ALGEBRA K4

  

If 48 + 24 =56; Find a.

            a

 

Solution

 

48 + 88 =56

        a

 

   88 = 56 - 48

    a

 

   88 =  8

    a

 

 

88 = 8a   after multiplying by a both sides.

 

88  = 8a

8       8

 

11 = a

 

Hence a=11.

 

 

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

 

 

22 + 72 = 30    ; find c.

         c

 

 

 

ALGEBRA K3

 

The sum of four consecutive numbers is 606. 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	1606

 

Then, n + (n+1) + (n+2) + (n+3) = 1606

 

4n + 1+2+3= 1606

 

4n + 6 = 1606

 

4n = 1606 – 6

 

4n = 1600 

 

4n = 1600

 4        4

 

n = 400

 

3rd number = n +2

 

                   = 400 + 2

 

                   = 402

 

Hence the 3rd number is 402

 

TRY THIS………………….. 

 

The sum of five consecutive numbers is 610. Find the largest number.

 

EXPONENTIALS K5

 Simplify (7a)² + 31a². 

  

Solution

 

= (7a)² + 31a²    

 

= 7².a² + 31a²              Since (mn)² =m² . n²

 

= 49a² + 31a²

 

= 80a²

 

 

TRY THIS………………

 

  

Simplify (6a)² + 50a².   

EXPONENTIALS K4

 

Find x if 128^x = 0.125; find x.

 

Solution

 

128^x = 0.125

 

(2⁷)^x = 0.125          [since 128=2⁷ ]

 

2^7x = 0.125      

    

2^7x = ¹²⁵/₁₀₀₀       [converting 0.125 into fraction]  

     

2^7x = ¹/₈             [after simplifying]

      

2^7x = 8^-1             [since ¹/_a=a^-1]

 

2^7x = (2³)^-1

 

2^7x = 2^{3 x-1}   

        

2^7x = 2^-3            [since (a^c)^d =a^cd]

 

2^7x = 2^-3            [same bases cancel out]

 

7x = -3         

 

7x = -3

7       7

 

x= ^-3/7

 

Hence x= ^-3/7  .

 

TRY THIS…………………….

 

 

Find x if 64^4m = 0. 5; find m.

SETS K2

 

In a certain school, 140 students take either Chemistry or physics. 160 take chemistry and 100 take both subjects. Find those who take physics.

 

Solution

 

In most cases, OR stands for union whereas AND/BOTH, stands for intersection.

 

Let Chemistry = n(C),

      physics = n(P).

 

n(C)=160 ,

n(P)=?

 

n(CuP)= 140,

n(CnP)=100

 

Now,the formula

 

n(CuP)= n(C) + n(P) – n(CnP)

 

140 =  160 + n(P)  – 100

 

140 =  160 – 100 + n(P) 

 

140 =  60+ n(P) 

 

140– 60  =  n(P)

 

n(P)=80

 

Hence those taking  physics are 80.  

 

TRY THIS………….

 

In a certain school, 185 students take either Chemistry or French. 160 take chemistry and 130 take both subjects. Find those who take French.

VARIATION K3

 x is directly proportional to y. x=32 while y=4. Find y when x is 152.

 

Solution

 

x ⍺ y

x = ky

 

32 = k x 4

 

32 = 4k    [dividing by 4 both sides]

4      4

 

k = 8

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

 

k = 8, y= ? x = 152.

 

x = ky

 

152 = 8 x y

 

152 = 8y

8       8

 

y = 19  .

 

TRY THIS…………….

 

x is directly proportional to y. x=40 while y=4. Find y when x is 180.

EXPONENTIALS K3

 If 8^10+2a = 32^a-2; find a

Solution

8^10+2a  = 32^a-2 

(2³)^10+2a  = (2⁵)^a-2 

2^30+6a = 2^5(a-2) 

2^30+6a  = 2^5a-10

2^30+6a   = 2^5a-10      [Same bases both sides, cancel out]

30+6a = 5a – 10

30 + 6a+10 = 5a        [after shifting 10 to the left]

40 + 6a = 5a        

40 = 5a-6a        [after shifting 6a to the left]

40 =    -1a

-1       -1

 

-40 = a

Hence a=-40

TRY THIS……………………….

 

If 4^7+2y = 64^y-2; find y

QUADRATICS K1

 

What must be added to x²  + 60x to make the expression a perfect square?

Solution

a=1, b=60,c=?

b² = 4ac

(60)² = 4 x 1 x c

3600 = 4c

3600 = 4c

 4        4

 

900 = c

Hence number to be added is 900

 

TRY THIS……………………

What must be added to x²  + 24x to make the expression a perfect square?  

LOGARITHMS K4

 If log₃(10x - 29)=4; find x.

 

Solution

 

log₃(10x - 29)=4

 

 (10x - 29)=3⁴       

                                            

 10x - 29=81

 

10x = 81 + 29

 

10x =  110

 

10x =  110¹¹   

10      10

 

x = 11

 

Hence x = 11

 

TRY THIS…………….

 

If log₃(5u + 16)=4; find u

VARIATION K2

 x is inversely proportional to y. x=12 while y=20. Find y when x is 15.

 

Solution

 

x ⍺ k

     y

 

x = k

     y

 

12 = k

      20

 

k = 12 x 20

 

k = 240

 

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

 

k = 240, y= ?, x = 15.

 

x = k

     y

 

15 = 240

        y

 

y x 15 = 240  x  y   (multiply by y on both sides)

              y

 

15y = 240

15     15

 

y = 16 .

 

TRY THIS…………….

 

x is inversely proportional to y. x=12 while y=40. Find y when x is 15.