**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. Example: We can’t compare the height of a person to the weight of another person to determine who is taller. The ratio of a to b is represented as: **a to b or a:b or a/b**.

**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 D

**Explanation –** Let Pintu’s present age = x, Chintu = 2x and Mintu = x + 2x = 3x

Two years later \([\frac{2x+2}{3x+2}]\) = \((\frac{7}{10})\)

20x + 20 = 21x + 14

x = 6

Pintu’s present age = 6

Five years ago, his age was 6 – 5 = 1 year

**Q2. In a school, students of class I and class II are going for a picnic to Surajkund and Badkal lake respectively. The ratio f number of students in class I and II is 5:3. Also the ratio of the contribution made by each student of class I and II is 19:17. If the total contribution made by the all the students of both the class is Rs.29200, then find the total contribution made by class II students only**

**Answer –** Option C

**Explanation –** Let the no. of students in class II be 5x and 3x respectively and contribution made by each student of class I and class II be 19y and 17y respectively.

Hence, ratio of total contribution of class I and class II = 5x × 19y : 3x × 17y = 95 : 51

Total contribution made by students of class II (\(\frac{51}{(95+51)}\)) × 29200 = 10200

**Q3. A donkey moves at a speed of 8 kmph, when no load is put on him. Reduction in the speed of donkey varies directly to the square root of the kgs of load put on him. When only 4 kgs of load is put the speed of the donkey becomes 6 kmph. Find the minimum load that can be put on the donkey with which it cannot move.**

**D.** Cannot be determined

**E.** None of these

**Answer –** Option A

**Explanation –** Speed of the donkey, Without any load = 8 kmph. With 4 kgs of load, speed becomes 6 kmph, hence speed is reduced by 2 kmph.

Reduction in speed varies directly with the square root of the load.

Hence (8 – 6) = k√4 = k ± 1

The donkey cannot move at zero speed. i.e. when his speed reduced by 8 kmph. So reduction in speed = 8 kmph

8 = 1√l => √l = 8 and l = 64, when √l = -8, l = 64

At 64 kg the donkey will stop.

**Q4. A container has a mixture of kerosene and castor oil in the ratio of 7:5 and another container contains kerosene and castor oil in the ratio of 5:3. Find the proportion in which the mixtures from two containers should be mixed so that the resultant mixture has ratio of kerosene and castor oil of 3:2.**

**Answer –** Option B

**Explanation –** Let quantity of mixture taken from first be x and second be y.

Amount of kerosene oil in the resultant mixture (x + y) is

(\(\frac{7}{12}\))x + (\(\frac{5}{8}\))y = (\(\frac{3}{5}\))(x + y)

(\(\frac{7}{12}\))x – (\(\frac{3}{5}\))x = (\(\frac{3}{5}\))y – (\(\frac{5}{8}\))y – (\(\frac{1}{60}\))x = -(\(\frac{1}{40}\))y => (\(\frac{x}{y}\)) = (\(\frac{6}{4}\)) = (\(\frac{3}{2}\)) = 3 : 2

**Q5. There are three vessels 1, 2 and 3. The ratio of the total capacity of vessels 1, 2 and 3 is 5:4:3 respectively. All the vessels are full of mixture of sugar syrup and water. In vessel 1, ratio of sugar syrup to water is 2:3. Similar ratio in case of vessel 2 and vessel 3 is 5:4 and 1:3 respectively. The mixture of all the three vessels is emptied into one bigger vessel. What is the resulting ratio of sugar syrup and water?**

**Answer –** Option B

**Explanation –** Let the total mixture in vessel 1,2 and 3 be 5 litres and 3 litres respectively. So, quantity of water in (5 + 4 + 3) = 12 litres of mixture is

= (\(\frac{3}{5}\))×5 + (\(\frac{4}{9}\))×4 + (\(\frac{3}{4}\))×3

= 3 + (\(\frac{16}{9}\)) + (\(\frac{9}{4}\)) = (\(\frac{108+64+81}{36}\)) = (\(\frac{253}{36}\))

Quantity of sugar syrup is

12 – (\(\frac{253}{36}\)) = (\(\frac{179}{36}\))

Ratio of sugar syrup to water = 179 : 253

**Answer –** Option A

**Explanation –** If the total amount be Rs. x, then (\(\frac{2x}{15}\)) = 4908

x = \(\frac{(4908 × 15)}{2}\) = Rs. 36810

Required difference = (\(\frac{(7 – 6)}{15}\)) × 36810 = Rs. 2454

**Q2. The average age of a man and his son is 30 years. The ratio of their ages four years ago was 10:3 respectively. What is the difference between the present ages of the man and his son?**

**Answer –** Option A

**Explanation –** Four years ago,

Father’s age = 10x years , son’s age = 3x years

10x + 3x + 8 = 60 (sum of their ages = 2 × 30 = 60)

= 13x = 60 – 8

= x = 4

Required difference

**Q3. The ratio between Gloria’s and Sara’s present ages is 4:7 respectively, Two years ago the ratio between their ages was 1:2 respectively. What will be Sara’s age three years hence?**

**Answer –** Option A

**Explanation –** Let Gloria’s and Sara’s present ages be 4x and 7x years respectively

Two years ago, \(\frac{(4x – 2)}{(7x – 2)}\) = \(\frac{1}{2}\)

8x – 4 = 7x – 2

x = 2

Sara’s age three years hence = 7x + 3 = 17 years

**Q4. The total number of students in a school is 31700. If the ratio of boys to the girls in the school is 743:842 respectively, what is the total number of girls
in the school?**

**Answer –** Option E

**Explanation –** .Boys : Girls = 743 : 842

Total number of students = 31700

Number of girls = (\(\frac{842}{(743 +842)}\)) × 31700 = (\(\frac{842}{1585}\)) × 31700

= 16840

**Q5. A sum of Rs. 10,980 is to be divided amongst A, B and C in the ratio 7:3:5 respectively. How much is C’s share?**

**Answer –** Option D

**Explanation –** C’s share = (\(\frac{5}{15}\)) × 10980 = Rs. 3660

**Answer –** Option C

**Explanation –** Girls : Boys = 2 : 1. Total no.of students is 60

Girls 60 × (\(\frac{2}{3}\)) = 40 and boys = 20. Kamal has been ranked 17 which means there are 16 students before him of which 9 are girls remaining 7 are boys.

7 boys are ahead of Kamal + Kamal is the 8\(^{th}\) boy, hence, there are (20 – 8) = 12 boys after him.

**Q2. Two varieties of rice at Rs. 10 per kg and Rs. 12 per kg are mixed together in the ratio 1 : 2. What is the price of the resulting mixture?**

**Answer –** Option D

**Explanation –** Let the amount of rice be 1 kg and 2 kg. (The given ratio is 1 : 2). So, total cost = 10 × 1 + 12 × 2 = 10 + 24 = 34

Rs. 34 is the cost of (1 + 2 = 3) kg of rice.

Cost per kg = (\(\frac{34}{3}\)) = 11.33 per kg

**Q3. The ratio of the number of boys and girls in a school is 3 : 2. If 20% of the boys and 25% of the girls are scholarship holders, then the percentage of the students who did not get scholarship is:**

**Answer –** Option B

**Explanation –** Number of students in school = 100 (let)

Boys = (\(\frac{3}{5}\)) × 100 = 60, Girls = 40

Students who did not get scholarship

Boys = 60 × (\(\frac{80}{100}\)) = 48

Girls = 40 × (\(\frac{75}{100}\)) = 30

Students who do not get scholarship = 78

Required percentage = 78

**Q4. In a mixture of 60 litres, the ratio of acid and water is 2 : 1. If the ratio of acid and water is to be 1 : 2, then the amount of water (in litres) to be added to the mixture is**

**Answer –** Option D

**Explanation –** In 60 litres of mixture,

Acid = (\(\frac{2}{3}\)) × 60 = 40 litres, Water = 20 litres

If x litres of water be mixed, then (\(\frac{40}{(20 + x)}\))= 80

x = 80 – 20 = 60 litres

**Q5. Salaries of Akash, Babloo and Chintu are in the ratio of 2 : 3 : 5. If their salaries were increased by 15%, 10% and 20% respectively, what will be the new ratio of their salaries?**

**Answer –** Option B

**Explanation –** .Required ratio = (\(\frac{(2 × 115)}{100}\)) : (\(\frac{(3 × 110)}{100}\)) : (\(\frac{(5 × 120)}{100}\)) 230 : 330 : 600 = 23 : 33 : 60

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