APPROXIMATIONS-1

Estimate the value of 87 x 71.

solution

round off to tens;

87 ≈ 90  [approximating to 90]

71 ≈ 70  [approximating to 70]

 = 90 x 70

 = 6300

Hence 87 x 71 ≈ 6300

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

Estimate the value of 61 x 69.[ans. 4200]

ARITHMETIC PROGRESSION-1

The 1st term of an A.P. is 78 and the common difference is 16. Find the 10th term.

Solution

A₁= 78, d = 20

A_n = A₁ + (n-1)d

A₁₀ = A₁ + (10-1)d

A₁₀ = A₁ + 9d

A₁₀ = 78 + (9x16) [ after substituting A₁= 78, d = 16 as given above]

∴ A₁₀ = 78 + 144

A₁₀ = 222

Hence the 10th term is 222.

TRY THIS……………

The 1st term of an A.P. is 30 and the common difference is 43. Find the 9th term.

ALGEBRA-1

Divide 68ax + 22ay - 10az by 2a.

solution

= 68ax + 22ay - 10az

              2a

= ³⁴68ax  +  ¹¹22ay  -  ⁵10az       [cancelling by 2a throughout]

     2a             2a           2a

= 34x + 11y - 5z

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

Divide 80am - 32an + 52ap by 4a.

FUNCTIONS-1

If F(x) = 2x + 6; 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= 2x + 6

y – 6 = 2x

y – 6  = 2x

  2        2

y – 6   = x

   2

x  =   y – 6     after rearranging

            2

 y^-1 = x – 6     after interchanging x and y variables.

            2

F^-1(x) = x – 6     after interchanging x and y variables.

               2

Hence, F^-1(x) = x – 6     

                            2

TRY THIS……………………………

If F(x) = 3x + 10; Find F^-1(x).

LOGARITHMS-1

If log 2= 0.3010; find the value of log 50,000,000 without using tables.

solution

log 50,000,000 = log(100,000,000÷ 2)

=log100,000,000- log2

=log10⁸ – log2

=8log10 –log2

=(8x1) – (0.3010)

=8– (0.3010)

= 7.699

Hence log5,000,000 = 7.699

TRY THIS……………

If log 2= 0.3010; find the value of log 500,000without using tables.

SETS - 1

If n(A)= 176 , n(B)= 164 and n(AnB)=150, find n(AuB).

Solution

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

             = 176 + 164 – 150

             = 340 – 150

             = 190

Hence n(AuB) = 190 answer

TRY THIS…………………………….

If n(A)= 116 , n(B)= 124 and n(AnB)=80, find n(AuB).

WORD PROBLEMS-1

Pedro sells 1 litre of milk for sh700/=. How many litres of milk should Pedro sell to get 42,700/=?

Solution

1litre = 700/=

  ?      = 42,700/=

after cross multiplication;

= 42,700x  1

         700

= 42700x  1

         700

= 427

     7

= 61

Hence Pedro should sell 61 litres.

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

Iniesta sells 1 litre of milk for sh200. How many litres of milk should Iniesta sell to get 72,600/= ?

BINARY OPERATIONS-1

If p*k = pk + p – 2k; find 4*2.

Solution

p*k = (pxk) + p – (2xk)   [rewriting the given expression more clearly]

4*2 = (4x2) + 4 – (2x2)   [substituting 4 for p and 2 for k]

4*2 = 8 + 4 – 4

4*2 = 12 – 4   [we add first before subtracting]

4*2 = 8.

Hence   4*2 = 8.

 TRY THIS........

If p*k = p + k – pk; find 6*2.

PRIME FACTORS-1

Write 440 as a product of prime factors.

Solution

2	440
2	220
2	110
5	55
11	11
	1

 Then, 440 = 2 x 2 x 2 x 5 x 11

Hence   440 = 2³ x 5 x 11.

TRY THIS........

Write 560 as a product of prime factors.

RATIOS-1

Divide 300 in the ratio 7:3.

solution

1st add the ratios. 7 +3 = 10

then;   7  x 300 = 210

           10

also;   3  x 300 = 90

          10

Hence the ratio parts are 210 and 90.

TRY THIS........

Divide 3200 in the ratio 5:3.

EXPONENTIALS - 1

If 2^(6m-7) x 3⁽ⁿ⁺⁹⁾ = (3⁴⁴) x (2⁹³); Find m and n.

solution

equating equal bases;

2^(6m-7) =  2⁹³

2^(6m-7) =  2⁹³

6m - 7 = 93

6m  = 93 + 7

6m  = 100

¹6m  = 100⁵⁰

₁6           6₃

m = ⁵⁰/₃

again for n we equate equal bases.

3⁽ⁿ⁺⁹⁾ = 3⁴⁴

3⁽ⁿ⁺⁹⁾ = 3⁴⁴

n + 9 = 44

n = 44 - 9

n = 35

Hence m = 20 and n = 35

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

If 8^(2m-9) x 11⁽ⁿ⁺⁴⁰⁾ = (11⁶⁰) x (8²³); Find m and n.