Monday 30 April 2018

LOGARITHMS 4


If log 2= 0.3010; find the value of log 160,000 without using tables.

solution

log160,000 = log(16 x 10,000)

=log16 + log10,000

=log24 + log104

=4log2 + 4log10

=4(0.3010) + (4 x 1)

= 1.2040+  4

=5.2040

Hence log 1,600,000 =5.2040


TRY THIS……………


If log 2= 0.3010; find the value of log 800,000,000 without using tables.

EXPAND 1


Expand 8w(5w + 7 - w)

Solution

= 8w(5w + 7 - w)

= (8w x 5w) + (8w x 7) - (8w x w)

= 40w2 + 56w - 8w2

= [40w2 - 8w2] + 56w collecting like terms

= 32w2 + 56w answer


TRY THIS………..


Expand 5a(8a + 14 - a)

EXPONENTIALS 3



If 2(6m-7) x 3(n+8) = (346) x (259); Find m and n.

solution

equating equal bases;

2(6m-7) =  259

2(6m-7)259

6m - 7 = 59

6m  = 59 + 7

6m  = 66

16m  =  6611
16         61

m =  11

again for n we equate equal bases.

3(n+8) = 346

3(n+8) = 346

n + 8 = 46

n = 46 - 8

n = 38

Hence m = 11 and n = 38


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


If 5(2m-17) x 13(5n+40) = (1370) x (543); Find m and n.

Sunday 29 April 2018

SIMPLE INTEREST 1



Ruta deposited the amount of 90,000/= in a bank for 5 years and got a profit of 27,000 /=. Find the interest rate?

Solution

I = 27,000 /=, P = 90,000/=, T = 5 years, R = ?

I  = PRT
      100

27,000  = 90,000 x R x 5
                      100

27,000  = 90,000 x R x 5
                      100

27,000  = 900 x R x 5
                      
27,000  = 4500R
           
627,000= 4500R
   4500       4500


Hence the interest rate was  6%

TRY THIS………….

Ruta deposited the amount of 90,000/= in a bank for 5 years and got a profit of 31,500/=. Find the interest rate?

PROBABILITY 1


A bag contains 12 red balls and 18 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) = 12, n(B) = 18, n(S) = 30

P(R) = n(R)
            n(S)

1st pick = 12/30
2nd pick = 12/30 as well.


P(RR) =  12      x    12
               30            30

P(RR) =         
                25          

Therefore Probability of drawing a red ball is 4/25     


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

A bag contains 16 purple stones and 14 blue stones. Two stones are taken from the bag. What is the probability that they are both blue?

SUM OF A.P 1


Find the sum of the 1st 20 terms of the following arithmetic progression: 5+11+17+…………………

Solution

d=6, A1 = 5,n=20

Sn = n[2A1 + (n-1)d]
       2

S20 = 20[2(5) + (20-1)d]
         2

S20 = 10[10 + 17d]
 
S20 = 10[10 + 17(6)]       since d=6

S20 = 10[10 + 102]  
         
S20 = 10 x 112 
          
S20 = 1120


Therefore the sum of 1st 20 terms is 1120.

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

Find the sum of the 1st 15 terms of the following arithmetic progression: 6+10+14+…………………

EXPONENTIALS 2


If a2 = 3; find the value of 10a8.

solution

10a8 = 10 (a2)4.

10a8 = 10 (3)4.   [ since a2 = 3]

10a8 = 10[3 x 3 x 3 x 3]

10a8 = 10[81]

10a8 = 810



10a8 = 810  answer


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



If 10a2 = 5; find the value of a4.

VARIATION 1


x is inversely proportional to y. x=12 while y=50. Find y when x is 3.

Solution

x k
      y

x = k
      y

12 = k
       50

k = 12 x 50

k = 600

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

k = 600, y= ?, x = 3.

x = k
      y

3 = 600
         y

y x 3 = 600  x  y   (multiply by y on both sides)
                y


3y = 600
3      3

y = 200  .


TRY THIS…………….


x is inversely proportional to y. x=200 while y=4. Find y when x is 2.

PERCENTAGES 1


A man got a profit of 7000/= 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% = 7000   X  100   
             B.P.

B.P. x 10 = 700,000   
         
B.P. x  101 =   700,000   
  110                  10

B.P. = 70,000
                      
Hence Buying Price was 70,000/=

TRY THIS………………….


A man got a profit of 8000/= after selling an item. Find the buying price if the percentage profit was 20%.

EXPONENTIALS 1


If 52w (400w) = 100,000 ; Find w.

Solution

52w (400w) = 100,000

(52)w (400w) = 100,000

(25)w (400w) = 100,000

(25 x 400)w = 100,000

(10000)w = 100,000

(104)w = 105   [since 105 = 100,000]

104w = 105   (Bases are alike, so they cancel out)

4w = 5

4w = 5
4       4

w = 5/4

TRY THIS…………………………….


NECTA 2004 QN. 4b

If 32t (4t) = 6 ; Find t.



G.P 1


The sum of the 1st five terms of a geometrical progression is 484. If the common ratio is 3 find the 4th term.

solution

S5 = 484, G1 = ?, r=3, n=5, A2=?

We use the summation formula to find the 1st term.
Sn = G1(rn-1)/r-1
               

S5 = G1(r5-1)/r-1
               

484= G1(35-1)/3-1
               

484= G1(243-1)
                  2

2 x 484= G1(242)
                  
968= 242 G1

4 968   =  G1
  242

G1 = 4

Now we solve for the 2nd term.

Gn = G1rn-1

G4 = G1r4-1

G4= G1r 3

G4 = 4 x (3)3

     = 4 x 27

     = 81


Hence the 4th term is 81.

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


The sum of the 1st five terms of a geometrical progression is 484. If the common ratio is 3 find the 3rd term.

Saturday 28 April 2018

SETS 2



If n(A)= 90 , n(AuB) = 140 and n(AnB)=30, find n(B)

Solution

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

  140     = 90 + n(B)  – 30

  140     = 90 -30 + n(B) 

  140     = 60 + n(B) 

  140 - 60        = n(B)

  80        = n(B)

Hence n(B) = 80 answer


TRY THIS…………………………….


If n(A)= 70 , n(AuB) = 137 and n(AnB) = 35, find n(B).

LOGARITHMS 3


Evaluate Log100,000,000 +  log 0.00001 + log3243

Solution

= Log100,000,000  +  log 0.00001 + log3243

= Log108 +  log 10-5 + log335

= 8Log10 +  (-5log 10) + (5log33)    [since logaa= nlogaa]

= (8x1) + (-5x1) + (5x1)             [since logaa = 1]

= 8 + (-5) + (5)

= 8

Hence Log100,000,000 +  log 0.00001 + log3243 = 7

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



Evaluate Log10,000,000,000 +  log 0.01 + log32187

POLYGONS 1


An interior angle of a regular polygon is 680 greater than an exterior angle. Find the interior angle.

Solution

Let i = interior angle, e = exterior angle.

Now i  + e=1800…………………(1)

But i = e+680 …………………(2)

Substitute (2) in (1) above.

e+680   + e=1800

e+ e+680   =1800

2e+ 680   =1800

2e=1800 - 680   

2e=1120

2e=1120          dividing by 2 both sides.
2      2

e = 560

But i  + e=1800…………………(1)

i  + 560=1800.

i  =1800 - 560

i = 1240

Hence i = 124


TRY THIS………………………   


An interior angle of a regular polygon is 840 greater than an exterior angle. Find the interior angle.

ALGEBRA 1


48 + 24 =56
        a

Solution

48 + 88 =56
        a

   88  = 56 - 48
   a

   88  =  8
    a

1 a  x  88  =  8 x a
        1 a

88 = 8a

88  = 8a
8       8

11 = a

Hence a=11.

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

22 + 72 =30    ; find c.
         c



FACTORIZE 1


Factorize 81-9m2

Solution

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

81-9m2 = 92 - 32m2       
             = 92 - (3m)2     
             = (9 - 3m)(9 + 3m)

Hence 81 - 9m2 = (9 - 3m)(9 + 3m)

TRY THIS…………….


Factorize 225 - 16c2