# QUESTIONS ON RATIO AND PROPORTION (PART-2)

## QUESTIONS ON RATIO AND PROPORTION (PART-2)

#### QUERY 11

A, B & C are three friends with their incomes in ratio 7 : 9 : 12 and expenditure in 8 : 9 : 15 . If savings of A is 1/4 of his income then find the ratio of their savings?

A) 56 : 99 : 69
B) 53 : 89 : 69
C) 56 : 89 : 72
D) 64 : 89 : 72

MAHA GUPTA

Ratio of incomes is 7 : 9 : 12
Now let their incomes of A, B and C restively are 7,000, 9,000, 12,000
Therefore A’s savings = 1/4 of 7,000 = 1,750
And his expenditure = 7,000 – 1,750 = 5.250

The ratio of their expenditure = 8 : 9 : 15
Now, let expenditure of A, B and C respectively = 8x, 9x and 15x

But A’s expenditure as above = 5,250
Therefore 8x = 5,250
=> x = 5250/8 = 656.25

Thus B’s expenditure is 9*656.25 = 5906.25
And C’s expenditure = 15*656.25 = 9843.75

Now, A’s savings = 1750 (calculated above)
B’s savings = 9000 – 5906.25 = 3093.75
C’s savings = 12000 – 9843.75 = 2156.25

Hence ratio of their savings = 1750 : 3093.75 : 2156:25
56 : 99 : 69 (option ‘A’)

TRICK TO FIND THE RATIO WITH THE HELP OF OPTIONS GIVEN
In the last step, to avoid calculation, we should roughly take the ratio of B and C as 3000 : 2000 = 3 : 2 = 99 : 66
NEAREST to this is only option ‘1’; hence option ‘A’ is correct.

#### QUERY 12

A mixture of milk and water is such that the quantity of milk is 3/5 that of water. The proportion of milk in the mixture is?

A) 3 : 5
B) 5 : 3
C) 2 : 5
D) 3 : 8

MAHA GUPTA
Quantity of milk is 3/5 that of water (given)
Now let the quantity of milk in the mixture = 3 units
Therefore the quantity of water in that mixture = 5 units

According to above quantity of mixture = 3+5 = 8 units
Hence the proportion of milk in the mixture = 3/8 = 3 : 8 (option ‘D’)

#### QUERY 13

A watermelon is cut into two pieces in the ratio of 3 : 5 by weight. The bigger of the two is further cut in the ratio of 5 : 7 by weight. Find the ratio of each of the three pieces.

A) 15 : 25 : 26
B) 5 : 7 : 9
C) 2 : 5 : 7
D) 36 : 25 : 35

MAHA GUPTA
For easier calculation assume the unit weight of the bigger of first two pieces be the LCM of 3, 5, 5+7=12 i.e. 60.

Now if weight of the bigger of first two pieces = 60 units; the weight of the smaller piece = 36 units                                     [because ratio of the first two pieces is given 3 : 5]

Ratio of the last two pieces which are cut out of the bigger of the first two pieces = 5 : 7

Therefore the weight of the smaller of these = 60*5/12 = 25 units
And the weight of the bigger piece = 60*7/12 = 35 units

Hence ratio of each of the three pieces = 36 : 25 ” 35 (option ‘D’)

#### QUERY 14

The prize money of Rs 1800 is divided among three students A, B and C in  such a way that 4 times the share of A is equal to 6 times the share of B, which is equal to 3 times the share of C. Find A’s share.

A) Rs 400
B) Rs 600
C) Rs 700
D) Rs 800

MAHA GUPTA
According to the question 4A = 6B = 3C
Now let 4A = 6B = 3C = k

From above, A = k/4; B = k/6; C = k/3
Hence A : B : C = 3 : 2 : 4

Therefore A’s share = 1800*(3/9) = 600 (option ‘B’)

#### QUERY 15

Find the third proportional to (x/y + y/x) and √(x² + y²)

A) xy
B) √(xy)
C) ∛(xy)
D) ∜(xy)

MAHA GUPTA
First let one know what the third proportional is
If a : b = b : c, then ‘c’ is called the third proportional to ‘a’ and ‘b’

Obviously in the question given above a = x/y + y/x = (x² + y²)/xy
And b = √(x² + y²)

Therefore the third proportional = b²/a = [√(x² + y²)]²/[(x² + y²)/xy]

= (x² + y²)*xy/(x² + y²) = xy (option ‘A’)

#### QUERY 16

Samir’s age is one fourth of his father’s age and two third of his sister Reema’s age. What is the ratio of the age of Samir, Reema and their father respectively.

A) 10 years
B) 7 years
C) 13 years
D) 8 years

MAHA GUPTA
Ratio of Samir’s age to his father = 1 : 4
At the same time Samir’s age is 2/3 of his sister’s age, means 1 : 3/2

Means Samir’s age : Reema’s age : father’s age = 1 : 3/2 : 4
=> 2 : 3 : 8 (option ‘D’)

#### QUERY 17

The total salary of A, B and C is Rs 666. If they spend 80%, 85%, 75% of their salaries respectively, their savings are 7 : 6 : 9. Then salary of C is?

A) Rs 300
B) Rs 250
C) Rs 216
D) Rs 245

MAHA GUPTA
Let their salaries respectively be x, y, z
Therefore their savings
A = (100 – 80)% of x = x/5
B = (100 – 85)% of y = 3y/20
C = (100 – 75)% of z = 3z/4

Also their savings ratio = 7 : 6 : 9
=> x/5 : 3y/20 : z/4 = 7 : 6 : 9
=> x = 35; y = 40; z = 36

Hence ratio of the salaries of A, B, C = 35 : 40 : 36
So, C’s salary = 666*(36/111) = 216 (option ‘C’)

#### QUERY 18

If (x – a) : (x – b) : (x – c) = 11 : 9 : 5; where x = (a+b+c)/2 then what is a : b : c

A) 5 : 6 : 9
B) 7 : 8 : 10
C) 8 : 9 : 12
D) 2 : 3 : 5

MAHA GUPTA
According to the first equation
(x – a)/11 = (x – b)/9 = (x – c)/5

Now let, (x – a)/11 = (x – b)/9 = (x – c)/5 = k

=> a = x – 11k; b = x – 9k; c = x – 5k

Hence, according to the second given equation = x = [(x – 11k) + (x – 9k) + (x – 5k)]/2
=> x = 25k

Putting this in: a = x – 11k; b = x – 9k; c = x – 5k
a = 14k; b = 16k; c = 20k
=> a : b : c = 14k : 16k : 20k
Therefore, a : b : c = 7 : 8 : 10 (option ‘B’)

NOTE: Such questions are done quickly with the answer options.

#### QUERY 19

If a/(b+c) = b/(c+a) = c/(a+b) and a+b+c is not equal to 0; then value of b/(a+b+c) is?

A) 1/3
B) 5/9
C) 2/7
D) 3/5

MAHA GUPTA
Using componendo and dividendo
a/(b+c) = b/(c+a) => (a + b + c)/(a – b – c) = (b + c + a)/(b – c – a)

=> a – b – c = b – c – a
=> a = b

Similarly it can be shown that a = b = c
Taking a = b = c = 1
b/(a+b+c) = 1/(1 + 1 + 1) = 1/3 (option ‘A’)

SHORT
Taking each of a, b, c as 1 satisfies a/(b+c) = b/(c+a) = c/(a+b)

So, b/(a+b+c) = 1/(1 + 1 + 1) = 1/3 (option ‘A’)

#### QUERY 20

The ratio of father’s age to his son’s age is 7 : 4. The product of of their ages is 1008. The ratio of their ages 6 years hence will be?

A) 5 : 3
B) 8 : 5
C) 7 : 4
D) 5 : 8

MAHA GUPTA
Let father’s present age is 7x years
Therefore his son’s age = 4x years

Now 7x*4x = 1008
=> 28x² = 1008
=> x = 6

Thus fathers age 6 years hence = 7x + 6 = 7*6 + 6 = 48
And his son’s age 6 years hence = 4x + 6 = 4*6 + 6 = 30

Hence the required ratio = 48 : 30 = 8 : 5 (option ‘B’)

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