Ratio & Proportion — Case Study Based Questions

A man distributes ₹21,600 among his three sons A, B and C in the ratio 5 : 6 : 7 respectively.

1. A's share is :
   (a) ₹7,200
   (b) ₹6,000
   (c) ₹3,600
   (d) ₹5,400

2. B's share is :
   (a) ₹7,200
   (b) ₹6,400
   (c) ₹5,600
   (d) ₹4,800

3. C's share is :
   (a) ₹7,600
   (b) ₹9,200
   (c) ₹8,400
   (d) ₹8,600

4. If ₹600 is added to each of their shares, then the new ratio of their shares would be :
   (a) 11 : 12 : 13
   (b) 11 : 13 : 15
   (c) 13 : 15 : 17
   (d) 19 : 21 : 23

Answer

1. Given,

Total money = ₹21,600

Ratio (A : B : C) = 5 : 6 : 7

Sum of ratio terms = 5 + 6 + 7 = 18.

A's share = $21{,}600 \times \dfrac{5}{18}$ = 5 × ₹1,200 = ₹6,000.

Hence, option (b) is the correct option.

2. Sum of ratio terms = 5 + 6 + 7 = 18.

B's share = $21{,}600 \times \dfrac{6}{18}$ = 6 × ₹1,200 = ₹7,200.

Hence, option (a) is the correct option.

3. Sum of ratio terms = 5 + 6 + 7 = 18.

C's share = $21{,}600 \times \dfrac{7}{18}$ = 7 × ₹1,200 = ₹8,400.

Hence, option (c) is the correct option.

4. Sum of ratio terms = 5 + 6 + 7 = 18.

A's share = $21{,}600 \times \dfrac{5}{18}$ = 5 × ₹1,200 = ₹6,000.

B's share = $21{,}600 \times \dfrac{6}{18}$ = 6 × ₹1,200 = ₹7,200.

C's share = $21{,}600 \times \dfrac{7}{18}$ = 7 × ₹1,200 = ₹8,400.

After adding ₹600 to each: A = ₹6,600, B = ₹7,800, C = ₹9,000.

New ratio = 6600 : 7800 : 9000

Dividing each term by 600:

$\dfrac{6600}{600} : \dfrac{7800}{600} : \dfrac{9000}{600} = 11 : 13 : 15$

Hence, option (b) is the correct option.

The ratio of the number of students in class VI of three schools P, Q and R is $\dfrac{1}{6} : \dfrac{1}{8} : \dfrac{1}{9}$ respectively. The total number of students of class VI in all the three schools is 725.

1. How many students are there in class VI of school P ?
   (a) 275
   (b) 300
   (c) 330
   (d) 280

2. How many students are there in class VI of school Q ?
   (a) 240
   (b) 235
   (c) 265
   (d) 225

3. How many students are there in class VI of school R ?
   (a) 200
   (b) 225
   (c) 185
   (d) 240

4. If the number of students studying in each of these schools is increased by 16%, 12% and 20% respectively, then the new ratio of number of class VI students in schools P, Q and R is :
   (a) 21 : 17 : 26
   (b) 29 : 21 : 20
   (c) 19 : 21 : 20
   (d) 9 : 11 : 7

Answer

1. Given,

Total number of students = 725

Ratio (P : Q : R) = $\dfrac{1}{6} : \dfrac{1}{8} : \dfrac{1}{9}$

Let us find the L.C.M. of 6, 8 and 9:

$\begin{array}{l|rrr} 2 & 6, & 8, & 9 \\ \hline 2 & 3, & 4, & 9 \\ \hline 2 & 3, & 2, & 9 \\ \hline 3 & 3, & 1, & 9 \\ \hline 3 & 1, & 1, & 3 \\ \hline & 1, & 1, & 1 \end{array}$

L.C.M. = 2 × 2 × 2 × 3 × 3 = 72.

Multiplying each term by 72:

$\dfrac{1}{6} : \dfrac{1}{8} : \dfrac{1}{9} = \left(\dfrac{1}{6} \times 72\right) : \left(\dfrac{1}{8} \times 72\right) : \left(\dfrac{1}{9} \times 72\right) = 12 : 9 : 8$

Sum of ratio terms = 12 + 9 + 8 = 29.

Number of students in school P = $725 \times \dfrac{12}{29}$ = 12 × 25 = 300.

Hence, option (b) is the correct option.

2. Sum of ratio terms = 12 + 9 + 8 = 29.

Number of students in school Q = $725 \times \dfrac{9}{29}$ = 9 × 25 = 225.

Hence, option (d) is the correct option.

3. Sum of ratio terms = 12 + 9 + 8 = 29.

Number of students in school R = $725 \times \dfrac{8}{29}$ = 8 × 25 = 200.

Hence, option (a) is the correct option.

4. Sum of ratio terms = 12 + 9 + 8 = 29.

Number of students in school P = $725 \times \dfrac{12}{29}$ = 12 × 25 = 300.

Number of students in school Q = $725 \times \dfrac{9}{29}$ = 9 × 25 = 225.

Number of students in school R = $725 \times \dfrac{8}{29}$ = 8 × 25 = 200.

New number of students in P = 300 + 16% of 300 = $300 + \dfrac{16}{100} \times 300$ = 300 + 48 = 348.

New number of students in Q = 225 + 12% of 225 = $225 + \dfrac{12}{100} \times 225$ = 225 + 27 = 252.

New number of students in R = 200 + 20% of 200 = $200 + \dfrac{20}{100} \times 200$ = 200 + 40 = 240.

New ratio = 348 : 252 : 240

Dividing each term by 12:

$\dfrac{348}{12} : \dfrac{252}{12} : \dfrac{240}{12} = 29 : 21 : 20$

Hence, option (b) is the correct option.