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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:

64 boys and 40 girls form a group for social work. During their membership drive, the same number of boys and girls joined the group. How many members does the group have now, if the ratio of boys to girls is 4:3 ?

A) 168 B) 201
C) 147 D) 154
 
Answer & Explanation Answer: A) 168

Explanation:

Let us say x boys and x girls joined the group.

(64 + x)/(40 + x) = 4/3

192 + 3x = 160 + 4x => x = 32

Number of members in the group = 64 + x + 40 + x

= 104 + 2x = 168.

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Q:

The ratio of boys and girls in a school is 9:5.If the total number of students in the school is 1050.Then number of boys is

Answer

Let the ratio be 'R'


Total number of students = 1050


Then,


9R + 5R = 1050


14R = 1050


=> R = 75


 


Hence, the number of boys = 9R = 9 x 75 = 675

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62 12742
Q:

The ratio of Pens and Pencils in a shop is 3 : 2 respectively. The average number of Pens and Pencils is 180. What is the number of Pencils in the shop? 

A) 444 B) 344
C) 244 D) 144
 
Answer & Explanation Answer: D) 144

Explanation:

Given ratio of pens and pencils = 3 :2

Number of Pens = 3x

Number of Pencils = 2x

Average number of pencils & Pens = 3x + 2x2 = 180

5x = 360

=> x = 72


Hence, the number of pencils = 2x = 72 x 2 = 144.

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Q:

The ratio of students in a coaching preparing for B.tech and MBA is 4 : 5. The ratio of fees collected from each of B.tech and MBA students is 25 : 16. If the total amount collected from all the students is 1.62 lakh, what is the total amount collected from omly MBA aspirants?

A) Rs. 62,000 B) Rs. 72,000
C) Rs. 82,000 D) None of these
 
Answer & Explanation Answer: B) Rs. 72,000

Explanation:

The ratio of fees collected from B.Tech : MBA = 4x * 25y : 5x * 16y 

                                                                  = 100xy : 80xy 

                                                                  = 5xy : 4 xy  = 5k : 4k

 

The amount collected only from MBA students = 49×1.62 lakh = Rs. 72,000

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18 12372
Q:

Paint needs to be thinned to a ratio of 2 parts paint to 1.5 parts water. The painter has by mistake added water so that he has 6 litres of paint which is half water and half paint. What must he add to make the proportions of the mixture correct?

A) 1 litre paint B) 1 litre water
C) ½ litre water and one litre paint D) ½ litre paint and one litre water
 
Answer & Explanation Answer: A) 1 litre paint

Explanation:

At the moment the paint is 3 liters of paint and 3 liters of water. We need to add to this to make the new ratio 2 liters paint to 1.5 liters water. As this is equivalent to 4 : 3 we can see that we have the right amount of water, and just need 1 liter of paint to make it correct.

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Q:

There are two containers, the first one contains 1 litre pure water and the second one contains 1 litre pure milk.Now 5 cups of water from the first container is taken out  is mixed well in the second container. Then, 5 cups of this mixture is taken out and is mixed in the first container. Let A denote the proportion of milk in the first container and B denote the proportion of water in the second container then:

A) A B) A=B
C) A>B D) can't be determined
 
Answer & Explanation Answer: B) A=B

Explanation:

Here the ratio of mixtures( i.e milk , water) doesnot matter. But the important point is that whether the total amount ( either pure or mixture ) being transferred is equal or not.Since the total amount ( i.e 5 cups) being transferred from each one to another , hence A =B.

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18 12128
Q:

The ratio of the incomes of Pavan and Amar is 4 : 3 and the ratio of their expenditures are 3:2. If each person saves Rs. 1889, then find the income of Pavan?

A) 6548 B) 5667
C) 7556 D) 8457
 
Answer & Explanation Answer: C) 7556

Explanation:

Let ratio of the incomes of Pavan and Amar be 4x and 3x

and Ratio of their expenditures be 3y and 2y

4x - 3y = 1889 ......... I

and

3x - 2y = 1889 ...........II

I and II

y = 1889

and x = 1889

Pavan's income = 7556

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41 11881
Q:

A, B and C have amounts in the ratio of  3 : 4 : 5. First B gives 1/4th to A and 1/4th to C then C gives 1/6th to A. Find the final ratio of amount of A, B and C respectively.

A) 4:3:5 B) 5:4:3
C) 3:4:5 D) 5:2:5
 
Answer & Explanation Answer: D) 5:2:5

Explanation:

A                   B                 C

 

3x                 4x                5x 

(3x+x)          2x               (5x+x)         (B Gives 1/4 to A and 1/4 to C) 

=4x              2x                 6x 

(4x+x)          2x                 5x              ( C Gives 1/6 to A) 

=5x              2x                 5x

 

Therefore, 5 : 2 : 5

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