Ratios and Proportions Questions

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, xy

(ii) We say that x is inversely proportional to y, if xy = k for some constant k and we write, x1y

Q:

Three cats are roaming in a zoo n such a way that when cat A takes 5 steps, B takes 6 steps and C takes 7 steps.But the 6 steps of A are equal to the 7 steps of B and 8 steps of C. what is the ratio of their speeds?

A) 140:144:147 B) 40:44:47
C) 15:21:28 D) 252:245:240
 
Answer & Explanation Answer: A) 140:144:147

Explanation:

Frequency of step of A:B:C = 5 : 6 : 7

But in terms of size of step, 6A = 7B = 8C

Therefore, Ratio of speeds of A, B and C = 5/6 : 6/7 : 7/8 = 140 : 144 : 147

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42 14773
Q:

In a class of 39 students the ratio of boys and girls is 2 : 1. Radhika ranks 15th among all the students from top and 8th among girls from bottom. How many boys are there below Radhika ?

A) 9 B) 17
C) 11 D) 15
 
Answer & Explanation Answer: B) 17

Explanation:

Total student= 39, boys : girls = 2:1, hence no of boys= 2/3(39)= 26
num of girls= 1/3(39) = 13 , since radhika ranked 15 among 39 student from top
and ranked 8 among girls from bottom , this means 7 girls are below radhika, rest 13-9= 5 girls are above her, now from 14 toper 5 are girls hence 14-5=9 boys.
Hence num of boys below radhika are 26-9=17

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35 14768
Q:

The sum of three numbers is 98. If the ratio of the first to second is 2 : 3 and that of the second to the third is 5 : 8, then the second number is:

A) 10 B) 20
C) 30 D) 40
 
Answer & Explanation Answer: C) 30

Explanation:

Let the three parts be A, B, C. Then,

A : B = 2 : 3 and B : C = 5 : 8 =5*35:8*35=  3:245  

A : B : C = 2 : 3 : =>  = 10 : 15 : 24

 

 B =245 = 30

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45 14129
Q:

There is certain numbers of toys in the box. They are divided into such a way that the person who gets 1/4 of the whole gets thrice of what the others get on an average. Find the number of people amongst whom the toys are distributed?

A) 8 B) 10
C) 12 D) 9
 
Answer & Explanation Answer: B) 10

Explanation:

If the person who gets 1/4 of the whole gets thrice of what the others get on an average, each one will get =1/3×1/4=112 of the whole.
Therefore, if there are k persons other than the person who gets one-fourth, then 1/4 + k/12 = 1
⇒ k=9

Hence, total number of people = 10.

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34 13998
Q:

What is the third proportional to 16 and 36 ?

Answer

Let  \inline 6:8::15:x


\inline 16x=36\times 36


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73 13688
Q:

In Maa Yatri Temple every devotee offers fruits to the orphans. Thus every orphan receives bananas, oranges and grapes in the ratio of 3:2:7 in terms of dozens. But the weight of a grape is 24 gm and  weight of a banana and an orange are in the ratio of 4:5, while the weight of an orangeis 150gm. Find the ratio of all the three fruits in terms of weight, that an orphan gets

A) 90:75:42 B) 180:150:82
C) 75:42:90 D) None of these
 
Answer & Explanation Answer: D) None of these

Explanation:

Ratio of fruits (by dozen) = 3:2:7

Ratio of fruits (by weight) =  120 :150: 24

Therefore, Ratio of fruits (combined) by weight = 3 x 120 : 2 x 150 : 7 x 24 = 30 : 25 : 14 

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16 13536
Q:

What is the ratio whose terms differ by 40 and the measure of which is 2/7 ?

A) 16:56 B) 14:56
C) 15:56 D) 16:72
 
Answer & Explanation Answer: A) 16:56

Explanation:

Let the ratio be x : (x + 40), then ,

 

 

 

xx+40=27 

 

 7x = 2x + 80 

 

 x = 16 

 

Required ratio = 16 : 56

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35 13288
Q:

A cubical block of metal weighs 6 pounds. How much will another cube of the same metal weigh if its sides are twice as long?

A) 48 B) 12
C) 36 D) 60
 
Answer & Explanation Answer: A) 48

Explanation:

If you double the sides of a cube, the ratio of the surface areas of the old and new cubes will be 1: 4. The ratio of the volumes of the old and new cubes will be 1: 8.

Weight is proportional to volume. So, If the first weighs 6 pounds, the second weighs 6x8 pounds =48.

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44 12852