RATIOS D1

The areas of two circles are in the ratio of 25:4 . Calculate the radius of the smaller circle when the radius of larger circle is 50 cm. 

solution

let R = radius of larger circle

let r = radius of smaller circle

R= 50cm, r = ? Ratio = 25:4.

For larger circle

A1 = ∏R² …………(i)

For smaller circle

A2 = ∏r² …………(ii)

A1 = ∏R²

A2    ∏r²

A1 = R²

A2     r²

A1 = (R/r)² 

A2

√(^A1/A2) =  (R/r)

√(²⁵/4) =  (50/r)

5 = 50                 cross multiply

2     r

(2 x 50) = (5 x r)

100 = 5r

100 = 5r

5         5

r = 20

∴ radius of smaller circle is 20cm.

TRY THIS………..

NECTA 1997 QN 5(c)

The areas of two circles are in the ratio of 16:9 . Calculate the radius of the smaller circle when the radius of larger circle is 24 cm.

EXPONENTS D1

If (2^x-7)(7^y+8) = (2²⁰)(7²⁸) find x+y.

solution

2^x-7 = 2²⁰

x – 7 = 20

x = 20+7

x = 27

…………………………

7^y+8 = 7²⁸

y + 8 = 28

y = 28 - 8

y = 20

……………………

x + y = 27 + 20 = 47.

TRY THIS………….

 NECTA 2008 QN. 3a

If (2^x-7)(7^y+8) = (2²⁰)(7²⁸) find (i)x+y.      (ii) y/x

LOGARITHMS D1

Simplify Log₂128 – Log₅625

Solution

= Log₂128 – Log₅625

= Log₂2⁷ – Log₅5⁴ 

= 7Log₂2 – 4Log₅5   because Log_aaⁿ =  nLog_aa

= (7 x 1) – (4 x 1)   

= 7 – 4

= 3

Hence Log₂128 – Log₅625 = 3 

TRY THIS…………..

NECTA 1995 QN 24b

Simplify Log₂32 – Log₃9.

ALJEBRA D3

Expand 10(y – 7)

solution

= 10(y – 7)

= (10 x y) – (10 x 7)

= 10y – 70 answer

TRY THIS………..

Expand 3(W– 9)

CONGRUENCE OF TRIANGLES D2

Prove that the following triangle is isosceles.

Solution

Given: triangle UTW with T produced to V.

Required to prove: ∆UTW is isosceles

Proof: 

UV = VW (given)

<UVT = <WVT = 90⁰ (given)

TV = VT (common line)

∆UVT ≡ ∆WVT by SAS rule 

TU = TW (Equal corresponding sides)

Hence ∆ UTW is isosceles (proved)