STATISTICS 1

In Biology test the following marks were recorded.

marks	10-19	20-29	30-39	40-49	50-59
No. of students	3	4	9	6	3

Calculate the mean.

Solution

In this question we are going to apply the method of assumed mean.

Here you are required to produce the frequency distribution table.

Let us take 44.5 as our assumed mean (A). Take away the assumed mean from each class mark.

Class interval	Class mark (x)	d=x-A	Frequency (f)	fx	fd
10-19	14.5	-30	3	43.5	-90
20-29	24.5	-20	4	98	-80
30-39	34.5	-10	9	310.5	-90
40-49	44.5	0	6	267	0
50-59	54.5	10	3	163.5	30
			∑f = 25	∑fx = 882.5	∑fd = -230

Mean = A +  ∑fd

                      ∑f

Mean = 44.5 + (-230)

                            25

Mean = 44.5 -  9.2

                           

         

Hence Mean =  35.3

TRY THIS………….

In Civics test the following marks were recorded.

marks	10-14	15-19	20-24	25-29	30-34
No. of students	6	10	14	12	8

SIMPLE INTEREST 1

Rugaya deposited the amount of 90,000/= in a bank for 5 years and got a profit of 22,500 /=. Find the interest rate?

Solution

I = 22,500 /=, P = 90,000/=, T = 5 years, R = ?

I  = PRT

      100

22,500  = 90,000 x R x 5

                      100

22,500  = 90,000 x R x 5

                      100

22,500  = 900 x R x 5

                      

22,500  = 4500R

           

⁵22,500  = 4500R

   4500       4500

Hence the interest rate was  5%

TRY THIS………….

Rweyendera deposited the amount of 12,000/= in a bank for 5 years and got a profit of 2100/=. Find the interest rate?

PROBABILITY 1

A bag contains 12 red balls and 8 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) = 8, n(S) = 20

P(R) = n(R)

            n(S)

1st pick = 12/20

2nd pick = 12/20 as well.

P(RR) =  12      x    12

               20            20

P(RR) =   9        

                25          

Therefore Probability of drawing a red ball is 16/81     

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

A bag contains 8 purple stones and 14 blue stones. Two stones are taken from the bag. What is the probability that they are both blue?

BODMAS 1

Evaluate 74 x (57 – 37) ÷ (845-843)

Solution

We apply BODMAS.

= 74 x (57 – 37) ÷ (845-843)

= 74 x 20 ÷ (845-843) [After dealing with 1^st brackets]

= 74 x 20 ÷ 2     [After dealing with 2^nd brackets]

= 74 x  10     [After dividing]

= 740     [After multiplying]

Hence  74 x (57 – 37) ÷ (845-843) = 740

TRY THIS……………….

Evaluate 60 x (93 – 13) ÷ (89-49)

LOGARITHMS-3

If log₆(2x + 17)=2; find x.

   Solution

Log₉(2x + 17)=2

(2x + 17)=6²           

                                            

2x + 17=36

2x =36 – 17

2x = 19

2x =   19   

2         2

x = 9.5

Hence x = 9.5

TRY THIS………………

If log₁₁(2u - 7)=2; find u

INEQUALITIES 1

PERCENTAGES 1

A man got a profit of 7300/= 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% = 7300   X  100   

            B.P.

B.P. x 10% = 730,000   x B.P.  [multiplying by B.P. both sides]

                       B.P.

B.P. x 10 = 730,000   

        

B.P. x 10¹ =   730,000     [dividing by 10 both sides]  

  ₁10                   10

B.P. =73,000

                      

Hence Buying Price was 73,000/=

TRY THIS………………

A man got a profit of 16,200/= after selling an item. Find the buying price if the percentage profit was 10%.

EXPAND 2

Expand 8w(5w + 4 - w)

Solution

= 8w(5w + 4 - w)

= (8w x 5w) + (8w x 4) - (8w x w)

= 40w² + 32w - 8w²

= [40w² - 8w²] + 32w collecting like terms

= 32w² + 32w answer

TRY THIS………..

Expand 4a(5a + 14 - a)

EXPONENTIALS 1

If 5^2w (400^w) = 1000 ; Find w.

Solution

5^2w (400^w) = 1000

(5²)^w (400^w) = 1000

(25)^w (400^w) = 1000

(25 x 400)^w = 1000

(10000)^w = 1000

(10⁴)^w = 10³   [since 10³ = 1000]

10^4w = 10³   (Bases are alike, so they cancel out)

4w = 3

4w = 3

4       4

w = ³/4

TRY THIS…………………………….

NECTA 2004 QN. 4b

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

VARIATIONS-1

x is inversely proportional to y. x=12 while y=5. Find y when x is 3.

Solution

x ⍺ k

      y

x = k

      y

12 = k

        5

k = 12 x 5

k = 60

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

k = 60, y= ?, x = 3.

x = k

      y

3 = 60

       y

y x 3 = 60  x  y   (multiply by y on both sides)

                y

  

3y = 60

3      3

y = 20  .

TRY THIS…………….

x is inversely proportional to y. x=20 while y=4. Find y when x is 2.

LOGARITHMS-2

If log_x 2401 – log₃729 = -4; find x

solution

log_x 2401 – log₃729 = -2

log_x 2401 – log₃3⁶ = -2

log_x 2401 – 6log₃3 = -2

log_x 2401 – 6 x 1 = -2

log_x 2401 – 6 = -2

log_x 2401 = -2 + 6

log_x 2401 = 4 

2401 = x⁴        2401=7x7x7x7=7⁴ by prime factors

7⁴ = x⁴      Powers cancel out

x = 7

Hence x = 7.

TRY THIS………….

If log_m2048 – log₆216 = 8; find m

EXPAND 1

Expand 8w²(5w – 7)

solution

= 8w²(5w – 7)

= (8w² x 5w) – (8w² x 7)

= 40w³ – 56w² answer

TRY THIS………..

Expand 4a²(7a+ 30)

VECTORS 1

Write (9,6) in terms of i and j.

Solution

= (9,6)

= (9,0) +  (0,6)

= 9(1,0) +  6(0,1)

= 9i + 6j   [since (1,0)=i and (0,1=j)

Hence (9,6) = 9i +  6j  

TRY THIS……………………..

Write (11,16) in terms of i and j.

FUNCTIONS 1

If F(x) = 8x + 20; Find F^-1(x).

Solution

HINT: F^-1(x) means inverse.

PROCEDURE:

Make x the subject and then interchange x and y variables.

Let y=F(x)

So,  y= 8x + 20

y – 20 = 8x

y – 20  = 8x

   8          8

y – 20   = x

   8

x  =   y – 20     after rearranging

           8

 y^-1 = x – 20     after interchanging x and y variables.

             8

F^-1(x) = x – 20     after interchanging x and y variables.

                8

Hence, F^-1(x) = x – 20     

                              8

TRY THIS……………………………

If F(x) = 2x - 26; Find F^-1(x).

PERFECT SQUARES 1

Evaluate 660² – 340²

Solution

660² – 340² = (660 + 340)( 660 - 340)

                   = (1000)( 320)

                   = 320000

Hence 660² – 340² = 320000

TRY THIS……………………

NECTA 2003 QN. 9a

Evaluate 365² – 135²