**What are Ratios?**[**a:b **or **a to b **or **a/b**]

Ratios are the mathematical numbers used to compare two entities which are similar to each other in terms of units. For example, we can compare the **height of student 1** to the **height of student 2**. ** FACT:** Properties/Elements that are not similar cannot be compared. Comparison of non-similar properties does not reveal any details of the entities being compared.

**What are Proportions**: [** a/b = c/d** or **a:b :: c:d** **Ex:** 5/7 = 30/42 ]

**Ratios** compare things similar to each other. The ratios are compared with each other using **proportions**. **Proportion** is an equation to represent that two ratios are equivalent. When **two ratios are the same**, they are said to be proportionate to each other or they are said to be in proportion. Proportions are represented by **‘::’ or ‘=’ sign**. Proportions are primarily used to find missing quantities by using the fact that one ratio is equal to the other.

In simple terms, a **ratio **is a way to compare two quantities by using division. A **proportion** on the other hand is an equation that says that two ratios are equivalent. If one number in a proportion is unknown you can find that number by solving the proportion.

**Answer –** Option C

**2. If \(\frac{x}{5} = \frac{x}{8}\), then (x+ 5) : (y + 8) is equal to**

**Answer –** Option D

**3. If x : y = 6 : 5, then ( 5x + 3y) : ( 5x – 3y) is equal to**

**Answer –** Option B

**4. What same number must be added to each term of the ratio 7 : 3 so that the ratio becomes 2 : 3 **

**Answer –** Option D

**5. The ratio of the two numbers is 3 : 4 & their sum is 420. The greater of the two numbers is**

**Answer –** Option C

**6. Five bananas and four apples cost as much as three bananas and seven apples. The ratio of the cost of one banana to that of one apple is**

**Answer –** Option A

**7. The speeds of three cars are in the ratio 5 : 4 : 6. The ratio between the times taken by them to travel the same distance is**

**Answer –** Option D

**8. A dog takes 3 leaps for every 5 leaps of a hare. If one leap of the dog is equal to 3 leaps of the hare, the ratio of the speed of the dog to that of the hare is**

**Answer –** Option B

**9. Rs 180 contained in a box consists of one rupee, 50 paise and 25 paise coins in the proportion of 2 : 3 : 4. What is the number of 50 paise coins?**

**Answer –** Option A

**10. In a school, 10% of the boys are same in number as\(\frac{1}{4}\) of the girls and 10% of the girls are same in number as 1/25 of the boys. What is the ratio of boys to girls in that school **

**Answer –** Option B

**11. Two numbers are in the ratio 3 : 4 and the product of their L.C.M. & H.C.F is 10800. The sum of the numbers is**

**Answer –** Option B

**12. The ages of x and y are in the ratio of 3 : 1. After 15 years the ratio will be 2 : 1. Their present ages(in years) are**

**Answer –** Option B

**13. Gold is 19 times as heavy as water and copper is 9 times as heavy as water. In what ratio should there be mixed to get an alloy 15 times as heavy as water**

**Answer –** Option D

**14. 85 liters of a mixture contains milk and water in the ratio 27 : 7. How much more water is to be added to get a new mixture containing milk and water in the ratio 3 : 1?**

**Answer –** Option A

**Answer –** Option A

**Explanation –**

S : R = 7X : 9x

9x – 7x = 4

or x = 2

Hence, S or age of Sachin = 7 × 2 = 14 years.

**2. The sum of two numbers is 40 and the difference of these two numbers is 4. Find the ratio of these two numbers.**

**Answer –** Option A

**Explanation –**

a +b = 40 and a – b = 4

a = 22 and b = 18

Hence, a : b = 11 : 9

**3. If A exceeds B by 40% and B is less than C by 20%. then A : C = ?**

**Answer –** Option D

**Explanation –**

Let C = 100, such that B = 80 and

A = 80 * 1.4 = 112

Hence, A : C = 112 : 100 = 28 : 25

**4. If a : b = 3 : 5 and b : c = 2 : 3, then a : c is**

**Answer –** Option B

**Explanation –**

Given: a : b = 3 : 5 and b : c = 2 : 3

a : b = 6 : 10 and b : c = 10 : 15

(making the value of ‘b’ equal in both the ratios)

a : c = 6 : 15 = 2 : 5.

**5. Which of the following numbers are in proportion**

**Answer –** Option C

**Explanation –**

Among the given options only 12, 27, 24 and 54 are in proportion i.e. 12 : 27 = 24 : 54 = 4 : 9.

**6. In two vessels A and B, milk and water are in the ratio of 4 : 3 and 3 : 5 respectively. The ratio in which these are to be mixed to obtain new mixture which contains half milk and half water**

**Answer –** Option A

**Explanation –**

Applying allegation for milk we get:

A | B |
---|---|

\(\frac{4}{7}\) | \(\frac{3}{8}\) |

\(\frac{1}{8}\) | \(\frac{1}{14}\) |

Hence, required ratio = \(\frac{1}{8}\) : \(\frac{1}{14}\) = 7 : 4

**7. If a : b = 4: 5 and b : c = 2 : 3, then c : a is**

**Answer –** Option C

**Explanation –**

a : b = 4 : 5 and b : c = 2 : 3

a : b = 8 : 10 and b : c = 10 : 15

c : a = 15 : 8

**8. If a : b = 5 : 6 and b : c = 3 : 4, then c : a is**

**Answer –** Option C

**Explanation –**

a : b = 5 : 6; b : c = 3 : 4

= 15 : 18 = 15 : 20

= 18 : 24

i.e, c : a = 24 : 15

= 8 : 5

**9. If a : b = 2 : 3 and a + b = 45. then a is equal to **

**Answer –** Option C

**Explanation –**

\(\frac{a}{b}\) = \(\frac{2}{3} a + b\) = 45

a : b = 2x : 3x

5x = 45

a = 2x = 18

**10. If 2, x, \({x}^{2}\), 4 are in proportion, then x is equal to**

**Answer –** Option A

**Explanation –**

2, x, \({x}^{2}\)x2, 4 are in proportion

\({2x}^{2}\) = x × 4

x = 2

**11. In two vessels A and B, spirit and water are in the ratio 2 : 1 and 2 : 3 respectively. The ratio in which these ire mixed which contains half water**

**Answer –** Option B

**Explanation –**

i.e, ratio = \({1}^{10}\) = \({1}^{6}\)

= 6 : 10 = 3 : 5

**12. The average of 20 observations is 18 and average of 30 observations is 25. The average of all 50 observations is**

**Answer –** Option C

**Explanation –**

Average = \(\frac{20 * 18 * 30 * 25}{50}\) = 22.2

**13. If 4, x, 2x, 32 are in proportion, then x is equal to**

**Answer –** Option B

**Explanation –**

4, x, 2x, 32 are in propertion

Product of extremes = product pf weavs

4 × 32 = 2x × x

x = 8

**14. If 8, 3x, 6, 27 are in proportion, then x is equal to**

**Answer –** Option C

**Explanation –**

Product of means = product of extremes

3x × 6 = 8 × 27

x = 12

**15. In two vessels A and B, the milk and water are in the ratio 5 : 4 and 3 : 5 respectively. The ratio in which these are mixed to obtain new mixture which Contains half milk and half water is**

**Answer –** Option A

**Explanation –**

Let quantity of liquid of A m mixture = x Let _____ B _____ = y

i.e, Milk from A + Milk from B = total milk

\(x\frac{5}{9} + y\frac{3}{8} = (x + y) \frac{1}{2}\)

\(\frac{x}{y} = \frac{9}{4}\) = 9 : 4

**Answer –** Option D

**Explanation –**

a : b = 4 : (3)* 7 = 4 *3 = 4 * 7 : 3 * 7

b : c 7 : (9)* 3 and 7 : 9 = 7 * 3 : 9 *3

a : b = 28 : 21

b : c = 21 : 27

i.e, a : b : c = 28 : 21 : 27

**2. Ravi spends \(\frac{2}{5}\) of his salary on House Rent; \(\frac{3}{10}\) of his salary on Food and \(\frac{1}{8}\) of his Salary on Conveyance. After this, he is left with Rs. 1400. Find his expenditure on Food**

**Answer –** Option C

**Explanation –**

Let the salary of Ravi be Rs. x

Total expenditure = \(x \frac{2}{5}\) + \(x \frac{3}{10}\) + \(x \frac{1}{8}\) = \(x \frac{33}{40}\)

Money left = 1 – \(x \frac{33}{40} = x \frac{7}{40}\) = 1400

x = 8000

i.e, Money spent on food

= \( \frac{3}{10}\)* 8000 = 2400

** 3. A sum of Rs. 312 was divided among 60 boys and some girls in such a way that each boy gets Rs.3.60 and each girl Rs. 2.40. The number of girls is**

**Answer –** Option B

**Explanation –**

No. of girls

\(\frac{3{\frac {3}{2}} – 60 * 3.6}{2.4}\) = 40

**4. Some students planned a trip. The budget for food was Rs. 500 But, 5 of them failed to go and thus the cost of food for each member increased by Rs.5. How many students attended the trip ?**

**Answer –** Option B

**Explanation –**

Let the no. of students initially = x

According to question \(\frac{\frac{500}{x – 5}}{\frac {500}{x}}\) = 5

x = 25

No. of students who attended the trip = x – 5 = 20

**5. In a class, there are two sections A and B. I f 10 students of section B shift over to section A, the strength of A becomes three times the strength of B. But, if 10 students shift over from A to B, both A and B are equal in strength. How many students are there in sections A and B ?**

**Answer –** Option A

**Explanation –**

Let the no. of students in class A = x

Let the no. of students in class B = y

According to question

(x + 10) = 3(y – 10) … (i)

also (x – 10) = y + 10 … (ii)

Solving eq. (i) and (ii), we get

x = 50 and y = 30

**6. Six persons went to a hotel for meals. Five of them spent Rs. 32 each on their meals while the \({6}^{th}\) person spent Rs. 80 more than the average expenditure of all the six. Total money spent by all the persons is :**

**Answer –** Option C

**Explanation –**

Let avg expenditure be x.

i.e, x = \(\frac{32 * 5 + (80 + 2)}{6}\)

x = 48

i.e, Total money spent = (32 × 5) + (80 + 48)

= 288

**7. X is 40 years old and Y is 60 years old How many years ago was the ratio of their ages 3:5 **

**Answer –** Option B

**Explanation –**

According to question,

\(\frac{40 – k}{60 – k}\) = \(\frac{3}{5}\)

k = 10 years

**8. Rs. 680 is divided among A, B, C such that A gets \(\frac{2}{3}\) of what B gets and B gets \(\frac{1}{4}\) of what C gets. Then their shares are respectively**

**Answer –** Option B

**Explanation –**

According to question,

\(A = \frac{2B}{3}\)

\(B =\frac{3A}{2}\)

and \(B = \frac{1C}{4}\)

C = 4B = 4\(\frac{3A}{2}\)

C = 6A

Now A + B + C = 680

A + \(\frac{3A}{2}\)+ 6A = 680

i.e, A = 80

i.e, B = 120

i.e, C = 480

**9. X. Y and Z start a business X invests 3 times as much as Y invests and Y invests \(\frac{2}{3}\)rd of what Z invests. Then the ratio of capitals of X. Y, Z is 3**

**Answer –** Option D

**Explanation –**

According to question, x = 3y

x : y = \(\frac{3}{1} * \frac{2}{2} * \frac{6}{2}\)

y = \( z \frac{2}{3}\)

y : x = 2 : 3

x : y = 6 : 2

x : y : z = 6 : 2 : 3

**10. Entry fee to an exhibition was Rs.80. Later, this was reduced by 25% which increased the sale by 20%. The percentage increased in the number of visitors is**

**Answer –** Option C

**Explanation –**

Entry fee × No. of visitors = sales.

Multiplying factor = \(\frac{3}{4} * x = \frac{6}{5}\)

x = \(\frac{8}{5}\) = 1 + \(\frac{3}{5}\) = increase of \(\frac{3}{5}\) in no. of persons

60 % increase

**11. If a : b = 8 : 15 , b : c = 5 : 8 and c : d = 4 : 5, then b : d is**

**Answer –** Option A

**Explanation –**

\(\frac{a}{b}\) = \(\frac{8}{15}\);\(\frac{b}{c}\) = \(\frac{5}{8}\);\(\frac{c}{d}\) = \(\frac{4}{5}\);

\(\frac{a}{b}\) * \(\frac{c}{d}\) = \(\frac{8}{15}\) * \(\frac{4}{5}\) = \(\frac{1}{2}\)

**12. A person gave \(\frac{1}{5}\) part his income to his son and 40 % part of his income to his daughter. He lent out the remaining money in three trusts A, B and C in the ratio of 5 : 3 : 2. I f the difference between the amount got by son and daughter is Rs. 50,000, how much amount did he invest in trust B?**

**Answer –** Option B

**Explanation –**

Amount received by son = 20%

Amount received by daughter = 40%

Amount given to trust = 40 %

Difference = 20% is Rs. 50,000

40% = Rs. 1,00,000.

A : B : C = 5 : 3 : 2

B gets \(\frac{3}{10}\) * 1,00,000 = Rs. 30,000

**13. Two alloys A and B contain zinc and copper in the ratio 5 : 6 and 7 : 8 respectively. If equal quantities of alloys are melted to form a third alloys C, then the ratio of copper and zinc in C will be**

**Answer –** Option B

**Explanation –**

Let quantity of alloy A = x

Let quantity of alloy B = y

Zinc in alloy (A + B) = \(\frac{5x}{11}\) + \(\frac{7y}{15}\)

copper in alloy (A + B) = \(\frac{6x}{11}\) + \(\frac{8x}{15}\)

\(\frac{Copper in C}{Zinc in C}\) = \(\frac{\frac{6x}11{} + \frac{8x}{15}}{\frac{5x}{11} + \frac{7y}{15}}\) = \(\frac{89}{76}\)

i.,e x = y (quantities are same)

**14. If A : B = 2 : 3, B : C = 5 : 6 and C : D 8 : 9, then A : D is**

**Answer –** Option D

**Explanation –**

\(\frac{A}{B}\) = \(\frac{2}{3}\),\(\frac{B}{C}\) = \(\frac{5}{6}\), \(\frac{C}{D}\) = \(\frac{8}{9}\)

\(\frac{A}{B}\) * \(\frac{B}{C}\) * \(\frac{C}{D}\) = \(\frac{2}{3}\) * \(\frac{5}{6}\) * \(\frac{8}{9}\)

\(\frac{A}{D}\) = \(\frac{40}{81}\)

**15. A sum of Rs.16500 is to be divided among A, B, C and D in such a way that the ratio of shares of A and B is 3:4, that of B and C is 1:3 and that of C and D is 6:7. Sum of shares of A and D is**

**Answer –** Option C

**Explanation –**

A : B : C = 3 : 4 : 12

C : D = 12 : 14

A : B : C : D = 3 : 4 : 12 : 14

sum of shares of A and D is \(\frac{3 + 14}{33}* 16500\) = 8500