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

Y₂ = -7
Y₁ = -3
X₂ = -4
X₁ = -5

Gradient = $\frac{\mathrm{-7-(-3)}}{\mathrm{-4-(-5)}}$ = $\frac{\mathrm{-7+3}}{\mathrm{-4\; +\; 5}}$ = $\frac{-4}{1}$ = -4

Scale factor = $\frac{\mathrm{Enlarged\; Side}}{\mathrm{Corresponding\; Object\; Side}}$

Enlarged side = 3 m + 6 m = 9 m

Corresponding object side = 3 m

Scale factor = $\frac{9}{3}$ = 3

2 × 3² × 3⁴ = 2 × 3² × 3⁴

Law of multiplication of indices

a^m x aⁿ = a^{m + m}

2 × 3² × 3⁴ = 2 × 3^{2 + 4}

2 × 3² × 3⁴ = 2 × 3⁶

Substitute the values and simplify

a = -4 and b = 3

$\frac{\mathrm{3a\; +\; 2b}}{\mathrm{ab}}$ = $\frac{\mathrm{3\; x\; (-4)\; +\; 2\; x\; (3)}}{\mathrm{(-4)\; x\; (3)}}$

$\frac{\mathrm{3a\; +\; 2b}}{\mathrm{ab}}$ = $\frac{\mathrm{-12\; +\; 6}}{-12}$

$\frac{\mathrm{3a\; +\; 2b}}{\mathrm{ab}}$ = $\frac{-6}{-12}$

$\frac{\mathrm{3a\; +\; 2b}}{\mathrm{ab}}$ = $\frac{1}{2}$

Express 16 and 8 as base 2 to a power

16 = 2 x 2 x 2 x 2 = 2⁴
8 = 2 x 2 x 2 = 2³

16² x 8² = (2⁴)² x (2³)²

From the law of indices, (a^m)ⁿ = a^{m x n}

(2⁴)² x (2³)² = 2^{4 x 2} x 2^{3 x 2}

(2⁴)² x (2³)² = 2⁸ x 2⁶

From the law of indices, a^m x aⁿ = a^m+n

2⁸ x 2⁶ = 2⁸⁺⁶ = 2¹⁴

16² x 8² = 2¹⁴

Summation of all fraction parts should give the whole (1)

When you sum known fractions and you subtract the result from 1, you will get the rest of the fraction.

Total known fraction spent = $\frac{1}{5}$ + $\frac{4}{7}$ = $\frac{\mathrm{7\; x\; 1\; +\; 5\; x\; 4}}{35}$ = $\frac{\mathrm{7\; +\; 20}}{35}$ = $\frac{27}{35}$

Fraction spent on Gari = 1 - $\frac{27}{35}$

Every number is divisible by 1.

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

Fraction spent on Gari = 1 - $\frac{27}{35}$ = $\frac{1}{1}$ - $\frac{27}{35}$ = $\frac{\mathrm{35\; x\; 1\; -\; 1\; x\; 27}}{35}$ = $\frac{\mathrm{35\; -\; 27}}{35}$ = $\frac{8}{35}$

The easiest way of finding the L.C.M is using the prime factors method.

Prime factors of 10

$$\begin{array}{cc}\mathrm{}& 10\\ 2& 5\\ 5& 1\end{array}$$

Prime factors of 10 = 2 x 5

Prime factors of 15

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

Prime factors of 15 = 3 x 5

Prime factors of 25

$$\begin{array}{cc}\mathrm{}& 25\\ 5& 5\\ 5& 1\end{array}$$

Prime factors of 25 = 5 x 5

Prime factors of 25 = 5²

The L.C.M is the product/multiplication of each of the prime factors in their highest power.

L.C.M = 2 x 3 x 5² = 6 x 5 x 5 = 150

Circumference of a circle = 2πr

Area of a circle = πr²

Where r = radius of the circle.

But area = 64 π cm²

πr² = 64 π cm²

π cancels each other

r² = 64 cm²

64 cm² = 8 cm x 8 cm
64 cm² = (8 cm)²

r² = (8 cm)²

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

r = 8 cm

Circumference of a circle = 2π x 8 cm

Circumference of a circle = 16 π cm

Length:Width = 5:3

Since you know the value of the length, you can use proportionality to get the width.

If the ratio 5 = 25 m
3 = $\frac{\mathrm{25\; m\; x\; 3}}{5}$ = 5 m x 3 = 15 m

4.5 km = $\frac{45}{10}$ km

If 1 cm = $\frac{45}{10}$ km

4 cm = $\frac{45}{10}$ km x 4 cm ÷ 1 cm = $\frac{\mathrm{45\; km\; x\; 4\; cm}}{\mathrm{10\; x\; 1\; cm}}$ = $\frac{\mathrm{180\; km}}{10}$ = 18 km

Intersection (∩) is the list of elements that can be found in both sets (A and B)

A∩B = {}

There are no element in both sets, hence the intersection is an empty set. The number of elements in the intersection of A and B is therefore 0.

Analyzing the x and its corresponding y, you will realize that the ys are obtained by squaring the xs.

Hence y → x²

You could as well test each of the rules by substituting the value of x into it and see if you will obtain its corresponding y.

Testing y → x + 2

When x = 1, y = 1+2 = 3

The result obtained isn't the value in the mapping. The expected value for x = 1 is 1. Hence y → x + 2 is not the rule.

Testing y → 2x

When x = 1, y → 2 x 1 = 2

The result obtained isn't the value in the mapping. The expected value for x = 1 is 1. Hence y → 2x is not the rule.

Testing y → x²

When x = 1, y = 1² = 1
When x = 2, y = 2² = 4
When x = 3, y = 3² = 9
When x = 4, y = 4² = 16
When x = 5, y = 5² = 25

As you can see, y → x² is the rule as each of the substituted x gave its corresponding y value.

You can still verify the last rule.

Testing y → 2x + 2

When x = 1, y = 2(1) + 2 = 4

The result obtained isn't the value in the mapping. The expected value for x = 1 is 1. Hence y → 2x + 2 is not the rule.

15 – 2x = 6
15 - 6 = 2x
9 = 2x

Divide both sides by 2

x = $\frac{9}{2}$ = 4.5

From pythagorean theorem,

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

|BD|² + |AD|² = 5²

|AD| = half of |AC| = $\frac{\mathrm{|AC|}}{2}$

|AD| = $\frac{8}{2}$ = 4

|BD|² + 4² = 5²

|BD|² = 5² - 4²

5² = 5 x 5 = 25

4² = 4 x 4 = 16

|BD|² = 25 - 16

|BD|² = 9

9 = 3 x 3 = 3²

|BD|² = 3²

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

|BD| = 3 cm

Law of multiplication of indices

a^m x aⁿ = a^{m + m}

Law of division of indices

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

$\frac{5^{7}x{5}^4}{5^2}$ = $\frac{5^{\mathrm{7+4}}}{5^2}$

$\frac{5^{7}x{5}^4}{5^2}$ = $\frac{5^{11}}{5^2}$

$\frac{5^{7}x{5}^4}{5^2}$ = 5^11-2

$\frac{5^{7}x{5}^4}{5^2}$ = 5⁹

Total No. of students = 215 + 185 = 400

Percentage of girls = 46.25%

Percentage of girls = 46% (nearest whole number)

NOTE: since the 2 after the decimal is less than 5, you don't have to add 1 to the whole number part (46)

Volume of water = Length x Width x Height of water

Volume of water when completely filled = 3 m x 2 m x 1.5 m

1.5 = $\frac{15}{10}$

Volume of water when completely filled = 3 m x 2 m x $\frac{15}{10}$ m = 3 m x 1 m x 3 m = 9 m³

Note:
1. 2 divides itself 1 and 10, 5 times
2. 5 divides itself 1 and 15, 3 times

Every 1 m³ cost ₵40,000

If 1 = ₵40,000

9 = $\frac{\mathrm{₵40,000\; x\; 9}}{1}$ = ₵40,000 x 9 = ₵360,000

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

Mean = $\frac{\mathrm{14\; +\; 20\; +\; 25\; +\; 15\; +\; 28\; +\; 16}}{6}$

Mean = $\frac{118}{6}$

Mean = 19.67

Before you express a ratio, make sure both entities are of the same units.

Change the weeks to days.

There are 7 days in 1 week.

3 weeks will have 3 x 7 days = 21 days

6 days : 3 weeks = 6 days : 21 days

Note: 3 divides 6, 2 times and 21, 7 times and the days cancel each other.

6 days : 3 weeks = 2 : 7

The region outside the two circles represent pupils who don't speak any of the two languages (neither Twi nor Ga)

The mode is the one which occurs the most/ the one with the highest frequency

The 3 mark occurred/appeared the most (4 times). Hence the mode is 3

Let n = the missing addend.

2345 + 1045 + n = 5110

3390 + n = 5110

n = 5110 - 3390

n = 1720

Substitute the values into the equation.

R = $\frac{h}{2}$ + $\frac{d^2}{\mathrm{8h}}$

d = 8 and h = 6

R = $\frac{6}{2}$ + $\frac{8^2}{\mathrm{8\; x\; 6}}$

8² = 8 x 8

R = $\frac{6}{2}$ + $\frac{\mathrm{8\; x\; 8}}{\mathrm{8\; x\; 6}}$

2 divides itself once and 6, 3 times. 8 at the denominator cancels one of the 8s at the numerator. 2 divides 8, 4 times and 6, 3 times.

R = 3 + $\frac{4}{3}$

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

R = 3 + 1$\frac{1}{3}$

R = (3+1)$\frac{1}{3}$

R = 4$\frac{1}{3}$

Scale factor = $\frac{\mathrm{Enlarged\; Side}}{\mathrm{Corresponding\; Object\; Side}}$

Side 1

Enlarged side = 3 m + 6 m = 9 m

Corresponding object side = 3 m

Side 2

Enlarged side = x

Corresponding object side = 1.5

Scale factor = $\frac{9}{3}$ = $\frac{\mathrm{x}}{1.5}$ = 3

$\frac{\mathrm{x}}{1.5}$ = 3

Get rid of the fraction by multiplying both sides by the denominator (1.5)

x = 3 x 1.5 = 4.5

Scale factor = $\frac{\mathrm{Enlarged\; Side}}{\mathrm{Corresponding\; Object\; Side}}$

Scale factor = $\frac{\mathrm{|ST|}}{\mathrm{|PQ|}}$ = $\frac{\mathrm{|QR|}}{\mathrm{|RT|}}$

|RT| = |QR| + |QT|

|RT| = 4 cm + 8 cm = 12 cm

$\frac{\mathrm{|ST|}}{2.4}$ = $\frac{12}{4}$

Cross multiply.

4 x |ST| = 2.4 x 12

Divide both sides by 4

|ST| = $\frac{\mathrm{2.4\; x\; 12}}{4}$ = 2.4 x 3

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

2.4 = $\frac{24}{10}$

|ST| = $\frac{24}{10}$ x 3

|ST| = $\frac{\mathrm{24\; x\; 3}}{10}$

|ST| = $\frac{72}{10}$ = 7.2 cm

Mean = Sum of numbers/ No. of numbers

Sum of numbers (ages) = 20+55+60+25 = 160

No. of numbers (ages) = 4

Mean = 160/4 = 40 years

3r²s – 9rs² = 3r²s – 9rs²

r² = r x r

s² = s x s

3r²s – 9rs² = 3r x r x s - 9r x s x s

3r²s – 9rs² = 3rs(r - 3s)

243_five = 2 x 5² + 4 x 5¹ + 3 x 5⁰

Note: a^o = 1, 5⁰ = 1

243_five = 2 x 25 + 4 x 5 + 3 x 1

243_five = 50 + 20 + 3

243_five = 73

Total number of students = 3 + 10 + 6 + 7 + 4
Total number of students = 30

The mode is the one which occurs the most/ the one with the highest frequency/number of times.

Sum means add.

one decimal place means there should be only 1 number after the decimal point. The number at the one decimal position is 6. The next number is less than 5, hence we don't have to add 1 to the 6

198.635 = 198.6 (one decimal place).

Express (2.3 × 82) × 7.9 in the form (23 × 82) × 79

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.

2.3 = 2.⌒3. = 23 x 10^-1

7.9 = 7.⌒9. = 79 x 10^-1

(2.3 × 82) × 7.9 = (23 x 10^-1 x 82) x 79 x 10^-1

The order in multiplication doesn't matter.

(2.3 × 82) × 7.9 = (23 x 82) x 79 x 10^-1 x 10^-1

Law of multiplication of indices

a^m x aⁿ = a^{m + m}

(2.3 × 82) × 7.9 = (23 x 82) x 79 x 10^{-1 + -1}

(2.3 × 82) × 7.9 = (23 x 82) x 79 x 10^{-1 - 1}

(2.3 × 82) × 7.9 = (23 x 82) x 79 x 10^-2

But (23 x 82) x 79 = 148,994

(2.3 × 82) × 7.9 = 148,994 x 10^-2

Change the whole number x 10^-n to decimal by moving the decimal point to the left n times.

Every whole number has .0 at the end which is not written.

148994 = 148994.0

n is 2 in 10^-2 hence we have to move the decimal point 2 times to the left.

148,994 x 10^-2 = 1489.⌒9⌒4.0 = 1489.94

(2.3 × 82) × 7.9 = 1489.94

A as a percentage of B = $\frac{\mathrm{A}}{\mathrm{B}}$ x 100%

25 as a percentage of 75 = $\frac{25}{75}$ x 100% = $\frac{\mathrm{1\; x\; 100\%}}{3}$ = $\frac{\mathrm{100\%}}{3}$ = 33.33%

Note: 25 divides itself 1 time and 75, 3 times.

There are 60 minutes to complete 360° (complete cycle)

Angle per minute = $\frac{\mathrm{360°}}{60}$ = 6°

The minutes between 5.25 p.m. and 5.15 p.m. is 10 minutes.

Angle for 10 minutes = 10 x 6° = 60°

Profit = Selling Price - Cost Price

Profit = ₵28,000 - ₵24,000

Profit = ₵4,000

The percentage of the cost price is 100%

If ₵24000 = 100%

₵4000 = $\frac{\mathrm{100\%\; x\; ₵4000}}{\mathrm{₵24000}}$

Note:
1. the zeros (0) at the end of 4000 and 24000 cancel each other
2. 4 divides itself 1 time and 24, 6 times
3. 2 divides 6, 3 times and 100, 50 times

₵4000 = $\frac{\mathrm{50\%}}{3}$ = 16.67% ≈ 16.7%

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

Perimeter = x + x + 7
Perimeter = 2x + 7

Before you can express as a percentage, both must be in the same unit. Is either you change the minutes to hours or hours to minutes.

Change the seconds to minutes. 60 seconds make 1 minute. Hence 30 seconds is half ($\frac{1}{2}$) minutes.

7 min. 30 sec. = 7 minutes + $\frac{1}{2}$ minutes.

7 min. 30 sec. = 7$\frac{1}{2}$ minutes

7 min. 30 sec. = $\frac{\mathrm{2\; x\; 7\; +\; 1}}{2}$ minutes

7 min. 30 sec. = $\frac{\mathrm{14\; +\; 1}}{2}$ minutes

7 min. 30 sec. = $\frac{15}{2}$ minutes

Percentage = $\frac{\mathrm{Number}}{\mathrm{Total}}$ x 100%

1 hour = 60 minutes

Percentage = $\frac{15}{2}$ minutes ÷ 60 minutes x 100%

The minutes cancel each other.

Percentage = $\frac{15}{2}$ ÷ 60 x 100%

Every number/letter is being divided by 1 but not written.

60 = $\frac{60}{1}$

Percentage = $\frac{15}{2}$ ÷ $\frac{60}{1}$ x 100%

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

Percentage = $\frac{15}{2}$ x $\frac{1}{60}$ x 100%

Note: 15 divides itself 1 and 60, 4 times.

Percentage = $\frac{1}{2}$ x $\frac{1}{4}$ x 100%

Note: 4 divides itself 1 and 100, 25 times.

Percentage = $\frac{1}{2}$ x 25%

Percentage = $\frac{\mathrm{1\; x\; 25\%}}{2}$

Percentage = $\frac{\mathrm{25\%}}{2}$

Percentage = 12.5%

n - $\frac{2}{3}$ > $\frac{1}{3}$ - n

Get rid of the fracttions by multiplying both sides by the L.C.M of the denominators (3).

3 x n - 3 x $\frac{2}{3}$ > 3 x $\frac{1}{3}$ - n x 3

3n - 2 > 1 - 3n

3n + 3n > 1 + 2

6n > 3

Divide both sides by 6

n > $\frac{3}{6}$

Note: 3 divides itself 1 time and 6, 2 times.

n > $\frac{1}{2}$

∴ the truth set is {n:n > $\frac{1}{2}$ }

When the denominator is 1 joined by a number of zeros such as 10, 100, 1000, ..., 1n0s, you simply move the decimal point of the numerator to the left n times.

Where n is the number of zeros.

If the number is whole number, you can add .0 to the whole number and move it by n times to the left.

$\frac{37}{100}$ x $\frac{7}{10}$ = $\frac{\mathrm{37\; x\; 7}}{\mathrm{100\; x\; 10}}$ = $\frac{259}{1000}$

Note: when multiplying by 1 joined by a number of zeros (1n0s), the result is simply 1 joined by the total number of zeros.

For example 100 x 10 = 1joined by three zeros = 1000

259 = 259.0

$\frac{259}{1000}$ = 0.⌒2⌒5⌒9.0 = 0.259

Note: the zeros in 1000 are three (3) hence we moved the decimal point in 259.0 three (3) times to the left to obtain the 0.259

2x + 4 > -6

2x > -6 - 4

2x > -10

Divide both sides by 2

x > $\frac{-10}{2}$

x > -5