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