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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:

Alloy A contains 40% gold and 60% silver. Alloy B contains 35% gold and 40% silver and 25% copper. Alloys A and B  are mixed in the ratio of 1:4 .What is the ratio of gold and silver in the newly formed alloy is?

A) 20% and 30% B) 36% and 44%
C) 25% and 35% D) 49% and 36%
 
Answer & Explanation Answer: B) 36% and 44%

Explanation:

Assume the weight of alloy a is 100 kg 

Therefore, The weight of alloy B is 400kg

 

             Gold          silver         copper

 

    A       40kg          60kg          0kg

 

    B      140kg         160kg        100kg

 

  total    180kg          220kg        100kg

 

Therefore, Ratio of gold and silver in new alloy = 180500:200500= 36% :44%

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100 29695
Q:

If 76 is divided into four parts proportional to 7, 5, 3, 4, then the smallest part is:

A) 12 B) 15
C) 16 D) 19
 
Answer & Explanation Answer: A) 12

Explanation:

Given ratio = 7 : 5 : 3 : 4, Sum of ratio terms = 19.

Smallest part =  (76*3/19)  = 12

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73 28645
Q:

If p : q : r = 2 : 3 : 5, then what will be the ratio of (p + q + r) : (3p + q – r)?

 

A) 5 : 2   B) 10 : 17  
C) 9 : 4 D) 5 : 4
 
Answer & Explanation Answer: A) 5 : 2  

Explanation:
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6 28588
Q:

Jo's collection contains US, Indian and British stamps. If the ratio of US to Indian stamps is 5 to 2 and the ratio of Indian to British stamps is 5 to 1, what is the ratio of US to British stamps?

A) 10 : 5 B) 15 : 2
C) 20 : 2 D) 25 : 2
 
Answer & Explanation Answer: D) 25 : 2

Explanation:

Indian stamps are common to both ratios. Multiply both ratios by factors such that the Indian stamps are represented by the same number.

US : Indian = 5 : 2, and Indian : British = 5 : 1. Multiply the first by 5, and the second by 2. 

Now US : Indian = 25 : 10, and Indian : British = 10 : 2

Hence the two ratios can be combined and US : British = 25 : 2

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Exam Prep: GRE

220 27439
Q:

The wages of labourers in a factory increases in the ratio 22:25 and there was a reduction in the number of labourers in the ratio 15:11. Find the original wage bill if the present bill is Rs. 5000 ?

A) Rs. 5500 B) Rs. 6000
C) Rs. 6200 D) Rs. 6350
 
Answer & Explanation Answer: B) Rs. 6000

Explanation:

The wages of labourers in a factory increases in the ratio 22:25 and there was a reduction in the number of labourers in the ratio 15:11. Find the original wage bill if the present bill is Rs 5000 ?
Ratio of increase of wages = 22:25
Ratio of decrease of labourers = 15:11
Compound ratio of wages of labourers = 22 x 15 : 25 x 11 = 330:275
Final bill = Rs. 5000
For 275 ratio wages = Rs. 5000
For 1 ratio wages = 5000/275
For 330 ratio wages = 5000/275 x 330 = Rs. 6000

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69 26322
Q:

If the ratio of the ages of two friends A and B is in the ratio 3 : 5 and that of B and C is 3 : 5 and the sum of their ages is 147, then how old is B?

A) 27 Years B) 75 Years
C) 45 Years D) 49 Years
 
Answer & Explanation Answer: C) 45 Years

Explanation:

The ratio of the ages of A and B is 3 : 5.
The ratio of the ages of B and C is 3 : 5.

B's age is the common link to both these ratio. Therefore, if we make the numerical value of the ratio of B's age in both the ratios same, then we can compare the ages of all 3 in a single ratio.

The can be done by getting the value of B in both ratios to be the LCM of 3 and 5 i.e., 15.

The first ratio between A and B will therefore be 9 : 15 and
the second ratio between B and C will be 15 : 25.

Now combining the two ratios, we get A : B : C = 9 : 15 : 25.

Let their ages be 9x, 15x and 25x.
Then, the sum of their ages will be 9x + 15x + 25x = 49x

The question states that the sum of their ages is 147.
i.e., 49x = 147 or x = 3.

Therefore, B's age = 15x = 15*3 = 45

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59 25687
Q:

A and B invests Rs.8000 and Rs.9000 in a business. After 4 months, A withdraws half of his capital and 2 months later, B withdraws one-third of his capital. In what ratio should they share the profits at the end of the year  ?

A) 21:29 B) 41:54
C) 32:45 D) 37:51
 
Answer & Explanation Answer: C) 32:45

Explanation:

             A              :               B
(8000x4)+(4000x8) : (9000x6)+(6000x6)
         64000           :      90000
=>        32 : 45

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28 25608
Q:

In a college, the ratio of the number of boys to girls is 8 : 5. If there are 160 girls, the total number of students in the college is

A) 100 B) 250
C) 260 D) 416
 
Answer & Explanation Answer: D) 416

Explanation:

Let the number of boys and girls be 8x and 5x.
Total number of students = 13x = 13 * 32 = 416.

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97 23859