\frac{1}{2}-\frac{1}{4})$ = $\frac{5}{4}$

$(1\frac{1}{2}+\frac{1}{4})$ ÷   $(1\frac{1}{2}-\frac{1}{4})$ = $\frac{7}{4}$ ÷ $\frac{5}{4}$

Reciprocate the fraction (numerator becomes denominator and denominator becomes numerator) at the right and change the division (÷) to multiplication (x) sign.

$(1\frac{1}{2}+\frac{1}{4})$ ÷   $(1\frac{1}{2}-\frac{1}{4})$ = $\frac{7}{4}$ x $\frac{4}{5}$

Note: The 4s cancels each other.

$(1\frac{1}{2}+\frac{1}{4})$ ÷   $(1\frac{1}{2}-\frac{1}{4})$ = $\frac{7}{5}$

$(1\frac{1}{2}+\frac{1}{4})$ ÷   $(1\frac{1}{2}-\frac{1}{4})$ = 1$\frac{2}{5}$

Triangle CBD is isosceles.

Angle CDB = CBD = 70

Sum of angles on a straight line is 180^o.

Angle CBA + Angle CBD = 180

Angle CBA + 70 = 180

Angle CBA = 180 - 70

Angle CBA = 110

Sum of interior angles of a triangle is 180^o

30^o + Angle CBA + P = 180

30^o + 110 + P = 180

140^o + P = 180

P = 180 - 140

P = 40^o

Difference means subtraction.

(2d)²

d=3

(2d)² = (2 x 3)²
(2d)² = (6)²
(2d)² = 6 x 6
(2d)² = 36

2d² = 2 x 3²
2d² = 2 x 3 x 3
2d² = 18

(2d)² - 2d² = 36 - 18
(2d)² - 2d² = 18

Scale factor = $\frac{\mathrm{Enlarged\; Size}}{\mathrm{Original\; Size}}$

Scale factor = $\frac{\mathrm{AC}}{\mathrm{AE}}$

|AC| = |AE| + |EC|
|AC| = 20 cm + 10 cm
|AC| = 30 cm

Scale factor = $\frac{\mathrm{30\; cm}}{\mathrm{20\; cm}}$

Scale factor = $\frac{3}{2}$

Change the decimal to whole number x 10^-n.

Where n is the number of times you have to move the decimal point to the right to obtain a whole number.

0.24 = 0.⌒2⌒4

0.24 = 24 x 10^-2

14.3 = 14.⌒3

14.3 = 143 x 10^-1

5.2 = 5.⌒2

5.2 = 52 x 10^-1

$\frac{\mathrm{0.24\; x\; 14.3}}{5.2}$ = $\frac{\mathrm{24\; x}{10}^{-2}x\mathrm{143\; x}{10}^{-1}}{\mathrm{52\; x}{10}^{-1}}$

10^-1 at the numerator and denominator cancels each other.

$\frac{\mathrm{0.24\; x\; 14.3}}{5.2}$ = $\frac{\mathrm{24\; x}{10}^{-2}x143}{52}$

Note:
1. 4 divides 24, 6 times and 52, 13 times
2. 13 divides itself 1 and 143, 11 times
3. 6 x 11 = 66

$\frac{\mathrm{0.24\; x\; 14.3}}{5.2}$ = 66 x 10^-2

Now change the whole number x 10^-n back to decimal by moving the decimal point to the left, n times.

Every whole number has a decimal point.

66 = 66.0

66 x 10^-2 = .⌒6⌒6.0

66 x 10^-2 = 0.66

Number of times she sold the oranges = $\frac{210}{3}$

Number of times she sold the oranges = 70

Total selling price = 70 x ₵20.00
Total selling price = ₵1400.00

Profit = Selling Price - Cost Price
Profit = ₵1400.00 - ₵650.00
Profit = ₵750