ACCOUNTS 1

Prepare a profit and loss account for the year ended 31^ST May, 2015 from the following balances:

Transport     sh. 69,000

Advertising Sh. 60,000

General expenses sh 35,000

Sundry expenses sh 4,000

Interest received sh. 32,000

Carriage sh 42,000

Gross profit  sh 122,000

Solution

Dr      PROFIT AND LOSS ACCOUNT FOR THE YEAR ENDED ON 31TH, MAY-2011               Cr
Particulars	Folio	Amount	Particulars	Folio	Amount
Transport		69,000	Gross profit	b/d	122,000
Advertising		60,000	Interest received		32,000
General expenses		35,000	Net Loss		56,000
Sundry expenses		4,000
Carriage		42,000
		210,000			210,000

TRY THIS………..

Prepare a profit and loss account for the year ended 31^ST September, 2016 from the following balances:

Transport -----------------------sh. 76,000

Advertising ---------------------Sh. 58,000

General expenses------------- sh 37,000

Sundry expenses ----------------sh 9,000

Interest received------------- sh. 32,000

Carriage -------------------------sh 23,000

Gross profit--------------------sh 116,000

EARTH AS SPHERE 1

Change 700 nautical miles into kilometers.

Solution

1 nautical mile= 1.852 km

700 nautical mile= ?

 = 1.852 x 700

          1

=1296.4

=1296.4 to 1 d.p

Hence 700 nautical miles= 1296km.

TRY THIS…………….

Change 60 nautical miles into kilometers

PROBABILITY 1

A bag contains 4 red balls and 6 blue balls. Two balls are taken from the bag. What is the probability that they are both red?

Solution

(This is a problem with replacement)

n(R) = 4, n(B) = 6, n(S) = 10

P(R) = n(R)

            n(S)

P(R) = 4

          10

1st pick = ⁴/10

2nd pick = ⁴/10 as well.

P(RR) =  4      x    4

             10           10

P(RR) =   ¹⁶ /100 = ⁴/25       

Therefore Probability of drawing a red ball is 4/25     

TRY THIS ..............

A bag contains 9 purple stones and 11 blue stones. Two stones are taken from the bag. What is the probability that they are both purple?

SEQUENCE&SERIES 1

The 1st term of arithmetic progression is 63 and the common difference is 40. Find the nth term

solution

A1=63, d= 40, n=?

An =A1 + (n-1) d

An=63 + (n-1)40 

An=63 + 40n -40

An=40n + 63-40

An=40n + 23

Hence the nth term is An=40n + 23  

TRY THIS………………….. 

The 1st term of arithmetic progression is 144 and the common difference is 77. Find the nth term

LOGARITHMS 5

Evaluate Log1,000,000,000 +  log 0.01 + log₃243

Solution

= Log1,000,000,000 +  log 0.01 + log₃243

= Log10⁹ +  log 10^-2 + log₃3^-5

= 9Log10 +  (-2log 10) + (-5log₃3)

= (9x1) + (-2x1) + (-5x1)

= 9 + (-2) + (-5)

= 2

Hence Log1,000,000,000 +  log 0.01 + log₃243 = 2

TRY THIS……………..

Evaluate Log100,000,000 +  log 0.0001 + log₃81

EXPONENTIALS 3

Simplify (6a)² - 3a². 

 

Solution

= (6a)² - 11a²    

= 6².a² - 11a²               Since (mn)² =m² . n²

= 36a² - 11a²   {since the square number of 6 is 36}

= 25a²

TRY THIS………………

Simplify (8m)² – 45m².   

WORD PROBLEMS 1

The sum of four consecutive numbers is 1358. Find the 3rd number.

Solution

Let the numbers be as shown in the table below

1st number	2nd number	3rd number	4th number	TOTAL
n	n+1	n+2	n+3	1358

Then, n + (n+1) + (n+2) + (n+3) = 1358

4n + 1+2+3= 1358

4n + 6 = 1358

4n = 1358 – 6

4n = 1352 

4n = 1352

 4        4

n = 338

3rd number = n +2

                   = 338 + 2

                   = 340

Hence the 3rd number is 340

TRY THIS………………….. 

The sum of four consecutive numbers is 426. Find the largest number.  

EXPAND 3

Expand (y – 4)(y – 11)

Solution

= (y – 4) (y – 11)

= y (y – 11) – 4 (y – 11)

= y² – 11y – 4y + 44 

= y² – 15y + 44 answer

TRY THIS………..

Expand (a – 7) (a – 10)

LOGARITHMS 4

If log 2= 0.3010; find the value of log 400,000 without using tables.

solution

log400,000=log(4 x 100,000)

=log4 + log100,000

=log2² + log10⁵

=2log2 + 5log10

=2(0.3010) + (5 x 1)

=(0.6020) +  5

=5. 6020

Hence log400,000=5. 6020

TRY THIS……………

If log 2= 0.3010; find the value of log 4,000,000 without using tables.

FACTORIZE 1

Factorize 625x² - 169y²

Solution

we use difference of two squares a² – b² = (a - b)(a + b)

625x²- 169y²  = 25²x² - 13²y²

                   = (25x)² - (13y)²

                   = (25x - 13y)(25x + 13y)

Hence 625x² - 169y² = (25x - 13y)(25x + 13y)

TRY THIS………………………….

factorize 121c²- 25d²

EXPONENTIALS 2

If 5^2w (40^w) = 10^w-40 ; Find w.

Solution

5^2w (40^w) = 10^w-40

(5²)^w (40^w) = 10^w-40

(25)^w (40^w) = 10^w-40

(25 x 40)^w = 10^w-40

(1000)^w = 10^w-40

(10³)^w = 10^w-40

10^3w = 10^w-40  (Bases are alike, so they cancel out)

3w = w-40

3w - w = -40

2w = -40

2w = -40²⁰

2       2

w = -20

TRY THIS…………………………….

If 3^2t (4^t) = 6^t-10 ; Find t.

STANDARD FORM 1

Evaluate the following giving your answer in standard form.

982.7 x 10^-9

   5 x 10^-30

Solution

=  982.7 x 10^-9

       5 x 10^-30

=  982.7  x   10^-9

       5         10^-30

=  196.54 x 10^{-9 –(-30)}

=  196.54 x 10^{-9 + 30}    

=  196.54 x 10²¹      then we change 196.54 into standard form as well

= 1.9654 x 10² x 10²¹   [when we have exponents with same base, we add the powers.]

 

= 1.9654 x 10²³

= 1.97 x 10²³ [correct to 2 d.p.]

Hence   982.7 x 10^-9   = 1.97 x 10²³

                5 x 10^-20

TRY THIS………………..

Evaluate the following giving your answer in standard form.

0.001765 x 10^-9

   2 x 10^-30

GRADIENT 1

A line passes through (2a, 7) and (3,5a). If its slope is 10, find a.

Solution

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

Slope(m) =    y2 – y1

                     x2 – x1

               10 =  5a – 7

                        3 – 2a

               10 =    5a – 7     [cross multiplying]

               1         3 – 2a

10(3 – 2a) = 5a – 7

30 – 20a = 5a – 7

30 – 8a + 7 = 5a

30 + 7 = 5a+ 8a

37 = 13a

37 = 13a

13    13

a= ³⁷/13

Hence a= ³⁷/13

TRY THIS………………….. 

A line passes through (2a, 6) and (3,5a). If its slope is 12, find a.

EXPAND 2

Expand (y – 7) (3y – 11)

Solution

= (y – 7) (3y – 11)

= y (3y – 11) – 7 (3y – 11)

= 3y² – 11y – 21y + 77 

= 3y² – 32y + 77 answer

TRY THIS………..

Expand (a – 8) (3a – 5)

SETS 1

If n(A)= 55 , n(B)= 90 and n(AnB)=30, find n(AuB).

Solution

n(AuB) = n(A) + n(B) - n(AnB)

            = 55 + 90 – 30

            = 145 – 30

            = 115

Hence n(AuB) = 115 answer

TRY THIS………….

If n(A)= 84 , n(B)= 95 and n(AnB)=77, find n(AuB).

INEQUALITIES 1

If 10x + 21 ≥ 7 + 3x ≥ 9x – 59; find x.

Solution

10x + 42 ≥ 7 + 3x and  7 + 3x ≥ 9x - 59

10x – 3x ≤ 7 - 42 and  7 +59 ≤ 9x - 3x

7x ≤ -35 and  66 ≤ 6x 

7x ≤ -35  and  66 ≤ 6x   [after dividing by 7 and 6 both sides respectively]

7        7            6       6

x ≤ -5 and  11 ≤ x

x ≤ -5 and  x  ≥ 11

TRY THIS…………….

If 10x - 49 ≥ 7 + 3x ≥ 9x – 53; find x

EXPAND 1

Expand 10w(15w – 3)

Solution

= 10w(15w – 3)

= (150w x 5w) – (10w x 3)

= 750w² – 30w answer

TRY THIS………..

Expand 4y(9y+ 10)

ALGEBRA 5

ALGEBRA 4

LOGARITHMS 3

If log₉(4x + 17)=2; find x.

   Solution

Log₉(4x + 17)=2

(4x + 17)=9²           

                                            

4x + 17=81

4x =81 – 17

4x = 64

4x =   64   

4        4

x = 16

Hence x = 16

TRY THIS………………

If log₇(2t - 25)=2; find t

RADICALS 1

VARIATIONS 2

x is inversely proportional to y. x=12 while y=5. Find y when x is 4.

Solution

x ⍺ k

      y

x = k

      y

12 = k

        5

k = 12 x 5

k = 60

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

k = 60, y= ?, x = 4.

x = k

      y

4 = 60

        y

y x 4 = 60  x  y   (multiply by y on both sides)

              y

4y = 60

4      4

y = 15  .

TRY THIS…………….

x is inversely proportional to y. x=200 while y=4. Find y when x is 8.

LOGARITHMS 2

If Log_ax = 0.74, find Log_a(¹/x)

Solution

= Log_a(¹/x)

= Log_ax^-1

= -1 x Log_ax

= -1 x 0.74

= -0.74

TRY THIS………..

If Log_ab= 0.41, find Log_a(¹/b)

POLYGONS 2

A regular polygon has 37 sides. Find the total angles of that polygon.

Solution

n = 37

Total angles = (n - 2)180⁰

                    = (37 – 2)180⁰

                    = 35 x 180⁰

                    = 6300⁰

Total angles = 6300⁰

TRY THIS……………………….

A regular polygon has 67 sides. Find the total angles of that polygon.

PERCENT PROFIT 1

A man got a profit of 1500/= after selling an item. Find the buying price if the percentage profit was 10%.

Solution

%’ge profit = Profit   X  100    where B. P. represents Buying Price.

                         B.P

10% = 1500   X  100   

            B.P.

10% = 150000   

               B.P.

B.P. x 10% = 150000   x B.P.

                       B.P.

B.P. x 10 = 150,000   

        

B.P. x 10¹ =   150000    [dividing by 10 both sides]

  ₁10                   10

B.P. = 15,000

                      

Hence Buying Price was 15,000/=

TRY THIS………………

A man got a profit of 2400/= after selling an item. Find the buying price if the percentage profit was 30%.

ALGEBRA 3

If  x+3y = 1            find  ^x/y             

    2x-3y    3

Solution

x+3y = 1           

2x-3y   3

1(2x-3y) = 3(x+3y)           [after cross multiplying]

2x-3y = 3x+9y

2x - 3x = 9y + 3y

-x = 12y

¹-x = 12y

₁-1    -1 ₁

 x = -12y

     

x = -12y

y       y

x = -12

y    

Hence ^x/_y = -12  answer.          

TRY THIS......................

If  x+2y = 1            find  ^x/y             

    x- 2y     5