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Comparison of two different numbers is done to find out how many times is one number greater than the other number. There is a requirement of the ratios for the comparison between the two terms. Hence the word ratio means the quantitative relationship of two amounts or different or sometimes similar numbers. This is usually derived with the help of the fractions, which on further simplification will reduce to decimals which of them are all included in the rational numbers but not integers.
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Key Terms: Percentage, Ratio, Number, Fraction, Quantity, Antecedent, Consequent
What is Ratio?
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In ratio, we compare two quantities of the same kind and consider what multiple or part one is of the other. In other terms, a ratio is a fraction that shows how many times a quantity is of another quantity of the same kind. The ratio of two quantities of the same kind and in the same units is a fraction that shows how many times one quantity is of the other.
The ratio is that relation between two numbers which is expressed by the fraction, the numerator of which is the measure of the first quantity and the denominator is the measure of the second quantity. The two numbers which form the ratio are called its terms and the first number is called the antecedent and the second number is called the consequent.
Read More: Ratio to Percentage
Simplest Form and Standard Form of Ratios
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A ratio is said to be in its lowest terms when the two terms of the ratio are prime numbers otherwise having one of the terms to be a prime number. i.e, their H.C.F is 1. Hence the ratio is said to be in the simplest form when the Antecedent and the consequent are having the same H.C.F.
If the antecedent and the consequent remain intact without simplification it will be said to be in the standard form. Generally, the simplest form is used in all the applications and is used to compare them easily.
Note: In a ratio, we compare two quantities. This type of comparison becomes meaningless if the quantities being compared are not of the same kind i.e. they are not measured in the same units. It is just meaningless comparing 20 bags with 200 crows. Therefore, to find the ratio of two quantities, they must be expressed in the same units.
How to Compare Ratios?
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Mentioned below are the steps to compare two ratios:
Step 1: Firstly, we make the consequent of both of the ratios equal, and calculate the LCM of both of the numbers. Once, the LCM was determined, divide both of them of the consequent and antecedent of both numbers and the corresponding values are multiplied with the quotient obtained.
Step 2: Comparing the antecedent of both the ratios with each other.
Step 3: After this, we can compare the two ratio values since they have the same consequence. So, they can be compared with each other. For example take 5:4 and 2:6. To determine which ratio is greater, firstly, calculate the LCM of the consequent of both ratios which is 12 for 4 and 6.
Secondly, finding the quotient as 12 ÷ 6 = 2 and 12 ÷ 4 = 3.
Therefore, (2 x 2) : (6 x 2) = 4 and 12 (5 x 3):(4 x 3) = 15 and 12 as 15 > 4, the ratio 2:6 is less than 5:4.
Methods Used to Compare Ratios
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There are primarily two methods of comparing ratios. They are the LCM method and the cross multiplication method.
- LCM Method: The LCM method is simple and can be described in the above solved example by calculating the LCM and then the quotient.
- Comparing Ratios by Cross Multiplication Method: Simply by cross multiplying, the ratios arranged side by side we get the product. And after comparing the products we can make the conclusions as shown in the below picture.
Comparison of Ratios
Ratio Between Three Quantities
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The comparison between the terms can be extended to the 3 terms. When we were given a ratio, the relationship between the income of the two employees A and B. Furthermore, there is another relationship between the income of employees B and C. Now while comparing the two ratios provided to you, one can easily present a single ratio between the income of employees A, B, and C.
For example, if there are three numbers in the ratio: '2', '5' and '3'. This implies that the amount is shared between the three people. 2 + 5 + 3 equals a total of 10 parts in the ratio. To find the value of each part, we divide the amount of 20 by the total number of parts.
20 / 10 = 2. Finally, we multiply each of the three numbers in the ratio by the value of each and every part. 2:5:3 multiplied by any numbers so that we can get the proportional quantities.
Ratio Between Three Quantities
The value of each of the parts is found by dividing the given amount by the sum of the parts present in the ratio Then multiplying each number in the ratio by the value of each part.
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Unitary Method
The method in which first we find the value of one unit and then the value of the required number of units is known as the unitary method. This method helps us to achieve the same thing. The below diagram gives a description of the working of the unitary method.
Unitary Method
From above image, we can conclude that:
Value of one article = (Value of the given number of articles) / (Number of articles)
Or
The Value of required number of articles = (Value of one article) × (Required number of articles)
Things to Remember
- The ratio of two quantities of the same kind determines how many times one quantity is contained by the other. So, the ratio of any two quantities of the same kind is an abstract quantity.
- A ratio has no unit or it is independent of the units used in the quantities compared.
- The order of the terms in a ratio a : b is very important. For example the ratio 3 : 2 is completely different from the ratio 2 : 3.
- Percentages is another important concept that depends highly on ratios and proportions.
- If the ratio is greater than 1, then the first term is greater than the second one. Also, if the ratio is less than 1, then the second number is greater than the first one.
Sample Questions
Ques. Given the ratio of P’s salary to Q’s salary is 2:3 also the ratio of Q’s salary to R’s salary is 4:5. What is the ratio of P’s salary to R’s salary? (3 marks)
Ans. Let’s start by considering two values of Q, they are the common values given in the ratio. Hence, 3 and 4 are those two values and take LCM of the values. The LCM will be 12. Now, convert Q’s value in each ratio to 12.
So, ratio 1 = 8/12 ; the ratio 2 = 12/15
Thus, P : Q : R = 8 : 12 : 15.
If it is given that P’s salary was 400, then we can find that R’s salary to be INR 750. Hence the R’s salary can be given as INR 750.
Ques. Divide 60 rupees in the ratio 1 : 2 between two persons Kriti and Kiran Mai? (3 marks)
Ans. The two parts are 1 and 2. Therefore, sum of the parts = 1 + 2 = 3. This implies if there are 3, Kriti will get 1, and Kiran Mai will get 2.
Or, we can say that Kriti gets 1 part and Kiran gets 2 parts out of all 3 parts. Therefore, Kriti’s share = 1/3(60) = 20 rupees.
And Kiran Mai’s share = 2/3(60) = 40 rupees.
Ques. The length and breadth of a rectangular field are 50 mts and 15 mts respectively. Also, find the ratio of the length to the breadth of the field (rectangle). (3 marks)
Ans. Length of rectangular field = 50 mts
The breadth of rectangular field = 15 mts
The ratio of the length to its breadth is 50: 15
The ratio can be written as 50/15 = 10/3 = 10 : 3
Thus, the required ratio is 10 : 3.
Ques. There are 45 people working in an office. If the number of females is given as 25 and remaining are males, calculate the ratio of: (5 marks)
The no. of females to a number of males.
The no. of males to a number of females.
Ans. Number of females = 25
Total number of workers = 45
Number of male persons = 45 – 25 = 20
Hence, the ratio of number of females to that of the number of males
= 25 : 20 = 5 : 4
And the ratio of males to that of females is
= 20 : 25 = 4 : 5.
We can clearly spot the difference between the two ratios 5 : 4 and 4 : 5 in this example.
Ques. A certain motorbike travels 220 km in 5 litres of petrol. How much of the distance will it cover in 1.5 litres of petrol? (3 marks)
Ans. In 5 litres of petrol, a motorbike can travel 220 km.
Therefore, in 1 litre of petrol, a motorbike travels = 220/5 kms.
Therefore, in 1.5 litres, motorbike travels = 220/5 × (1.5) km
= (220/5) (15/10) km = 66 km.
Thus, a motorbike can travel 66 km in 1.5 litres of petrol.
Ques. The cost of 105 envelopes is 35 rupees. How many envelopes could be purchased with 10 rupees? (3 marks)
Ans. In 35 rupees, the number of envelopes that can be purchased = 105
Therefore, in 1 rupee, the number of envelopes that can be purchased = (105/35)
Therefore, in 10 rupees, the number of envelopes that can be purchased
= 105/35 × 10 = 30.
Thus, 30 envelopes can be purchased for 10 rupees.
Ques. A car travels 90 km in 2 1/2 hours. (5 marks)
(a) How much time is required to cover 30 km with the same speed?
(b) Find the distance covered in 2 hours with the same speed?
Ans. (a) In this case, time is unknown and distance is known. Therefore, we proceed as follows:
2 x 1/2 hours = 5/2 hours = 5/2 × 60 minutes = 150 minutes.
90 km is being covered in 150 minutes Therefore, 1 km can be covered in (150/90) minutes
Therefore, 30 km can be covered in (150 /90) × 30 minutes i.e. 50 minutes
Therefore, 30 km can be covered in 50 minutes.
(b) In this case, distance is unknown and time is known. Therefore, we proceed as follows :
Distance covered in 2 1 2 hours (i.e. 5/2 hours) = 90 km
Hence, distance covered in 1 hour = 90 × 2 5 = 36 kms.
Hence, distance covered in 2 hours = 36 × 2 = 72 km.
Therefore, in 2 hours, the distance covered is 72 km.
Ques. Compare given ratios and find out which of the following is greater: 12:16 or 18 : 20. (3 marks)
Ans. LCM of 16 and 20 is 80.
Dividing the LCM with the consequences,
we get 80 ÷ 16 = 5 ; 80 ÷ 20 = 4.
Multiplying the answers with the ratios.
(12 x 5) : (16 x 5) = 60 and 80
(18 x 4) : (20 x 4) = 72 and 80
As 72 > 60, the ratio 18:20 is greater than 12:16.
Ques. Use a cross multiplication method for comparison of the ratios and find which ratio is greater, 5:18 or 9:25? (3 marks)
Ans. Given ratios are 5:18 and 9:25.
By rearranging them as 5/18 or 9/25. using the cross-multiplication method,
we get 5 x 25 and 9 x 18 = 125 and 162
Since 162 > 125. Therefore, 9:25 is greater.
Ques. Compare the ratio 3:5 and 2:3 by two different methods of comparing the ratios. (3 marks)
Ans. The given ratio 3:5 and 2:3. By the lcm method, we have to change the denominator to a common term as
By cross multiplication process:
Hence it can be clearly seen from the above picture that the ratio 2:3 is greater than 3:5.
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