RATIONS

1.
(a)
(3y - 2x)/(x - 4y) = 2
3y - 2x = 2x - 8y
-4x = -11y
x : y = 11 : 4

(b)
x ; y = 11 : 4
Let x = 11k and y = 4k

√(x + y) : √(x - y)
= √(11k + 4k) : √(11k - 4k)
= √15 : √7

2.
(a)
(m - 1)/n = 5/3
3m - 3 = 5n
3m = 5n + 3
m = (5n + 3)/3

(b)
Let m - 1 = 5k and n = 3k
Then, m = 5k + 1

3m + 4n = 57
3(5k + 1) + 4(3k) = 57
k = 2
Hence, m = 5(2) + 1 = 11
and n = 3(2) = 6

3.
(a)
5a = 4b
a : b = 4 : 5

3b = 5c
b : c = 5 : 3

Hence, a : b : c = 4 : 5 : 3
Let a = 4k, b = 5k, c = 3k

(2a + 3b - 4c) : (3a - 4b + 4c)
= [2(4k) + 3(5k) - 4(3k)] : [3(4k) - 4(5k) + 4(3k)]
= 11 : 4

(b)
a + b + c = 48
4k + 5k + 3k = 48
k = 4
Hence, b = 5(4) = 20

4.
(a)
Since 2x = 3y, then y = 2x/3
Since 2x = 4z, then z = x/2

x : y : z = x : 2x/3 : x/2
x : y : z = 6 : 4 : 3

(b)
Let x = 6k, y = 4k, z = 3k
(6k)2 + 2(4k)2 + 3(3k)2 = 95
k2 = 1
k = 1 or k = -1
Then, x = 6 or x = -6

5.
(a)
6x2 - 11xy + 4y2 = 0
(3x - 4y)(2x - y) = 0
3x = 4y or 2x = y
x/y = 4/3 or x/y = 1/2

(b)
When x/y = 4/3
Let x = 4k, y = 3k
(x2 + 3xy + y2)/(x2 - 3xy + y2)
= [(4k)2 + 3(4k)(3k) + (3k)2]/[(4k)2 - 3(4k)(3k) + (3k)2]
= (61k2)/(-11k2)
= -61/11

When x/y = 1/2
Let x = m, y = 2m
(x2 + 3xy + y2)/(x2 - 3xy + y2)
= [(m)2 + 3(m)(2m) + (2m)2]/[(m)2 - 3(m)(2m) + (2m)2]
= (11m2)/(-m2)
= -11

6.
(a)
water : alcohol
= 25% : (1 - 25%)
= 1 : 3

(b)
(213 L) : alcohol = 1 : 3
Volume of alcohol = 639 L

7.
(a)
R.F.
= 6 cm : 12 km
= 6 cm : (12 x 1000 x 100 cm)
= 1 : 200000

(b)
Actual area
= (4 x 200000 cm)2
= (800000 cm)2
= (8 km)2
= 64 km2

8.
(a)
Let the angles be ko, 2ko and 3ko.
k + 2k + 3k = 180
k = 30
The angles of the triangle are 30o, 60o and 120o

(b)
By sine law,
Ratio of the sides of the triangle
= sin 30o : sin60o : sin90o
= 1/2 : (√3)/2 : 1
= 1 : √3 : 2

9
(a)
(i)
Speed of car A
= 210/3 km/h
= 70 km/h

Speed of car B
= 200/2.5 km/h
= 80 km/h

(ii)
Speed of car A
= (70 x 1000 m)/ (3600 s)
= [19 and (4/9)] m/s

Speed of car B
= (80 x 1000 m)/ (3600 s)
= [22 and (2/9)] m/s

(b)
Time taken by car A
= (280/70) h
= 4 h

Time taken by car B
= (280/80) h
= 3.5 h
=