Ration & proportion
A ratio is a mathematical expression of comparing two similar or different quantities by division. It helps compare two things of similar or different dimensions. For example, in a box, there are 10 green balls and 5 yellow balls.
The ratio of green balls to yellow balls means how many numbers of green balls are there compared to yellow balls. Here, many green balls to yellow balls can be written as 10:5 which is same as 2:1 (simplified form). It tells us how much of the first quantity is there compared to that of second quantity.
Quantities like height, weight, length, objects, the population can be compared using the ratio method. It is expressed using the word “to” For example green balls to Yellow balls. There are two notations of expressing ratio –- 10:5 which is called odds notation and the fractional notation using 10/5.
It should be noted and kept in mind that the ratio of green balls to Yellow balls is not the same as that of yellow balls to green balls.
It not only compares to quantities of similar dimensions but also of different dimensions. For example, women to men in a population, boys to girls in a class. If the number of rectangles is given as 4 and squares is given as 7 the ratio of rectangles to squares is 4:7.
Numbers have various applications in daily life, and so does ratio. The ratio can be used in business, sports, construction of buildings and even cooking.
The comparison of ratios concept is used for the design of airplanes, tall buildings, and large ships.
When two ratios are equal in value, they are said to be in proportion. Thus, it compares two ratios. Proportions are denoted by the symbol ‘::’ or ‘=.’
For example –-a/b= c/d or a: b:: c: d
Ratio and Proportion Problems
Example 1. Are the ratios 4:8 and 5:10 in Proportion?
Since both ratios are equal, they are said to be in proportion.
Example 2. Are the ratios 3:5 and 4:5 in Proportion?
Since the ratios are unequal, they are not in proportion.
In the Unitary method, the value of a unit is found and then the value of the required number of units.
Example1- If a shopkeeper is selling 14 oranges for Rs.56, the cost of one orange can be found easily. It is Rs.56/14=Rs4.So if you want to buy 10 oranges, you must spend Rs.4×10= Rs. 40.
Example2-A bus travels 400 Km on 80 liters of petrol. How many kilometers will it run on 15 liters of fuel?
So, we know
80 liters is used to travel 400 Km
1 liter is used to travel 400/80 Km=5Km
So, with 15 liters of fuel, it travels=15×5=75Km
Therefore, the bus travels 75Km using 15 liters.
Unitary method has practical applications ranging from speed problems, distance, time to problems calculating the cost of materials.
Example3-A car is traveling at a speed of 120 kmph and covers 600 Km. How much time will it take to cover 720 km?
600 km=5 hours
720 km=5/600×720=6 hours
1) A: B: C is in the ratio of 3: 2: 5. How much money will C get out of Rs 1260?
- None of these
2) If a: b is 3: 4 and b: c is 2: 5. Find a: b: c.
- 3: 2: 5
- 3: 6: 5
- 3: 4:10
- 2: 3: 4
3) A: B is 1: 2; B: C is 3: 2 and C: D is 1:3. Find A: B: C: D.
- 3: 6: 4: 12
- 2: 3: 5: 7
- 3: 5: 7: 6
- 2: 1: 6: 13
4. 5600 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).
- Rs 2400, Rs 3000
- Rs 2000, Rs 3000
- Rs 2400, Rs 3200
- Rs 2000, Rs 3200
5) The ratio of the total amount distributed in all the males and females as salary is 6: 5. The ratio of the salary of each male and female is 2: 3. Find the ratio of the no. of males and females.
6) The number of employees is reduced in the ratio 3: 2 and the salary of each employee are increased in the ratio 4: 5. By doing so, the company saves Rs. 12000. What was the initial expenditure on the salary?
7) The ratio of the salary of A and B, one year ago is 3: 2. The ratio of original salary to the increased salary of A is 2: 3 and that of B is 3: 4. The total present salary of A and B together is Rs. 21500. Find the salary of B.
8) The ratio of income of two workers A and B are 3: 4. The ratio of expenditure of A and B is 2: 3 and each saves Rs 200. Find the income of A and B.
- 500, 600
- 600, 800
- 600, 900
- 800, 1000
9) The ratio of the expenditure of Pervez, Sunny, and Ashu are 16: 12: 9 respectively and their savings are 20%, 25%, 40% of their income. The sum of the income is Rs 1530, find Sunny’s salary.
10) The ratio of income of Pervez, Sunny, and Ashu is 3: 7: 4 and the ratio of their expenditure is 4: 3: 5 respectively. If Pervez saves Rs 300 out of 2400, find the savings of Ashu.
11) The ratio of income in two consecutive years is 2: 3 respectively. The ratio of their expenditure is 5: 9. Income of second-year is Rs 45000 and Expenditure of first-year is Rs 25000. Find the Savings in both years together.
12) Pervez, Sunny, and Ashu Bhati alone can complete a piece of work in 30, 50, and 40 days. The ratio of their salaries of each day is 4: 3: 2 respectively. The total income of Parvez is Rs 144. Find the total income of Sunny.
13) A person covers the different distances by train, bus, and car in the ratio of 4: 3: 2. The ratio of the fair is 1: 2: 4 per km. The total expenditure as a fair is Rs 720. Find the total expenditure as fair on the train.
14) The price of silver-biscuit is directly proportional to the square of its weight. A person broke down the silver-biscuit in the ratio of 3: 2: 1, and faces a loss of Rs 4620. Find the initial price of silver-biscuit.
15. B is inversely proportional to the cube of A. If B=3, A=2. If B = 8/9. Find the value of A.
16) Rs 7800 are distributed among A, B, and C. The share of “A” is the ¾ of the share of B, and the share of B is the 2/3 of the share of C. Find the difference between the share of B and C.
17) A bag contains Rs 410 in the form of Rs 5, Rs 2, and Rs 1 coins. The number of coins is in the ratio 4: 6: 9. So, find the number of 2 Rupees coins.
18) The ratio of copper and zinc in a 63 kg alloy is 4: 3. Some amount of copper is extracted from the alloy, and the ratio becomes 10: 9. How much copper is extracted?
The ratio of land and water on earth is 1: 2. In the northern hemisphere, the ratio is 2: 3. What is the ratio in the southern hemisphere?1:11
20) Vessels A and B contain mixtures of milk and water in the ratios 4: 5and 5: 1respectively. In what ratio should quantities of the mixture be taken from A and B to form a mixture in which milk to water is in the ratio 5: 4?
- 2: 5
- 4: 3
- 5: 2
- 2: 3