Intersection (∩) are elements you can find in both sets.

n in n(A∩B) means the number of elements. So you count how many elements are in A∩B once you are done finding the elements.

A = {1, 3, 5, 7, 9, 11, 13, 15}
B = {3, 6, 9, 12, 15}

As you can see 3, 9 and 15 are the only elements that can be seen in both the sets A and B

Hence A∩B = {3,9,15}

Since you are asked how many elements (n), you count how many elements are in {3,9,15} and I believe you've counted three

Hence n(A∩B) = 3

Since the set P is equal to the set Q, thus P = Q, it means all the elements in P is in Q and all the elements in Q is also in P. So you simply have to find which element is in P but not in Q.

P = {4, 8, 12, 16, 20}

Q = {16, 4, 12, k, 20}

Since the elements of P are arranged from the smallest to the largest, let's arrange Q too as well

⇒ Q = {4, k, 12, 16, 20}

P = {4, 8, 12, 16, 20}

As you can see the missing element (k) in Q is 8

The significant digits of a number are the digits that have meaning or contribute to the value of the number. Sometimes they are also called significant figures.

Which digits are significant?

There are some basic rules that tell you which digits in a number are significant:
• All non-zero digits are significant
• Any zeros between significant digits are also significant
• Trailing zeros to the right of a decimal point are significant

7352.4658 has 8 significant figures (7,3,5,2,4,6,5 and 8). Count from the left, three times to land on the three significant figure. If the next digit is 5 or more, you add 1 to the digit the count ends if not you keep that digit and replace the remaining digits by zeros till the decimal point.

⇒ 7352.4658 = 7350 three significant figures

Difference in mathematics means subtraction.

Product means multiplication.

Sum means addition.

Hence difference between (2d)² and 2d² implies (2d)² - 2d²

NOTE: In (2d)² the square (2) is affecting both 2 and d since they are both in a bracket but in 2d², the square (2) only affects the d.

2d means 2 x d

2d² means 2 times d square, thus 2 x d²

d² = d x d

Hence 2d² = 2 x d x d

In (2d)², you can simplify the 2d and when done square the result, thus multiply the result by itself.

Since d = 3, substitute 3 into wherever you see d and simplify

2d = 2 x d, but d = 3

⇒ 2d = 2 x 3 = 6

(2d)² = (2 x 3)² = 6² = 6 x 6 = 36

2d² = 2 x 3² = 2 x 3 x 3 = 2 x 9 = 18

(2d)² - 2d² = 36 - 18 = 18

Property: Rotations by Multiples of 90 Degrees about the Origin

For a point with coordinates (𝑥,𝑦),the following is true:

From point 4, rotation of point through 360^o about the origin is the same point, hence the image of rotating the point (-3, 5) through 360^o will be the same, (-3, 5)

First determine the type of the sequence

A sequence is an enumerated collection of objects in which repetitions are allowed and order matters.

Types of Sequences

1. Linear/Arithmetic sequences

Sequences where the difference between each term is the same. Terms add (or subtract) the same number each time.

Example

The number added or subtracted is called common difference. To find the common difference, substrate the (n-1)^th term from the nth term. The common difference is supposed to be the same for all substations if it’s a linear sequence. If it’s not the same, then the sequence is not a linear sequence.

To find the n^th term of a linear/arithmetic sequence, the formula below is used:

a_n = a₁ + (n-1)d
where a₁ = first term
n = the position you are finding
d = the common difference (a_n –a_(n-1))

For example to find the 5^th term for the sequence, 3,5,7,9,...,... we can apply the above formula.

n = 5
a₅ = a₁+(5-1)d
d = 5 -3 = 7 – 5 = 9 – 7 = 2
a₁ = first term = 3

Hence the 5^th term = 3 +(5-1) x 2 = 3 + 4 x 2 = 11

Let's determine if the given sequence is linear or not.

2, 5, 10, 17,....., ......

Common Difference = 5 – 2 = 3 or 10 -5 = 5 or 17 – 10 = 7, since the common differences are not the same, the sequence can’t be a linear sequence.

2. Geometric progressions

Geometric sequences / progressions are similar to linear sequences, but instead of adding/subtracting the same number each time, you multiply (or divide) by the same number each time.

Example

The term multiplied or divided is called common ratio. The common ratio is calculated by dividing the nth term by the (n-1)th term.

Thus for the above example 2, 4, 8, 16, the common ratio can be calculated as
4/2 = 8/4 = 16/8 = 2

The common ratio for each consecutive terms must be the same. If not, then the sequence is not geometric.

Let’s determine if the sequence in the question is geometric

2, 5, 10, 17, ....., ......

Common ratio = 5/2 or 10/5 or 17/10

As you can see the common ratios are not the same so the sequence can’t a geometric sequence.

3. Fibonacci sequences

Fibonacci sequences are sequences where each term is the sum of the previous two terms. In a Fibonacci sequence, the first two numbers must be chosen before starting the sequence. The most common sequence begins with 0 and 1. This specific sequence of numbers generated by 0 and 1 are known as the "Fibonacci numbers".

Examples

Let's determine if the sequence in the question is a Fibonacci sequence

2, 5, 10, 17, ....., ......

The 3^rd term is supposed to be 2 + 5 = 7 but its 10, hence the sequence is not a Fibonacci sequence.

4. Quadratic sequences

Quadratics are expressions in which the highest power of x is an x² term. The general term of a quadratic sequence is ax² + bx + c. For example, 2x² + 3x + 2 is a quadratic because the highest power of x is x². However, x³ – 2x² is not a quadratic, because although it contains an x² term, there is a higher power of x (the x³).

In quadratic sequences, the differences between the terms are not the same, however the difference of the differences are the same. This is the only way you can identify them.

Take the examples below. You will see that the differences between the terms (in red) are different.

However the differences of the red terms (in green) are the same.

Let's check if the sequence in the question is quadratic or not.

2, 5, 10, 17, ....., ......

2      5      10      17   

   +3   +5    +7

      +2     +2

As you can see, the sequence is a quadratic sequence. Hence we have to find the quadratic equation that generates the sequence.

Finding the formula for a quadratic sequence

The general formula for a quadratic sequence is U_n = an² + bn + c

The a, b and c can be obtained by the formula below:
2a = the second difference
3a + b = U₂ – U₁
a + b + c = U₁

Where U₂ – U₁ is the difference between the first and the second terms of the sequence.
U₁ is the first term

We can now apply the above formula to find the equation for the sequence

2, 5, 10, 17, ....., ......

2      5      10      17   

   +3   +5    +7

      +2     +2

The second difference (the one in green color) is 2.

Hence 2a = 2, solving for a, a = 2/2 = 1 ---------- equation (1)

3a + b = U₂ – U₁

U₁ (the first term) = 2

U₂ (the second term) = 5, hence the difference (U₂ – U₁) = 5 – 2 = 3

Hence 3a + b = 3, substitute a = 1 from equation 1 in this equation.

3(1) + b = 3, b = 3-3 = 0, hence b = 0

Since we know a and b, we can find c

a + b + c = U₁

U₁ (the first term) = 2

Hence 1 + 0 + c = 2, c = 2-1 = 1, hence c = 1

U_n = an² + bn + c

Hence U_n = (1)n² + (0)n + 1

1 x n² = n² , 0 x n = 0, hence

U_n = n² + 1

Now we can find the 5^th and 6^th terms

U₅ = 5² + 1 = 25 + 1 = 26

U₆ = 6² + 1 = 36 + 1 = 37

Hence the 5^th and 6^th terms are 26 and 37 respectively.

You can verify the already given terms with the obtained formula to prove the formula is correct

ROUNDING WHOLE NUMBERS

If the number you are rounding is followed by 5, 6, 7, 8, or 9, round the number up. Example: 38 rounded to the nearest ten is 40

If the number you are rounding is followed by 0, 1, 2, 3, or 4, round the number down.

Example: 33 rounded to the nearest ten is 30.

All the numbers to the right of the place you are rounding to become zeros.

ROUNDING FRACTIONS (DECIMAL POINTS)

Rounding fractions works exactly the same way as rounding whole numbers. The only difference is that instead of rounding to tens, hundreds, thousands, and so on, you round to tenths, hundredths, thousandths, and so on after the decimal point.

ROUNDING TO DECIMAL PLACE(S)

Count the numbers after the decimal point to the number of places you are rounding to. If the next number after where you land is 5 or more, you add 1 to the number where the number of the decimal places land.

In the question above we are to express 134.78 correct to the nearest tenth, hence correct to 1 decimal place. When we count from the decimal point (.) one, it lands on 7 and the next number is 8. Since 8 is more than 5, we add 1 to where our decimal point ended (7) and we get 8 instead of 7.

Hence the 134.8 is the nearest tenth of 134.78

Kofi’s ratio is 2 and Ama’s ratio is 3 from the Kofi : Ama = 2:3

Sum of ratios = 2 + 3 = 5

NOTE: 5 goes into 30, 6 times then you multiply 2 by 600 to get the 1,200

ASSIGNMENT

Calculate the amount received by Ama

SOLUTION

Ama received 1,800. Alternatively you can subtract 1200 (Amount received by Kofi) from 3000 (Total amount shared) to get Ama’s amount.

Solve for the value of x (Make x the subject)

2x – 1 = 5
2x = 5 + 1
2x = 6

Divide both sides by 2

2x/2 = 6/2

x = 3

NOTE: When negative crosses the equal to sign, it becomes positive, hence the -1 becomes +1 when it moves to the right hand side.

Solve: (1-x)÷3<4

Get rid of the denominators by multiplying both sides of the equation by L.C.M. Every number is divisible by 1.

The L.C.M of 3 and 1 is 3

Divide both sides by -1.

NOTE: When dividing x by negative, the sign must change x > -11

Alternatively you can take the x to the right hand side and the numbers to the left.

Which reads -11 is less than x

Hence if -11 is less than x means x is also greater than – 11.

Hence x > -11

Let’s test

Hence x > -11 is correct

Testing x < -11 is option too. Numbers less than -11 are -12, -13, -14, -15 etc.

when x = -12

Area of a rectangle = length x breadth

= L x B

Let L = 2 and B = ?

From the question the area is 18cm²

L x B = 18

But L = 2

2 x B = 18

Divide both sides by 2

Perimeter of a rectangle = L + L + B + B

= 2L + 2B

L = 2 and B = 9

Perimeter = 2 x 2 + 2 x 9

= 4 + 18

= 22 cm

5 red

7 black

ASSIGNMENT

What’s the propability of picking a red ball?

NOTE: When you sum all the probabilities, you are supposed to get 1

1⁄3 = 5 km

18 cm = ?

Let 18 cm = xkm and cross multiply to solve for the value of x

1⁄3 x x = 5 x 18

Applying the principles of multiplying fractions

Every number is being divided by 1 hence x = x/1

1⁄3 x x = 5 x 18

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

The L.C.M of 3 and 1 is 3, hence multiply both sides of the equation by 3

x = 270

Alternatively

You can cross multiply and solve for the value of x

x x 1 = 90 x 3

x = 270

Alternatively, you can calculate how many times 1⁄3 can go into 18 and multiply the result by 5 km

Applying the principles of dividing fractions

Thus when dividing fractions, you reverse the fraction at the right and change the division to multiplication

Now you can use the principles of multiplying fractions to simplify

Every number is being divided by 1

Check if the mapping is a linear/arithmetic sequence by finding the difference between each consecutive value of y. A linear sequence has the common difference between consecutive numbers the same

If it is linear, use the equation of linear formula to find the rule of the mapping

y₂ - y₁ = 9 - 5 = 4

y₃ - y₂ = 13 - 9 = 4

y₄ - y₃ = 17- 13 = 4

y₅ - y₄ = 21 - 17 = 4

As you can see, the rule of the mapping is a linear since the difference is the same, hence use the linear sequence formula to find the rule

U_n = U₁ + (n-1)d

Where U_n is the value at the n^th position, in the rule of mapping, the y value

U₁ is the first term, in the mapping above, 5

d is the common difference between each consecutive terms

n is the position, in the case of the above mapping, x

y = 5 + (x-1)4

y = 5 + (x-1) x 4

y = 5 + x x 4 -1x4

y = 5 + 4x-4

y = 4x+5-4

y = 4x+1

We can test with when x = 3

y = 4x+1

y = 4 x 3+1

y = 12+1 = 13

You can try the values of x to see if you will get the corresponding y value to prove the rule is correct

Find the rule of the mapping first

Check if the mapping is a linear/arithmetic sequence by finding the difference between each consecutive value of y. A linear sequence has the common difference between consecutive numbers the same

If it is linear, use the equation of linear formula to find the rule of the mapping

y₂ - y₁ = 9 - 5 = 4

y₃ - y₂ = 13 - 9 = 4

y₄ - y₃ = 17- 13 = 4

y₅ - y₄ = 21 - 17 = 4

As you can see, the rule of the mapping is a linear since the difference is the same, hence use the linear sequence formula to find the rule

U_n = U₁ + (n-1)d

Where U_n is the value at the n^th position, in the rule of mapping, the y value

U₁ is the first term, in the mapping above, 5

d is the common difference between each consecutive terms

n is the position, in the case of the above mapping, x

y = 5 + (x-1)4

y = 5 + (x-1) x 4

y = 5 + x x 4 -1x4

y = 5 + 4x-4

y = 4x+5-4

y = 4x+1

We can test with when x = 3

y = 4x+1

y = 4 x 3+1

y = 12+1 = 13

You can try the values of x to see if you will get the corresponding y value to prove the rule is correct

y = 4x+1

y = 37

37 = 4x+1

Make x the subject

37 - 1 = 4x

36 = 4x

Divide both sides by 4

36/4 = 4x/4

9 = x

Profit = Selling Price - Cost Price

Profit percent (%) is the amount of the profit expressed in terms of percentage. The profit is based on the cost price, hence the formula to profit % is (Profit/Cost Price) x 100

But profit = Selling Price - Cost Price

Hence profit % = ([Selling Price-Cost Price]/Cost Price) x 100

From the question, cost price (bought) = 1500

Profit % = 20%

Selling price = ?

Let the selling price be x

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

Every number is divisible by 1, hence 20 = 20/1

The L.C.M of 1500 and 1 is 1500, hence multiply both sides by 1500 and cancel out

100 x (x-1500) = 20 x 1500

100 x x - 100 x 1500 = 30000

100x - 150000 = 30000

100x = 30000 + 150000

100x = 180000

Divide both sides by 100

x = 1800

Properties

Alternate Angles

Alternate angles are equal

From the diagram above, angle DCB and angle OBA are alternate angles

Hence angle DCB = OBA

Sum of angles in triangle

Sum of angles in a triangle add up to 180^o

A^o + B^o + C^o = 180^o

From the diagram in the question, angle x^o + angle OBA + angle BAO = 180^o (Sum of angles in triangle)

x^o + OBA^o + 20^o = 180^o

x^o = 180^o - 20^o - OBA^o

x^o = 160^o - OBA^o

Sum of angles on a straight line

x + y = 180 (sum of angles on a straight line add up to 180)

From the diagram in the question, 120 + angle DCB = 180 (Sum of angles on a straight line)

120 + DCB = 180

DCB = 180 - 120

DCB = 60

Angle DCB = OBA, alternate angles

If angle DCB = 60, then angle OBA is also 60

From x^o = 160^o - OBA^o

OBA = 60

x = 160 - 60

x = 100

Divide both sides of the equation by πr² since its multiplying the h

Since the left hand side is equal to the right hand side, you can switch positions

ASSIGNMENT

Make r the subject of the relation V = πr²h.

SOLUTION

Since π and h are multiplying r², divide both sides by πr

Take square root of both sides so that the square root cancels the square power on top of the r

Principal is the amount invested or loan taken

Time is the duration of the investment or the time for the loan to be paid. Its mostly in years (annum). If you are given months or weeks, convert to years

Rate is the percentage rate (%) that will be used to calculate the interest earned on the investment or the interest to pay on the loan

The principal = 600

The rate = 5

Since the time was given in months, convert to years since the rate was per annum

There are 12 months in a year so divide the 8months by 12 months to get the yearly equavalent of 8 months

4 goes into 8, 2 times and 12, 3 times

Substitute the values into the simple interest formula

3 goes into 60, 20 times and itself once

The zeros cancels each other

Simple interest = 2 x 2 x 5 = 4 x 5 = 20

Priscila's age = k

Mary's age = b

Mary is 15 years older than Priscila. This means that if you add 15 years to Priscila's age, you will get Mary's age or if you subtract 15 years from Mary's age you will get Priscila's age.

Mary's age = Priscila's age + 15

b = k + 15

Now let 15 be alone at the right hand side of the equation, thus move the k to the left hand side. Its positive(+) so it becomes negative (-)

b - k = 15

A factor is a number that fits exactly into another number. For example, 2 is a factor of 4 because 2 goes into 4, 2 times

The highest common factor is the largest whole number which is shared by given numbers.

To find the highest common factor, list the factors of all the numbers and select the highest among them

Write down the multiples which gives the numbers you are looking for their factors. Pick each of the numbers from the list

You can verify if you've list all the factors, by picking the first and multiplying with the last then the second and the last but two and so on. If none is left without a pair, then you've listed all. Only numbers which multiplied by itself gives the number you are looking for its factor doesn't have a pair since multiplying itself gives the number

From the above list, 8 is the highest number which appears in all the list of factors of 24, 32 and 64

Hence the Highest Common Factor of 24, 32 and 64 is 8

Let's group them

The 3ax + 6a Group

Bring out which numbers or letters are common in both expressions

You can see 3 can go into 6, hence 3 is common

a also appeared at both sides, hence a is also common

3a(x+2)

The - x - 2 Group

-x = -1 x x

-2 = -1 x 2

Hence - x - 2 = -1 x x -1 x 2

- 1 is common

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

3ax + 6a - x - 2 = 3a(x+2)-1(x+2)

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

(x+2) is also common, hence bring it out

(x+2)(3a-1)

(x+2)(3a-1) is the same as (3a-1)(x+2), you can change the order of multiplication and the result will still be the same

You can make time the subject of the equation

Since time is dividing, you can multiply both sides of the equation by time to cancel the time at the right hand side

Time x Speed = Distance

Divide both sides by Speed

From the question, distance covered = 150 km

Speed was 18 km/h

Time = 150/18

3 goes into 150, 50 times and 18, 6 times

Time = 50/6

2 goes into 50, 25 times and 6, 3 times

Time = 25/3

3 can go into 25, 8 times remainder 1

Time = 8 1⁄3 hours, thus 8 whole number 1 over 3

You can change the fraction part of the hours to minutes by multiplying by 60 minutes. There are 60 minutes in an hour

1⁄3 hours = 1⁄3 x 60 minutes = 20 minutes

Therefore the time taken is 8 hours 20 minutes

Find the L.C.M of 1 and 4 and use it to divide the entire expression to simplify

The L.C.M of 1 and 4 is 4, hence divide the entire expression by 4

NOTE: - x - = +, hence -1 x -3 = + 3

In -1(2y-3), the -1 multiplies everything in the bracket

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

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

In 12y - 2y = 10y because they are like terms and can be simplified further

Change the decimals to whole numbers with powers.

Count how many numbers are after the decimal points and express the whole number to 10 raise to the power of the negative of that number

0.5445

There are 4 numbers after the decimal points thus 5,4,4 and 5

0.5445 = 5445 x 10^-4

0.09

There are two numbers after the decimal points, thus 0 and 9

0.09 = 9 x 10^-2

9 goes into 5445, 605 times and itself once

Apply the law of indices to simplify the powers

Since base 10 in each of the indices, we can apply the low of indices

Thus if the bases are the same and dividing, you subtract the power of the denominator from the power of the numerator

NOTE: to change the result back to decimal, you move the decimal points as many times as the negative power to the left

Every whole number has a decimal point at the end. Hence 605 = 605.0

Since the power is negative 2 (-2), we move the decimal point to the left 2 times, landing after 6

Median is the middle number or the average of the middle numbers when two numbers happen to be in the middle of an ordered numbers.

12 and x happens to be in the middle and so the median is the average (the sum of the numbers divided by 2) of 12 and x

The median is therefore calculated as:

Now calculate for the value of x

Every number is being divided by 1

We can therefore write the above equation as:

You can cross multiply to solve for the value of x

1 x (12+x) = 14 x 2

12 + x = 28

x = 28 - 12

x = 16

Expand simply means multiply and simplify

Principle

(a+b)(c+d) = a(c+d)+b(c+d)

(a-b)(c+d) = a(c+d)-b(c+d)

(7r-5)(3r+4) = 7r(3r+4)-5(3r+4)

7r(3r+4)-5(3r+4) = 7rx3r + 7rx 4 - 5x3r-5x4

7rx3r + 7rx 4 - 5x3r-5x4 = 21r² + 28r-15r-20

21r² + 28r-15r-20 = 21r² + 13r-20

NOTE: r x r = r²

a = a¹

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

r x r = r¹ x r¹ = r¹⁺¹ = r²

Thus the difference of y coordinates over the x coordinates

A(-3,5) B(7,-2)

For a point (x,y) the first coordinate is x and the second is y

y2 = -2, y1 = 5, x2 = 7, x1 = -3

The area of trapezium could be calculated by the formula below

The distance between the two parallel sides is the height.

a = 9.5 and b = 10.5

The area, A = 36

9.5 = 9+0.5

10.5 = 10+0.5

9.5 + 10.5 = 9+10+0.5+0.5

0.5 = half

0.5 + 0.5 = half + half = 1

9.5 + 10.5 = 9+10+1 = 20

36 = 10 x h

Divide both sides by 10

36/10 = h

3.6 = h

Sum of observations (numbers) = 5+6+7+x

Total number of observations (numbers) = how many numbers are in the observation. In this case 4, thus (5,6,7 and x)

Every number is divisible by 1

Re-write the equation

Simplify like terms and cross multiply to solve for the value of x

1 x (18+x) = 4 x 8

1 times any number is the same number, hence 1 x (18+x) = 18+x

4 x 8 = 32

18+x = 32

x = 32 - 18

x = 14

Amount Left = Total Amount Needed - (Amount contributed by the 12 towns + Amount contributed by the district assembly

Each (Each and every) community contributed GH₵ 25,000.00, to get the total amount contributed by the communities, you have to multiply the amount contributed by each by 12 (Total number of communities which contributed)

Amount contributed by the community = GH₵ 25,000.00 x 12 = GH₵ 300,000.00

Total amount contributed = Amount contributed by the communities + Amount contributed by the District Assembly

Total amount contributed = GH₵ 300,000.00 + GH₵ 30,500.00 = GH₵ 330,500.00

Amount left for the project = Total Amount Needed - Amount Contributed

Amount left for the project = GH₵ 375,000.00 - GH₵ 330,500.00 = GH₵ 44,500.00

Commutative Property

States that changing the order of addends does not change the sum.

For example, 3 + 5 = 5 + 3, thus three plus five equals five plus 3.

Associative Property

States that changing the grouping of addends does not change the sum.

For example, (2 + 5) + 1 = 2 + (5 + 1), thus when you group the first two terms and when you group the last two terms and the answer is the same, then its associative property

From the question,(6 + x) + 5 = 6 + (x + 5), the first two terms are group at the left and its the same as grouping the last two terms at the right

Difference Between Commutative & Associative Property

The commutative property states that it's okay to change of the order of the numbers in addition and multiplication operations because the result will be the same, no matter the order but the associative property, on the other hand, states that the result will be the same, no matter how you group the number or variables in addition/multiplication operations.

Distributive Property

To "distribute" means to divide something or give a share or part of something.

According to this property, multiplying the sum of two or more addends by a number will give the same result as multiplying each addend individually by the number and then adding the products together.

Example: a(b+c) = a x b + a x c, a(b-c) = a x b - a x c

This indicates that operand a is shared between the other two operands.

Profit = Selling Price - Cost Price

Total cost price for the 100 yams = 100 x GH₵ n = GH₵ 100n

Total selling price for the 100 yams = 100 x GH₵ m = GH₵ 100m

Profit = Selling Price - Cost Price

Profit = GH₵ 100m - GH₵ 100n

Factorize GH₵ 100 out

Profit = GH₵ 100(m - n)

12 goes into 156, 13 times

Number of boxes = 13

ALTERNATIVELY

Let x = Number of boxes

12 pencils = 1 box

156 pencils = x box

Cross multiply to solve for the value of x

12 x x = 156 x 1

Divide both sides by 12

12 goes into 156, 13 times

Number of boxes = 13

ALTERNATIVELY

Using ratio and proportion

If 12 pencils = 1 box

156 pencils = $\frac{\mathrm{1\; box\; x\; 156\; pencils}}{\mathrm{12\; pencils}}$ = 13 boxes

Note:
1. Pencils cancel each other
2. 12 divides itself 1 time and 156, 13 times

x = 4, y = 7

2xy + 3(x+y) = 2 x 4 x 7 + 3(4+7)

2xy + 3(x+y) = 56 + 3(11)

2xy + 3(x+y) = 56 + 3 x 11

2xy + 3(x+y) = 56 + 33 = 89

Applying the indices principle

The expression can therefore be rewritten as:

Cancel out

x = 8

(9x + 8)^o = (9 x 8 + 8)^o

(9x + 8)^o = (72 + 8)^o

(9x + 8)^o = (80)^o

80^o is less than 90^o, hence the angle is acute

Change the decimal to the form a x 10^-n

where a = whole number

n = how many times you have to move the decimal point to the right to get a whole number

0.725 = 725 x 10^-3

From the law of indices:

Cancel out

25 goes into 725, 29 times and 1000, 40 times

Expand the brackets first

Concept

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

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

3(5a² + 2c) = 3 x 5a² + 3 x 2c

3(5a² + 2c) = 15a² + 6c

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

NOTE: - x - = + and a x a = a²

- 2a(1 - 3a) = - 2a + 6a²

3(5a² + 2c) - 2a(1 - 3a) - 6c = 15a² + 6c - 2a + 6a² - 6c

Group like terms

3(5a² + 2c) - 2a(1 - 3a) - 6c = 15a² + 6a² - 2a + 6c - 6c

3(5a² + 2c) - 2a(1 - 3a) - 6c = 21a² - 2a + 0

3(5a² + 2c) - 2a(1 - 3a) - 6c = 21a² - 2a