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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:

In a competitive exam, the number of passed students was four times the number of failed students. If there had been 35 fewer appeared students and 9 more had failed, the ratio of passed and failed students would have been 2 : 1, then the total number of students appered for the exam ?

A) 175 B) 165
C) 155 D) 145
 
Answer & Explanation Answer: C) 155

Explanation:

Let the number of failed students be x

=> Number of passed students = 4x 

So total number of students was 5x

From the given data,

If total number of students be 5x – 35

=> 4x-35-9/x+9 = 2/1

=> 4x – 44 = 2(x + 9)

=> 4x – 2x = 18 + 44

=> x = 31

Total number = 31×5

= 155

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43 17762
Q:

A child has three different kinds of chocolates costing  Rs.2, Rs.5, Rs.10.  He spends total Rs. 120 on the chocolates. what is the minimum possible number of chocolates he can buy, if there must be  atleast one chocolate of each kind?

A) 22 B) 19
C) 17 D) 15
 
Answer & Explanation Answer: C) 17

Explanation:

Minimum number of chocolates are possible when he purchases maximum number of costliest chocolates.

Thus,          2 x 5 + 5 x 2 =Rs.20

Now Rs.100 must be spend on 10 chocolates as 100 = 10 x 10.

Thus minumum number of chocolates = 5 + 2 + 10 = 17

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48 17715
Q:

The ratio of male and female in a city is 7 : 8 respectively and percentage of children among male and female is 25 and 20 respectively. If number of adult females is 156800, what is the total population of the city?

A) 4,12,480 B) 3,67,500
C) 5,44,700 D) 2,98,948
 
Answer & Explanation Answer: B) 3,67,500

Explanation:

Let the total population be 'p'

Given ratio of male and female in a city is 7 : 8

In that percentage of children among male and female is 25% and 20%

=> Adults male and female % = 75% & 80%

But given adult females is = 156800

=> 80%(8p/15) = 156800

=> 80 x 8p/15 x 100 = 156800

=> p = 156800 x 15 x 100/80 x 8

=> p = 367500

Therefore, the total population of the city = p = 367500

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73 17709
Q:

Vinod have 20 rupees. He bought 1, 2, 5 rupee stamps. They are different in numbers by the reason of no change, the shop keeper gives 3 one rupee stamps. So how many stamps Vinod have ?

A) 10 B) 18
C) 12 D) 15
 
Answer & Explanation Answer: A) 10

Explanation:

Given total rupees = 20 Rs
No. of one rupee stamps = 3
Now, remaining money = Rs. 17
With that he buys only 2 and 5 rupee stamps
Let number of Rs. 5 stamps = K
Let number of Rs. 2 stamps = L
5K + 2L = 17
K = 3, L = 1 (possible)
L = 6, K = 1 (possible)
=> But given that they are different in number so, K is not equal to 3
one rupee stamps = 3
2 two stamps = 6
5 rupee stamps = 1
Total number of stamps = 10.

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45 17643
Q:

Pencils, Pens and Exercise books in a shop are in the ratio of 10: 2 : 3. If there are 120 pencils, the number of exercise books in the shop is:

A) 26 B) 36
C) 46 D) 56
 
Answer & Explanation Answer: B) 36

Explanation:

Let Pencils = 10x, Pens = 2x & Exercise books = 3x. Now, 10x = 120 hence x = 12.
Number of exercise books = 3x = 36.

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44 17495
Q:

For any two numbers m , n ; (m+n) : (m-n) : mn = 7: 1: 60 . Find the value of 1/m : 1/n

A) 4:3 B) 8:7
C) 3:4 D) 7:8
 
Answer & Explanation Answer: C) 3:4

Explanation:

m+nm-n=7xx ⇒mn=4x3x  

 

Again   mn=12x2 

 

and         mn =60x  

 

so, 60x=12x2 ⇒x=5    

 

=>  m= 20  and    n= 15

 

Hence,    1m:1n=120:115=3:4

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47 16244
Q:

In a fort, there are 1200 soldiers. If each soldier consumes 3 kg per day, the provisions available in the fort will last for 30 days. If some more soldiers join, the provisions available will last for 25 days given each soldier consumes 2.5 kg per day. Find the number of soldiers joining the fort in that case ?

A) 693 B) 741
C) 528 D) 654
 
Answer & Explanation Answer: C) 528

Explanation:

Assume x soldiers join the fort. 1200 soldiers have provision for 1200 (days for which provisions last them)(rate of consumption of each soldier)
= (1200)(30)(3) kg.

Also provisions available for (1200 + x) soldiers is (1200 + x)(25)(2.5) k

As the same provisions are available
=> (1200)(30)(3) = (1200 + x)(25)(2.5)

x = ([(1200)(30)(3)] / (25)(2.5)) - 1200 => x = 528.

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28 16040
Q:

The third proportional tox2-y2 and x-y is :

A) (x + y ) B) (x - y)
C) (x + y) / ( x - y) D) (x -y) / ( x + y)
 
Answer & Explanation Answer: D) (x -y) / ( x + y)

Explanation:

Let the third proportional to x2-y2 and x-y be z. Then, 

 

x2-y2:x-y::x-y:z  

 

=>x2-y2*z=x-y2  

 

=> z=x-y2x2-y2=x-yx+y

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