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:

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 14560
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 14336
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 14250
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 14030
Q:

If a=2b/3, b=2c/3, and c=2d/3, then find the ratio of b and d:

 

A) 8/9 B) 4/9
C) 4/3 D) 5/27
 
Answer & Explanation Answer: B) 4/9

Explanation:
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3 13583
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 13576
Q:

The ratio of the angles of a triangle is 3 : 4 : 5. The three angles of a quadrilateral is equal to three angles of this triangle. What is the sum of the largest angle and second smallest angle of the quadrilateral ?

A) 225 deg B) 210 deg
C) 205 deg D) 245 deg
 
Answer & Explanation Answer: A) 225 deg

Explanation:

Angle of triangle = 180 × 3/12

= 45 deg

Other angles of the triangle are = 60 deg, 75 deg

Angle of quadrilateral

= 360 - (45 + 60 + 75)

= 180 deg

Therefore, the four angles of the quadrilateral are 45, 60, 75 & 180 degrees.

Hence, the sum of largest and smallest angle of the quadrilateral are 180 + 45 = 225 deg.

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15 13460
Q:

One year ago the ratio between Maneela’s and Shanthi’s salary was 3 : 4. The ratios of their individual salaries between last year’s and this year’s salaries are 4 : 5 and 2 : 3 respectively. At present the total of their salary is Rs. 4160. The salary of Maneela, now is?

A) Rs. 1600 B) Rs. 1700
C) Rs. 1800 D) Rs. 1900
 
Answer & Explanation Answer: A) Rs. 1600

Explanation:

Let the salaries of Maneela and Shanthi one year before be M1, S1 & now be M2, S2 respectively.

Then, from the given data,

M1/S1 = 3/4  .....(1)

M1/M2 = 4/5 .....(2)

S1/S2 = 2/3  .....(3)

and M2 + S2 = 4160  .....(4)

 

Solving all these eqtns, we get M2 = Rs. 1600.

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