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 Ratios and Proportions Questions

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FACTS  AND  FORMULAE  FOR  RATIO  AND  PROPORTION  QUESTIONS

 

 

1. RATIO: The ratio of two quantities a and b in the same units, is the fraction a/b and we write it as a:b.

In the ratio a:b, we call a as the first term or antecedent and b, the second term or consequent.

Ex. The ratio 5: 9 represents 5/9 with antecedent = 5, consequent = 9.

Rule: The multiplication or division of each term of a ratio by the same non-zero number does not affect the ratio. Ex. 4: 5 = 8: 10 = 12: 15 etc. Also, 4: 6 = 2: 3.

 

2. PROPORTION : The equality of two ratios is called proportion. 

If a: b = c: d, we write, a: b :: c : d and we say that a, b, c, d are in proportion . Here a and d are called extremes, while b and c are called mean terms.

Product of means = Product of extremes.

Thus, a: b :: c : d <=> (b x c) = (a x d).

 

 3. (i) Fourth Proportional : If a : b = c: d, then d is called the fourth proportional to a, b, c.

     (ii) Third Proportional : If a: b = b: c, then c is called the third proportional to a and b.

     (iii) Mean Proportional : Mean proportional between a and b is ab

 

 4. (i) COMPARISON OF RATIOS : We say that (a: b) > (c: d) ⇔ (a/b)>(c /d).

     (ii) COMPOUNDED RATIO : The compounded ratio of the ratios (a: b), (c: d), (e : f)  is    (ace: bdf)

 

5. (i) Duplicate ratio of (a : b) is a2:b2.

    (ii) Sub-duplicate ratio of (a : b) is (√a : √b).

    (iii)Triplicate ratio of (a : b) is a3:b3.

    (iv) Sub-triplicate ratio of (a : b) is a13: b13.

    (V) If ab=cd, thena+ba-b=c+dc-d (Componendo and dividendo)

 

 6. VARIATION:

(i) We say that x is directly proportional to y, if x = ky for some constant k and we write, x∝y

(ii) We say that x is inversely proportional to y, if xy = k for some constant k and we write, x∝1y

Q:

Two vessel P and Q contain milk and water in ratio 3 : 2 and 5 : 3 respectively. If 10 liters of the mixture is removed from vessels P and poured in vessel Q then ratio of milk to water in vessel Q becomes 8 : 5. Find the initial quantity of water in vessel Q.

A) 10 lit B) 6 lit
C) 16 lit D) 12 lit
 
Answer & Explanation Answer: A) 10 lit

Explanation:

Given ratio of initial mixture of milk and water in Q = 5 : 3

Let the initial quantity of mixture in vessel Q = 8x

Let quantity of Milk = 5x and

Let quantity of water = 3x

According to the question,

5x + 10 x 353x + 10 x 25  = 85

=> 25x + 30 = 24x + 32

=> x = 2

Required Initial quantity of milk = 5x = 5 x 2 = 10 lit.

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18 8662
Q:

If Rs. 782 be divided into three parts, proportional to 1/2:2/3:3/4, then the first part is:

A) Rs. 408 B) Rs. 204
C) Rs. 412 D) Rs. 220
 
Answer & Explanation Answer: B) Rs. 204

Explanation:

Given ratio = 1/2:2/3:3/4 = 6:8:9

1st part = 782 x 6/23 = Rs. 204.

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10 8641
Q:

Pooja, Shipra and Mounika are three sisters .Pooja and Shipra are twins. The ratio of sum of the ages of Pooja and Shipra is same as that of Mounika alone. Three years earlier the ratio of age of  Pooja and Mounika was 2:7 . what will be the age of Shipra 3 years hence?

A) 21 years B) 16 years
C) 8 years D) 12 years
 
Answer & Explanation Answer: C) 8 years

Explanation:

Since Pooja and Shipra are twins so their ages be same. Let their ages be x and age of Mounika be y , then 

           x+x = y    -------(1) 

 and   x-3y-3=27⇒7x-2y=15 

 

Now, from  equation (1), 

         7x - 4x = 15  =>  x = 5

So the age of Shipra 3 years hence will be 5+3 = 8 years

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12 8605
Q:

If 12 men can reap 120 acres of land in 36 days, how many acres of land can 54 men reap in 54 days?

A) 710 acres B) 760 acres
C) 810 acres D) 860 acres
 
Answer & Explanation Answer: C) 810 acres

Explanation:

12 men120 acres36 days54 men?54 days

 As 12 men can reap 120 acres, 54 men will be able to reap more acres in 36 days, 120 acres of land was reaped, so in 54 days, more land will be reaped.

 

Thus, the numbers of acres that can be reaped by 54 men in 54 days = 5412x120x5436 = 810 acres

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19 8467
Q:

Based on the analogy given, find the missing number from the given options.

 

11 : 13 :: 17 : ___

 

A) 29 B) 21
C) 19 D) 23
 
Answer & Explanation Answer: D) 23

Explanation:
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1 8466
Q:

Rs. 7444 is divided among 5 ladies, 3 gents and 3 girls. The ratio of share of a lady, a gents and a girl is 7: 4: 3. What is the share of gents ?

A) Rs. 1595.14 B) Rs. 1793.4
C) Rs. 595.14 D) Rs. 1551.5
 
Answer & Explanation Answer: A) Rs. 1595.14

Explanation:

Ratio of the share of a lady, a gents and a girl is 7 : 4 : 3
No of ladies, gents and girls are 5, 3, 3
Thus effective ratio of ladies, gents and girls is 7 x 5 : 4 x 3 : 3 x 3 = 35 : 12 : 9
so part of gents = (12/56) x 7444 = Rs. 1595.142
so part of 1 gents = 1595.142/3 = Rs. 531.71.

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9 8167
Q:

A man spend Rs. 810 in buying trouser at Rs. 70 each and shirt at Rs. 30 each. What will be the ratio of trouser and shirt when the maximum number of trouser is purchased ?

A) 1:3 B) 2:1
C) 3:2 D) 2:3
 
Answer & Explanation Answer: C) 3:2

Explanation:

Let us assume S as number of shirts and T as number of trousers
Given that each trouser cost = Rs.70 and that of shirt = Rs.30
Therefore, 70 T + 30 S = 810
=> 7T + 3S = 81......(1)
T = ( 81 - 3S )/7

We need to find the least value of S which will make (81 - 3S) divisible by 7 to get maximum value of T
Simplifying by taking 3 as common factor i.e, 3(27-S) / 7
In the above equation least value of S as 6 so that 27- 3S becomes divisible by 7

Hence T = (81-3xS)/7 = (81-3x6)/7 = 63/7 = 9

Hence for S, put T in eq(1), we get
S = 81-7(9)/3 = 81-63 / 3 = 18/3 = 6.
The ratio of T:S = 9:6 = 3:2.

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14 8150
Q:

What is the equivalent compound ratio of 5:6::7:10::6:5  ?

A) 7 : 9 B) 9 : 7
C) 7 : 4 D) 7 : 10
 
Answer & Explanation Answer: D) 7 : 10

Explanation:

Given 5:6 :: 7:10 :: 6:5

We know that compound ratio is calculated as
a:b :: c:d :: e:f
(axcxe):(bxdxf)

=> (5x7x6):(6x10x5)
=> (210):(300)
= 7:10

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11 7889