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.

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) =   4        

                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

S_n = 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 5^2w (400^w) = 100,000 ; Find w.

Solution

5^2w (400^w) = 100,000

(5²)^w (400^w) = 100,000

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

(25 x 400)^w = 100,000

(10000)^w = 100,000

(10⁴)^w = 10⁵   [since 10⁵ = 100,000]

10^4w = 10⁵   (Bases are alike, so they cancel out)

4w = 5

4w = 5

4       4

w = ⁵/4

TRY THIS…………………………….

NECTA 2004 QN. 4b

If 3^2t (4^t) = 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 1^st term.

S_n = 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 2^nd term.

G_n = G₁r^n-1

G₄ = G₁r^4-1

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.

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 log_aaⁿ  = nlog_aa]

= (8x1) + (-5x1) + (5x1)             [since log_aa = 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 = 124⁰ 

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