320_five = 3 x 5² + 2 x 5¹ + 0 x 5⁰

Any number raised to the power 0 is 1.

Any number raised to the power 1 is the same number.

5² = 5 x 5 = 25

320_five = 3 x 25 + 2 x 5 + 0 x 1

320_five = 75 + 10 + 0

320_five = 85

P = {2, 4, 6, 8}

Q = {even counting numbers less than 12}

Q = {2, 4, 6, 8, 10}

All the members of P can be found in Q hence P is a subset (⊂) of Q

Intersection (∩) is the region both shapes join/intersect.

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.

628.7 = 628.⌒7. = 6287 x 10^-1

5.43 = 5.⌒4⌒3. = 543 x 10^-2

628.7 x 5.43 = 6287 x 10^-1 x 543 x 10^-2

628.7 x 5.43 = 6287 x 543 x 10^-1 x 10^-2

But 6287 x 543 = 3413841

628.7 x 5.43 = 3413841 x 10^-1 x 10^-2

Law of multiplication of indices

a^m x aⁿ = a^{m + m}

628.7 x 5.43 = 3413841 x 10^{-1 + -2}

628.7 x 5.43 = 3413841 x 10^{-1 - 2}

628.7 x 5.43 = 3413841 x 10^-3

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

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

3,413,841 = 3413841.0

628.7 x 5.43 = 3413.⌒8⌒4⌒1.0 = 3,413.841

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

Change the metres to centimetres.

1 m = 100 cm

2 m = 2 x 100 cm = 200 cm

Percentage = $\frac{\mathrm{30\; cm}}{\mathrm{200\; cm}}$ x 100% = 15%

Note:
1. the zeros (0) at the end of 200 and 100 cancel each other
2. 2 divides itself 1 time and 30, 15 times

The base of the pyramid determines the type of pyramid. The base is triangular hence it is for triangular pyramid.

3(6b – 9a) + 7(6a – 5b) = 3(6b – 9a) + 7(6a – 5b)

3(6b – 9a) + 7(6a – 5b) = 18b - 27a + 42a - 35b

3(6b – 9a) + 7(6a – 5b) = 18b - 35b - 27a + 42a

3(6b – 9a) + 7(6a – 5b) = -17b + 15a

3(6b – 9a) + 7(6a – 5b) = 15a - 17b

Change the fractions to decimals and compare all.

$\frac{9}{16}$ = 0.5625

$\frac{5}{8}$ = 0.625

Descending means from the highest to the lowest. The order of the decimals in descending order is: 0.625, 0.62, 0.5625 hence the order of the fractions is:

$\frac{5}{8}$, 0.62, $\frac{9}{16}$

The L.C.M is the product of each of the prime factors in their highest powers.

L.C.M = 2 x 3² x 5 x 7

The Highest Common Factor (H.C.F) is the product of the common prime factors in their lowest powers.

H.C.F of 18, 42 and 90 = 2 x 3 = 6

$\left(\begin{array}{c}4\\ 5\end{array}\right)$ + $\left(\begin{array}{c}-2\\ 2\end{array}\right)$ + $\left(\begin{array}{c}1\\ 6\end{array}\right)$ = $\left(\begin{array}{c}4\\ 5\end{array}\right)$ + $\left(\begin{array}{c}-2\\ 2\end{array}\right)$ + $\left(\begin{array}{c}1\\ 6\end{array}\right)$

$\left(\begin{array}{c}4\\ 5\end{array}\right)$ + $\left(\begin{array}{c}-2\\ 2\end{array}\right)$ + $\left(\begin{array}{c}1\\ 6\end{array}\right)$ = $\left(\begin{array}{c}\mathrm{4\; +\; -2\; +\; 1}\\ \mathrm{5\; +\; 2\; +\; 6}\end{array}\right)$

$\left(\begin{array}{c}4\\ 5\end{array}\right)$ + $\left(\begin{array}{c}-2\\ 2\end{array}\right)$ + $\left(\begin{array}{c}1\\ 6\end{array}\right)$ = $\left(\begin{array}{c}\mathrm{4\; -\; 2\; +\; 1}\\ 13\end{array}\right)$

$\left(\begin{array}{c}4\\ 5\end{array}\right)$ + $\left(\begin{array}{c}-2\\ 2\end{array}\right)$ + $\left(\begin{array}{c}1\\ 6\end{array}\right)$ = $\left(\begin{array}{c}3\\ 13\end{array}\right)$

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

Get rid of the fraction by multiplying both sides of the equation by the denominator (2)

2 x R = 2 x $\frac{1}{2}$(a + b)k

2R = (a + b)k

Divide both sides by (a + b)

k = $\frac{\mathrm{2R}}{\mathrm{(a\; +\; b)}}$

k = $\frac{\mathrm{2R}}{\mathrm{a\; +\; b}}$

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

Scale factor = -$\frac{\mathrm{6\; cm}}{\mathrm{3\; cm}}$ = -2

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

The scale factor value is the same for each corresponding sides.

$\frac{6}{3}$ = $\frac{9}{\mathrm{OM}}$

Cross multiply

OM x 6 = 3 x 9

Divide both sides by 6

OM = $\frac{\mathrm{3\; x\; 9}}{6}$ = $\frac{9}{2}$ = 4.5

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

Translation by a vector

Translation = Point + Vector

P₁ = P + V

If P(x,y) translated by the vector $\left(\begin{array}{c}\mathrm{a}\\ \mathrm{b}\end{array}\right)$

P₁ = $\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)$ + $\left(\begin{array}{c}\mathrm{a}\\ \mathrm{b}\end{array}\right)$ = $\left(\begin{array}{c}\mathrm{x\; +\; a}\\ \mathrm{y\; +\; b}\end{array}\right)$

$\left(\begin{array}{c}6\\ 3\end{array}\right)$ → $\left(\begin{array}{c}6\\ 3\end{array}\right)$ + $\left(\begin{array}{c}-4\\ -1\end{array}\right)$ = $\left(\begin{array}{c}\mathrm{6\; +\; -4}\\ \mathrm{3\; +\; -1}\end{array}\right)$ = $\left(\begin{array}{c}\mathrm{6\; -\; 4}\\ \mathrm{3\; -\; 1}\end{array}\right)$ = $\left(\begin{array}{c}2\\ 2\end{array}\right)$ = (2, 2)

The total fraction for the pocket money is 1

Remainder after spending on transport and fruits = 1 - $\frac{17}{35}$

Every whole number is being divided by 1

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

Remainder after spending on transport and fruits = $\frac{1}{1}$ - $\frac{17}{35}$

Remainder after spending on transport and fruits = $\frac{\mathrm{35\; x\; 1\; -\; 1\; x\; 17}}{35}$

Remainder after spending on transport and fruits = $\frac{\mathrm{35\; -\; 17}}{35}$

Remainder after spending on transport and fruits = $\frac{18}{35}$

of means multiplication.

He spends $\frac{5}{6}$ of the remainder on sweets

Amount spent on sweets = $\frac{5}{6}$ x the remainder

Amount spent on sweets = $\frac{5}{6}$ x $\frac{18}{35}$ = $\frac{3}{7}$

Note:
1. 6 divides itself 1 time and 18, 3 times
2. 5 divides itself 1 time and 35, 7 times

Triangle PRQ is isosceles triangle. The base angles are the same.

Angle RPQ = Angle RQP = 46°

a + angle RPQ = 180°(sum of angles on a straight line)

angle RPQ = 46°

a + 46° = 180°

a = 180° - 46°

a = 134°

Triangle PRQ is isosceles triangle. The base angles are the same.

Angle RPQ = Angle RQP = 46°

Angle RPQ + Angle RQP + b = 180°(sum of interior angles of a triangle is 180°)

46° + 46° + b = 180°

92° + b = 180°

b = 180° - 92°

b = 88°

Angle PQR + c + 29° = 180°(sum of angles on a straight line)

46° + c + 29° = 180°

c + 46° + 29° = 180°

c + 75° = 180°

c = 180° - 75°

c = 105°

p = 7, q = 5 and r = 3

p² + q – r³ = 7² + 5 - 3³

7² = 7 x 7 = 49

3³ = 3 x 3 x 3 = 27

p² + q – r³ = 49 + 5 - 27

p² + q – r³ = 27

Law of division of indices

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

Every number/variable is raised to the power 1

x = x¹

$\frac{35}{7}$ = 5

35x⁵y³ ÷ 7xy² = 5x^{5 - 1}y^{3 - 2}

35x⁵y³ ÷ 7xy² = 5x⁴y¹

35x⁵y³ ÷ 7xy² = 5x⁴y

Let x = the other number

Sum means addition.

x + y = 240

x = 240 - y

∴ the other number is 240 - y

2n + 5 = 7a

2n = 7a - 5

Divide both sides by 2

n = $\frac{\mathrm{7a\; -\; 5}}{2}$

n = $\frac{1}{2}$(7a - 5)

Bearing starts from the north.

Bearing of Q from P is angle NPQ

Angle NPE is a right angle (90°)

angle NPQ + 30° = 90°

angle NPQ = 90° - 30°

angle NPQ = 60°

The bearing of Q from P is ∴ 060°

The area is the sum of the area of the rectangle and the triangle

Area of a rectangle = Length x Breadth

Area of the rectangle = 6 cm x 8 cm = 48 cm²

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

Base of triangle = 6 cm
Height of triangle = 3 cm

Area of the triangle = $\frac{1}{2}$ x 6 cm x 3 cm

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

Area of the triangle = 3 cm x 3 cm = 9 cm²

Total area of the figure = 48 cm² + 9 cm²

Total area of the figure = 57 cm²

The median is the middle number of an ordered distribution. If there are two numbers at the middle, add the two numbers and divide the sum by 2.

There are two numbers at the middle, 5 and 7

Median = $\frac{\mathrm{5\; +\; 7}}{2}$ = $\frac{12}{2}$ = 6

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

Mean = $\frac{\mathrm{0\; +\; 1\; +\; 3\; +\; 3\; +\; 5\; +\; 7\; +\; 8\; +\; 9\; +\; 9\; +\; 9}}{10}$

Mean = $\frac{54}{10}$ = 5.4

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

Number of boys who scored 3 = 2

Number of boys who took the test = 10

P(scored 3) = $\frac{2}{10}$ = $\frac{1}{5}$

57 base 10 to base 2

$$\begin{array}{cc}\mathrm{}& 57\\ 2& \mathrm{28\; r\; 1}\\ 2& \mathrm{14\; r\; 0}\\ 2& \mathrm{7\; r\; 0}\\ 2& \mathrm{3\; r\; 1}\\ 2& \mathrm{1\; r\; 1}\\ 2& \mathrm{0\; r\; 1}\end{array}$$

You list the remainders from the bottom to the top.

57_ten = 111001_two

If 1 cm = 100,000

4 cm = $\frac{\mathrm{100,000\; x\; 4\; cm}}{\mathrm{1\; cm}}$ = 400,000

The 400, 000 is in cm. Hence convert it to metres then to kilometres.

100 cm = 1 m

400, 000 cm = $\frac{\mathrm{1\; m\; x\; 400,\; 000\; cm}}{\mathrm{100\; cm}}$ = 4,000 m

To change metres to kilometres, you divide by 1000

4,000 m = $\frac{\mathrm{4,000}}{1000}$ = 4 km

The full length is 100%

If 22% = 55 m

100% = $\frac{\mathrm{55\; m\; x\; 100\%}}{\mathrm{22\%}}$ = 5 m x 50 = 250 m

Note:
1. 2 divides 100, 50 times and 22, 11 times
2. 11 divides itself 1 time and 55, 5 times

The sum of the ratios represent the total amount shared.

Sum of ratios = 3 + 2 = 5

Adwoa's ratio = 3

If 5 = ₵6,000

3 = $\frac{\mathrm{₵6,000\; x\; 3}}{5}$ = ₵1200 x 3 = ₵3,600

Note: 5 divides itself 1 time and 60, 12 times.

5x – 8 ≤ 2x + 4

5x - 2x ≤ 4 + 8

3x ≤ 12

Divide both sides by 3

x ≤ $\frac{12}{3}$

x ≤ 4

∴ the truth set is {x ≤ 4}

Area of a circle = π x radius x radius

radius = $\frac{\mathrm{diameter}}{2}$ = $\frac{\mathrm{28\; cm}}{2}$ = 14 cm

Area of the tray = $\frac{22}{7}$ x 14 cm x 14 cm

Note: 7 divides itself 1 time and one of the 14s, 2 times

Area of the tray = 22 x 2 cm x 14 cm

Area of the tray = 616 cm²

a = $\left(\begin{array}{c}3\\ 3\end{array}\right)$ and b = $\left(\begin{array}{c}5\\ -5\end{array}\right)$

a + b = $\left(\begin{array}{c}3\\ 3\end{array}\right)$ + $\left(\begin{array}{c}5\\ -5\end{array}\right)$

a + b = $\left(\begin{array}{c}\mathrm{3\; +\; 5}\\ \mathrm{3\; +\; -5}\end{array}\right)$

a + b = $\left(\begin{array}{c}8\\ \mathrm{3\; -\; 5}\end{array}\right)$

a + b = $\left(\begin{array}{c}8\\ -2\end{array}\right)$

As you can observe, y is obtained by squaring the x

0→0² = 0 x 0 = 0

1→1² = 1 x 1 = 1

2→2² = 2 x 2 = 4

3→3² = 3 x 3 = 9

4→4² = 4 x 4 = 16

If ₵2500 = 50 oranges

₵15000 = $\frac{\mathrm{50\; oranges\; x\; ₵15000}}{\mathrm{₵2500}}$ = 50 oranges x 6 = 300 oranges

Note:
1. the zeros (0) at the end of 15000 and 2500 cancel each other
2. 25 divides itself 1 time and 150, 6 times

Number of rows = $\frac{\mathrm{Number\; of\; pineapple\; suckers}}{\mathrm{Number\; of\; pineapples\; in\; a\; row}}$

Number of rows = $\frac{1853}{17}$ = 109

The triangle is an isosceles triangle. The base angles are the same.

Angle QRP = Angle QPR = x

Angle QRP + Angle QPR + Angle PQR = 180°(sum of interior angles of a triangle)

x + x + 90 = 180

2x = 180 - 90

2x = 90

Divide both sides by 2

x = $\frac{90}{2}$ = 45

a × b × c = 1197

a = 21 and b = 3

21 × 3 × c = 1197

63 × c = 1197

Divide both sides by 63

c = $\frac{1197}{63}$ = 19