RATIO AND PROPORTION

 RATIO AND PROPORTION

Ratio 

            The ratio of two quantities let a and b in the same units, is the fraction of a/b and it can be  written as a ∶ b .

            Here, a is the first term also called as antecedent and 

                      b is the second term also called as consequent.

Proportion

             The equality of two ratios is called proportion.

If a ∶ b = c ∶ d, then we can write a ∶ b ∶∶ c ∶ d and we can say a, b, c, d are in proportion.

Here a and d are called extremes & b and c are called mean terms.

Products of extremes = Products of means

         Then,  a ∶ b ∶∶ c ∶ d ⟹ a × d = b × c

• If a ∶ b = c ∶ d, then d is called fourth proportional to a, b, c.
• if a ∶ b = b ∶ c, then c is called third proportional to a and b.
• mean proportional between a and b is √ab .

If a ∶ b ＞ c ∶ d  ⟹ a/b ＞ c/d  - Comparison of ratios

1. Duplicate ratio of  a ∶ b is  a² :  b²    
2. Sub-duplicate ratio of  a ∶ b is √a ∶ √b
3. Triplicate ratio of a ∶ b is a³ :  b³   
4. Sub-triplicate ratio of a ∶ b is ∛a ∶ ∛b 

Componendo and Dividendo of ratios : 

                            

     If a/b = c/d, 

              then (a + b) / (a − b) = (c + d) / (c − d)

Examples

1. If a ∶ b = 5 ∶ 9 and b ∶ c = 4∶ 7, find a ∶ b ∶ c.

  Solution:

                Here b is common in both ratios. So, same the part of b in both ratios.

                Then take LCM of b in both ratios, accordingly same the part b value.

                So, LCM (9, 4) = 36

                In a ∶ b = 5 ∶ 9 = 20 ∶ 36 & b ∶ c = 4 ∶ 7 = 36 ∶ 63, so b is same in both parts.

                Then, a ∶ b ∶ c = 20 ∶ 36 ∶ 63 (Ans.)

2. Find the 3rd proportional to 16 & 36.

   Solution:

              Let 3rd proportional to 16 & 36 be x.

                16 ∶ 36 ∶∶ 36 ∶ x ⟹ x = 36 × 36 / 16 = 81 (Ans.)

3. Divide Rs.1162 among A, B, C in the ratio 35 ∶ 28 ∶ 20.

   Solution :

            A ' share = [35 / (35 + 28 + 20)] × 1162 = Rs.490

             B ' share = [28 / (35 + 28 + 20)] × 1162 = Rs.392

             C ' share = [20 / (35 + 28 + 20)] × 1162 = Rs.280

4. A mixture contains alcohol and water in the ratio 4  ∶ 3. If 5 liters of water is added to the mixture, the ratio becomes 4  ∶ 5. find the quantity of alcohol in the given mixture.

    Solution:

                In the given mixture, alcohol = 4x & water =3x

                then, 4x / (3x + 5) = 4 /5

                 ⟹ x = 2.5

      Alcohol = 4x = 4 × 2.5 = 10 liter (Ans.)

Do it Yourself                 

1. If x ∶ y = 3 ∶ 4, then what is the value of (3x − y) ∶ (x + 2y) ?

                    A. 5 ∶ 11                                  B. 11 ∶ 5

                   C. 16 ∶ 55                                 D. 55 ∶ 16

2. The salary of X, Y and Z are in the ratio 5 ∶ 3 ∶ 1. The salary of X and Y together is Rs.7200. By what percent is salary of X more than that of Z ?

                    A. 100%                                  B. 200%

                   C. 300%                                   D. 400%

3. If a  ∶ b  ∶ c = 3  ∶ 4  ∶ 7, then the ratio (a + b + c)  ∶ c is ?

                    A. 2  ∶  1                               B. 14  ∶ 3

                    C. 7  ∶   2                              D. 1  ∶ 2

4.  If x ∶ y = 1 ∶ 2, then what is the value of (2x + 3y) ∶ (x + 4y) ?

                    A. 8 ∶ 9                                   B. 7 ∶ 8

                    C. 5 ∶ 7                                   D. 7 ∶ 5

5. Rs.13,400 are divided among three workers A, B & C in the ratio of 4/5 ∶ 2 ∶ 5/3 . Find the shares of each worker.

                     A. 2400, 6000, 5000                                  

                     B. 2400, 5000, 6000

                     C. 3600, 8000, 1800                                 

                     D. 5000, 6000, 2400

6. A box has 210 coins of denominations one-rupee and fifty paise only. The ratio of their respective values is 13 ∶ 11. The number of one rupee coin is 

                     A. 65                                      B. 66

                    C. 77                                       D. 78

7. If  A ∶  B = 2 ∶ 3 and B ∶ C = 4 ∶ 5, then A ∶ B ∶ C is 

                     A. 2 ∶ 3 ∶ 5                                   B. 5 ∶ 4 ∶ 6

                    C. 8 ∶ 12 ∶ 15                               D. 6 ∶ 4 ∶ 5

8. If A and B are in the ratio 3 ∶ 4 and B and C in the ratio 12 ∶ 13, then A and C will be in the ratio

                     A. 3 ∶ 13                                B. 9 ∶ 13

                    C. 36 ∶ 13                               D. 13 ∶ 9

9. If 2/3 of A = 75% of B = 0.6 of C, then A ∶ B ∶ C is 

                     A. 2 ∶ 3 ∶ 3                                B. 3 ∶ 4 ∶ 5

                    C. 4 ∶ 5 ∶ 6                                 D. 9 ∶ 8 ∶ 10

10. If A ∶ B ∶ C = 2 ∶ 3 ∶ 4, then A/B ∶ B/C ∶ C/A is 

                     A. 4 ∶ 9 ∶ 16                               B. 8 ∶ 9 ∶ 12

                    C. 8 ∶ 9 ∶ 16                                D. 8 ∶ 9 ∶ 24

11. Two numbers are in the ratio 3 ∶ 5. If 9 is subtracted from each, the new numbers are in the ratio 12 ∶ 23. The smaller number is 

                     A. 27                                      B. 33

                    C. 49                                       D. 55

12. A fraction which bears the same ratio to 1/27 that 3/11 does to 5/9, is equal to :

                     A. 1/55                                      B. 1/11

                    C. 3/11                                       D. 55 

13. Two numbers are respectively 20% and 50% more than a third number. The ratio of two numbers is 

                     A. 2 ∶ 5                                 B. 3 ∶ 5

                    C. 4 ∶ 5                                  D. 6 ∶ 7

14. The sum of three numbers is 98. If the ratio of the first to the second is 2 ∶ 3 & that of the second to the third is 5 ∶ 8, then the second number is :

                    A. 20                                      B. 30

                    C. 48                                      D. 58

15. If 40%  of a number is equal to two-third of another number, what is the ratio of first number to the second number?

                     A. 2 ∶ 5                                 B. 3 ∶ 7

                    C. 5 ∶ 3                                  D. 7 ∶ 3

16. A bag contains 50p, 25p & 10p coins in the ratio 5 ∶ 9 ∶ 4, amounting to Rs.206. Find  the number of coins of each type.

17. The incomes of A, B, C are in the ratio 12 ∶ 9 ∶ 7 and their spending are in the ratio 15 ∶ 9 ∶ 8. If A saves 25% of his income, then what is the ratio of savings of A, B and C?

                    A. 11 ∶ 13 ∶ 18                       B. 15 ∶ 18 ∶ 11

                    C. 16 ∶ 17 ∶ 21                       D. 13 ∶ 16 ∶ 11

Some Practice Questions.

1. Rs.366 are divided amongst A, B & C so that A may get 1/2  as much as B and C together, B may get 2/3 as much as A and C together, then the share of A is______. (Ans.Rs.122)

2. A and B together have Rs.1210. If 4/15 of A's amount is equal to 2/5 of B's amount, how much amount does B have? 

3. Rs.3200 is to be divided into A, B, C and D in such a way that the ratio of share of A ∶ B is 1 ∶ 2, B ∶ C is 3 ∶ 1 and C ∶ D is 2 ∶ 3. Find the sum of (A and C) and (B and C) ?

4. Sum of three numbers is 98. If the ratio between the first and second be 2 ∶ 3 and that, between second and 3rd is 5 ∶ 8, then find the second number. (Ans.30) 

5. (4x² - 3y²) ∶ (2x² + 5y²) = 12 ∶ 19, find x/y. (Ans.1.5)

6. Ratio between the values of tickets of I, II and IIIrd class is 3 ∶ 2 ∶ 1 and the ratio in number of passengers corresponding is 2 ∶ 3 ∶ 6. Find the ratio of total collection each class? (Ans.1 ∶ 1 ∶1)

7. Speeds of three cars are in ratio 4 ∶ 3 ∶ 2. The ratio between the time taken by the cars to cover the same distance is _____. (Ans. 3 ∶ 4 ∶ 6)

8. Duplicate ratio of 3 ∶ 4 is ________. (Ans. 9 ∶ 16)

9. 1 man can do as much work as 2 women, 2 women can do as much as 3 boys. What is their efficiency ratio? (Ans. 6 ∶ 3 ∶ 2)

10. If Rs.510 be divided among A, B, C in such a way that A gets 2/3 of what B gets and B gets 1/4 of what C gets, then their shares are respectively___________. 

11. An amount of Rs.1050 is distributed among A , B & C in the ratio of 2 ∶ 5 ∶ 8. What is difference between the shares of C & B? (Ans.210)

12. Zinc and Cooper are in the ratio of 6 ∶ 5 in 220 gm of alloy. How much gm of copper can be added so as to make it in the ratio of 5 ∶ 6? (Ans.44)

13. X varies inversely as square of Y. If Y = 2, when X = 1, then value of X for Y = 6, is ? (Ans. 1/9)

14. A college has admitted 40% additional number of students during the current year than previous year. Due to Government stipulation, girls have been exempted from fees this year and hence the college has enhanced the fees of the boys by 25% to equalise the total fee collection with last year.  What is the ratio of boys and girls this year? (Ans. 4 : 3)

Solution:-

Let no. of students = x
Fees collected per student = y
Total fee = xy
Now, no. of students increased by 40%.
Then, total no. of students = 1.4x
Here, enhanced fee for boys = 25%
Total fee = y + y25% = 1.25y
No. of boys = xy / 1.25y = 0.8x
No. of girls = 1.4x - 0.8x = 0.6x
Ratio = Boys : Girls = 0.8 : 0.6 = 4 : 3 (Ans.)

15. In a school of 1440 students,  boys and girls are in a ratio of 7 : 5. If 40 new boys are admitted,  then how many new girls need to be admitted to make this ratio 5 : 4 ?

16. The ratio of syrup and water in a mixture is 3 : 1, then the percentage of syrup in this mixture is: _________.