RATIO PROPORTION AND VARIATION
8. If we have two equations containing three unknowns as a1x + b1y + c1z = 0 ....(1)
and a2x + b2y + c2z = 0 ....(2)
Then, the value of x, y and z cannot be resolved without having a third equation. However, in the absence of a third equation,
we can find the proportion x : y : z. This will be given by b1c2–b2c1: c1a2–c2a1: a1b2–a2b1. This can be remembered by writing as follows.
Multiply the coefficients across the arrow indicated always taking a multiplication as positive if the arrow points downwards and taking it as negative if the arrow points upwards.
Thus x corresponds to b1c2 – b2c1 and so on.
9. If the ratio a/b > 1 (called a ratio of greater inequality) and if k is a positive number.
(a + k)/ (b + k) < a/b and (a – k)/(b – k) > a/b Similarly if a/b < 1 then
(a + k)/(b + k) > a/b and (a – k)/(b – k) < a/b
[The student should try assuming certain values and check the results]
10 . Maintenance of equality when numbers are added in both the numerator and the denominators.
This if best illustrated through an example.
20/30 = (20 + 2)/(30 + 3)
i.e. a/b = (a + c)/(b + d) if and only if c/d = a/b. In other words, the ratio of the additions should be equal to the original
ratio to maintain equality of ratios when two different numbers are added in the numerator and denominator.
Ex. 1 Rs. 3650 is divided among 4 engineers, 3 MBAs and 5 CAs such that 3 CAs get as much as 2 MBAs and 3 Engineers as much as 2 CAs. Find the share of an MBA.
(A) 300 (B) 450
(C) 475 (D) None of these
Sol.
ASSIGNMENT
1. Divide Rs. 1870 into three parts in such a way that half of the first part, one-third of the second part and one-sixth of the third part are equal.
(A) 241, 343, 245 (B) 400, 800, 670 (C) 470, 640, 1160 (D) 420, 600, 850
(E) None of these
2. Divide Rs. 500 among A, B, C and D so that A and B together get thrice as much as C and D together, B gets four times of What C gets and C gets 1.5 times as much as D. Now the value of What B gets is
(A) 300 (B) 75 (C) 125 (D) 150
(E) None of these
3. 
4. If 6x2 + 6y2 = 13xy, what is the ratio of x to y ?
(A) 2 : 3 (B) 3 : 2
(C) 4 : 5 (D) 1 : 2
5. 
6. A crew can row a certain course up the stream in 84 minutes; they can row the same course down stream in
9 minutes less than they can row it in still water. How long would they take to row down with the stream.
(A) 45 or 23 minutes (B) 63 or 12 minutes (C) 60 minutes (D) 19 minutes
(E) 25 minutes
7. The speeds of three cars in the ratio 2 : 3 : 4. The ratio between the times taken by these cars to travel the same distance is
(A) 2 : 3 : 4 (B) 4 : 3 : 2
(C) 4 : 3 : 6 (D) 6 : 4 : 3
(E) 3 : 4 : 6
8. If a, b, c and d are proportional then the mean proportion between a2 + c2 and b2 + d2 is
(A) ac/bd (B) ab + cd
(C) a/b + d/c (D) a2/b2 + c2/d2
(E) None of these
9. After an increment of 7 in both the numerator and denominator, a fraction changes to 3/4. Find the original fraction.
(A) 5/12 (B) 7/9
(C) 2/5 (D) 3/8
(E) 7/12
10. The difference between two positive numbers is 10 and the ratio between them is 5 : 3. Find the product of the two numbers.
(A) 375 (B) 175
(C) 275 (D) 125
(E) 250
11. A varies jointly as B and C; and A = 6 when
B = 3; C = 2; find A when B = 5, C = 7.
(A) 17.5 (B) 35
(C) 70 (D) 105
12. If x varies as y directly, and as z inversely, and x = 14 when y = 10; find z when x = 49, y = 45.
(A) 14/10 (B) 10
(C) 10/14
(D) Cannot be determined
13. A cask contains a mixture of 49 litres of wine and water in the proportion 5 : 2. How much water must be added to it so that the ratio of wine to water may be 7 : 4 ?
(A) 3.5 (B) 6
(C) 7 (D) None of these
14. A cask contains 12 gallons of mixture of wine and water in the ratio 3 : 1. How much of the mixture must be drawn off and
water substituted, so that wine and water in the cask may become half and half.
(A) 3 litres (B) 5 litres
(C) 6 litres (D) None of these
15. The total number of pupils in three classes of a school is 333. The number of pupils in classes I and II are in the ratio 3 : 5 and
those in classes II and III are in the ratio 7 : 11. find the number of pupils in the class that had the highest number of pupils.
(A) 63 (B) 105
(C) 165 (D) 180
Directions for Question 16 - 18: Read the following and answer the questions that follow.
Tuliram runs in a triathlon consisting of three phases in the following manner. Running 12 km, cycling 24 km and swimming 5 km. His speeds
in the three phases are in the ratio 2 : 6 : 1. He completes the race in n minutes. Later, he changes his strategy so that the distances he covers in each
phase are constant but his speeds are now in the ratio 3 : 8 : 1. The end result is that he completes the race taking 20 minutes more than the earlier speed
. It is also known that he had not changed his running speed when hechanges his strategy.
16. What is his initial speed while swimming ?
(A) 1/2 km/min (B) 0.05 km/min (C) 0.15 km/min (D) 0.2 km/min
(E) None of these
17. If his speeds are in the ratio 1 : 3 : 1, with the running time remaining unchanged, what is his finishing time ?
(A) 500/3 min (B) 250/3 min (C) 200/3 min (D) 350/2 min
(E) None of these
18. What is tuliram’s original speed of running ?
(A) 9 kmph (B) 18 kmph
(C) 54 kmph (D) 12 kmph
(E) None of these
19. There are two alloys of gold and silver. In the first alloy, there is twice as much gold as silver, and in the second alloy there is 5 times less gold
than silver. How many times more must we take of the second alloy than the first in order to obtain a new alloy in which there would be twice as much silver as gold?
(A) Two times (B) Three times (C) Four times (D) Ten times
(E) Five times
ANSWER KEY
1) E 2) A 3) B 4) A 5) D 6) B 7) D 8) B 9) C 10) A 11) B
12) D 13) B 14) D 15) C 16) E 17) E 18) B 19) B