Ratio and Proportion

Introduction

In our daily life, many a times we compare two quantities of the same type. For example, Avnee and Shari collected flowers for scrap notebook. Avnee collected 30 flowers and Shari collected 45 flowers. So, we may say that Shari collected 45 – 30 = 15 flowers more than Avnee.

Also, if height of Rahim is 150 cm and that of Avnee is 140 cm then, we may say that the height of Rahim is 150 cm – 140 cm = 10 cm more than Avnee. This is one way of comparison by taking difference.

If we wish to compare the lengths of an ant and a grasshopper, taking the difference does not express the comparison. The grasshopper’s length, typically 4 cm to 5 cm is too long as compared to the ant’s length which is a few mm. Comparison will be better if we try to find that how many ants can be placed one behind the other to match the length of grasshopper. So, we can say that 20 to 30 ants have the same length as a grasshopper.

Consider another example. Cost of a car is ₹ 2,50,000 and that of a motorbike is ₹ 50,000. If we calculate the difference between the costs, it is ₹2,00,000 and if we compare by division;

\(\text{i}.\text{e}.\text{ }\frac{\text{2},\text{5}0,000}{\text{5}0,000}=\text{ }\frac{\text{5}}{1}\text{ }\)

We can say that the cost of the car is five times the cost of the motorbike. Thus, in certain situations, comparison by division makes better sense than comparison by taking the difference. The comparison by division is the Ratio. In the next section, we shall learn more about ‘Ratios’.

Ratio and Proportion

Ratio

Consider the following:

Isha’s weight is 25 kg and her father’s weight is 75 kg. How many times Father’s weight is of Isha’s weight? It is three times. Cost of a pen is ₹ 10 and cost of a pencil is ₹ 2. How many timesthe cost of a pen that of a pencil? Obviously it is five times.

In the above examples, we compared the two quantities in terms of ‘how many times’. This comparison is known as the Ratio. We denote ratio using symbol ‘:’

Consider the earlier examples again. We can say, The ratio of father’s weight to Isha’s weight

\(\frac{\text{75}}{25}\text{ =}\frac{3}{1}\)=  3:1

The ratio of the cost of a pen to the cost of a pencil = \(\frac{10}{2}=\frac{5}{1}\)= 5:1

 Let us look at this problem. In a class, there are 20 boys and 40 girls. What is the ratio of (a) Number of girls to the total number of students. (b) Number of boys to the total number of students.

First we need to find the total number of students, which is,

Number of girls + Number of boys = 20 + 40 = 60.

Then, the ratio of number of girls to the total number of students is \(\frac{40}{60}=\frac{2}{3}\)= 2 : 3

Find the answer of part (b) in the similar manner.

Now consider the following example. Length of a house lizard is 20 cm and the length of a crocodile is 4 m.

“I am 5 times bigger than you”, says the lizard. As we can see this is really absurd. A lizard’s length cannot be 5 times of the length of a crocodile. So, what is wrong? Observe that the length of the lizard is in centimetres and length of the crocodile is in metres.

So, we have to convert their lengths into the same unit.

Length of the crocodile = 4 m = 4 × 100 = 400 cm.

Therefore, ratio of the length of the crocodile to the length of the lizard

\(\frac{400}{20}=\frac{1}{20}=1:20\)

Two quantities can be compared only if they are in the same unit.

Now what is the ratio of the length of the lizard to the length of the crocodile?

It is \(\frac{20}{400}=\frac{1}{20}=1:20\)

Observe that the two ratios 1 : 20 and 20 : 1 are different from each other. The ratio 1 : 20 is the ratio of the length of the lizard to the length of the crocodile whereas, 20 : 1 is the ratio of the length of the crocodile to the length of the lizard.

Now consider another example. Length of a pencil is 18 cm and its diameter is 8 mm. What is the ratio of the diameter of the pencil to that of its length? Since the length and the diameter of the pencil are given in different units, we first need to convert them into same unit.

Thus, length of the pencil = 18 cm = 18 × 10 mm = 180 mm.

 The ratio of the diameter of the pencil to that of the length of the pencil \(\frac{8}{180}=\frac{2}{45}=2:45\)

Think of some more situations where you compare two quantities of same type in different units. We use the concept of ratio in many situations of our daily life without realising that we do so. Compare the drawings A and B. B looks more natural than A. Why?

The legs in the picture A are too long in comparison to the other body parts. This is because we normally expect a certain ratio of the length of legs to the length of whole body.

Compare the two pictures of a pencil. Is the first one looking like a full pencil? No. Why not? The reason is that the thickness and the length of the pencil are not in the correct ratio.

Ratio and Proportion

Same ratio in different situations :

 Consider the following :

● Length of a room is 30 m and its breadth is 20 m. So, the ratio of length of the room to the breadth of the room = \(\frac{30}{20}=\frac{3}{2}=3:2\)

● There are 24 girls and 16 boys going for a picnic. Ratio of the number of girls to the number of boys = \(\frac{24}{16}=\frac{3}{2}=3:2\)

 The ratio in both the examples is 3 : 2.

 ● Note the ratios 30 : 20 and 24 : 16 in lowest form are same as 3 : 2. These are equivalent ratios.

● Can you think of some more examples having the ratio 3 : 2? It is fun to write situations that give rise to a certain ratio. For example, write situations that give the ratio 2 : 3.

● Ratio of the breadth of a table to the length of the table is 2 : 3.

● Sheena has 2 marbles and her friend Shabnam has 3 marbles. Then, the ratio of marbles that Sheena and Shabnam have is 2 : 3. Can you write some more situations for this ratio? Give any ratio to your friends and ask them to frame situations. Ravi and Rani started a business and invested money in the ratio 2 : 3. After one year the total profit was ₹ 4,00,000.

Ravi said “we would divide it equally”, Rani said “I should get more as I have invested more”. It was then decided that profit will be divided in the ratio of their investment.

Here, the two terms of the ratio 2 : 3 are 2 and 3.

Sum of these terms = 2 + 3 = 5 What does this mean?

This means if the profit is ₹ 5 then Ravi should get ₹2 and Rani should get ₹3. Or, we can say that Ravi gets 2 parts and Rani gets 3 parts out of the 5 parts

i.e., Ravi should get \(2 \over 5\) of the total profit and Rani should get \(3 \over 5\) of the total profit.

 If the total profit were ₹ 500

Ravi would get ₹\(2 \over 5\) × 500 = ₹200

 and Rani would get \(3 \over 5\) × 500 = ₹300

 Now, if the profit were ₹ 4,00,000 could you find the share of each?

Ravi’s share = ₹\(2 \over 5\) × 4,00,000 =₹1,60,000

And Rani’s share = ₹\(3 \over 5\) × 4,00,000 = ₹ 2,40,000

Can you think of some more examples where you have to divide a number of things in some ratio? Frame three such examples and ask your friends to solve them. Let us look at the kind of problems we have solved so far.

Example : Length and breadth of a rectangular field are 50 m and 15 m respectively. Find the ratio of the length to the breadth of the field.

Solution : Length of the rectangular field = 50 m

Breadth of the rectangular field = 15 m

The ratio of the length to the breadth is 50 : 15

The ratio can be written as =\(\frac{50}{15}=\frac{50\div 5}{15\div 5}=\frac{10}{3}\)= 10 : 3

Thus, the required ratio is 10 : 3.

Example : Find the ratio of 90 cm to 1.5 m.

Solution : The two quantities are not in the same units.

Therefore, we have to convert them into same units. 1.5 m = 1.5 × 100 cm = 150 cm.

Therefore, the required ratio is 90 : 150 =\(\frac{90}{150}=\frac{90\times 3}{150\times 3}=\frac{3}{5}\)

Required ratio is 3 : 5.

Example  : There are 45 persons working in an office. If the number of females is 25 and the remaining are males, find the ratio of:

(a) The number of females to number of males. (b) The number of males to number of females.

 Solution : Number of females = 25

Total number of workers = 45

 Number of males = 45 – 25 = 20

Therefore, the ratio of number of females to the number of males = 25 : 20 = 5 : 4

 And the ratio of number of males to the number of females = 20 : 25 = 4 : 5.

(Notice that there is a difference between the two ratios 5 : 4 and 4 : 5).

Example : Give two equivalent ratios of 6 : 4.

Solution : Ratio 6 : 4 = \(\frac{6}{4}=\frac{6\times 2}{4\times 2}=\frac{12}{8}\)

Therefore, 12 : 8 is an equivalent ratio of 6 : 4

Similarly, the ratio 6 : 4 = \(\frac{6}{4}=\frac{6\times 2}{4\times 2}=\frac{3}{2}\)

So, 3:2 is another equivalent ratio of 6 : 4.

Therefore, we can get equivalent ratios by multiplying or dividing the numerator and denominator by the same number.

Write two more equivalent ratios of 6 : 4

Example : Ratio of distance of the school from Mary’s home to the distance of the school from John’s home is 2 : 1.

(a) Who lives nearer to the school?

(b) Complete the following table which shows some possible distances that Mary and John could live from the school

(c) If the ratio of distance of Mary’s home to the distance of Kalam’s home from school is 1 : 2, then who lives nearer to the school?

Solution : (a) John lives nearer to the school (As the ratio is 2 : 1)

(b)

(c) Since the ratio is 1 : 2, so Mary lives nearer to the school.

Example  : Divide ₹60 in the ratio 1 : 2 between Kriti and Kiran.

Solution : The two parts are 1 and 2.

Therefore, sum of the parts = 1 + 2 = 3.

This means if there are ₹3, Kriti will get ₹ 1 and Kiran will get ₹ 2.

Or, we can say that Kriti gets 1 part and Kiran gets 2 parts out of every 3 parts.

Therefore, Kriti’s share = \(\frac{1}{3}\)×60 = ₹20

And Kiran’s share = \(2\over 3\) ×60 =₹40.

Ratio And Proportion

Sharing a ratio means something!

Consider the following examples :

● Raj purchased 3 pens for ₹15 and Anu purchased 10 pens for ₹50. Whose pens are more expensive?

   Ratio of number of pens purchased by Raj to the number of pens purchased by Anu = 3 : 10.

   Ratio of their costs = 15 : 50 = 3 : 10

   Both the ratios 3 : 10 and 15 : 50 are equal.  Therefore, the pens were purchased for the same price by both.

● Rahim sells 2 kg of apples for ₹180 and Roshan sells 4 kg of apples for ₹360. Whose apples are more expensive?

    Ratio of the weight of apples = 2 kg : 4 kg = 1 : 2

    Ratio of their cost = ₹180 : ₹360 = 6 : 12 = 1 : 2

    So, the ratio of weight of apples = ratio of their cost.

    Since both the ratios are equal, hence, we say that they are in proportion. They are selling apples at the same rate.

If two ratios are equal, we say that they are in proportion and use the symbol ‘::’ or ‘=’ to equate the two ratios.

     For the first example, we can say 3, 10, 15 and 50 are in proportion which is written as 3 : 10 :: 15 : 50 and is read as 3 is to 10 as 15 is to 50 or it is written as 3 : 10 = 15 : 50.

     For the second example, we can say 2, 4, 180 and 360 are in proportion which is written as 2 : 4 :: 180 : 360 and is read as 2 is to 4 as 180 is to 360.

      Let us consider another example.

      A man travels 35 km in 2 hours. With the same speed would he be able to travel 70 km in 4 hours?

      Now, ratio of the two distances travelled by the man is 35 to 70 = 1 : 2 and the ratio of the time taken to cover these distances is 2 to 4 = 1 : 2 .

      Hence, the two ratios are equal i.e. 35 : 70 = 2 : 4.

      Therefore, we can say that the four numbers 35, 70, 2 and 4 are in proportion.

      Hence, we can write it as 35 : 70 :: 2 : 4 and read it as 35 is to 70 as 2 is to 4.

      Hence, he can travel 70 km in 4 hours with that speed.

      Now, consider this example.

      Cost of 2 kg of apples is  ₹180 and a 5 kg watermelon costs  ₹45.  

      Now, ratio of the weight of apples to the weight of watermelon is 2 : 5.

      And ratio of the cost of apples to the cost of the watermelon is 180 : 45 = 4 : 1.

      Here, the two ratios 2 : 5 and 180 : 45 are not equal, i.e. 2 : 5 ≠ 180 : 45

      Therefore, the four quantities 2, 5, 180 and 45 are not in proportion

If two ratios are not equal, then we say that they are not in proportion. In a statement of proportion, the four quantities involved when taken in order are known as respective terms. First and fourth terms are known as extreme terms. Second and third terms are known as middle terms. For example, in 35 : 70 : : 2 : 4; 35, 70, 2, 4 are the four terms. 35 and 4 are the extreme terms. 70 and 2 are the middle terms.

Example: Are the ratios 25g : 30g and 40 kg : 48 kg in proportion?

Solution: 25 g : 30 g = \(\frac{25}{30}\) = 5 : 6

40 kg : 48 kg = \(\frac{40}{48}\)= 5 : 6

So, 25 : 30 = 40 : 48.

Therefore, the ratios 25 g : 30 g and 40 kg : 48 kg are in proportion,

 i.e. 25 : 30 :: 40 : 48

The middle terms in this are 30, 40 and the extreme terms are 25, 48.

Example: Are 30, 40, 45 and 60 in proportion?

Solution: Ratio of 30 to 40 = \(\frac{30}{40}\)= 3 : 4.

Ratio of 45 to 60 =\(\frac{45}{60}\)= 3 : 4.

Since, 30 : 40 = 45 : 60.

Therefore, 30, 40, 45, 60 are in proportion.

Example: Do the ratios 15 cm to 2 m and 10 sec to 3 minutes form a proportion?

Solution: Ratio of 15 cm to 2 m = 15 : 2 × 100 (1 m = 100 cm) = 3 : 40

Ratio of 10 sec to 3 min             = 10 : 3 × 60 (1 min = 60 sec) = 1 : 18

Since, 3 : 40 ≠ 1 : 18, therefore, the given ratios do not form a proportion.

Ratio And Proportion

Proportion

 Consider this situation : Raju went to the market to purchase tomatoes. One shopkeeper tells him that the cost of tomatoes is ₹40 for 5 kg. Another shopkeeper gives the cost as 6 kg for ₹42. Now, what should Raju do? Should he purchase tomatoes from the first shopkeeper or from the second? Will the comparison by taking the difference help him decide? No. Why not? Think of some way to help him. Discuss with your friends.

Consider another example.

 Bhavika has 28 marbles and Vini has 180 flowers. They want to share these among themselves. Bhavika gave 14 marbles to Vini and Vini gave 90 flowers to Bhavika. But Vini was not satisfied. She felt that she had given more flowers to Bhavika than the marbles given by Bhavika to her

What do you think? Is Vini correct? To solve this problem both went to Vini’s mother Pooja. Pooja explained that out of 28 marbles, Bhavika gave 14 marbles to Vini.

Therefore, ratio is 14 : 28 = 1 : 2. And out of 180 flowers, Vini had given 90 flowers to Bhavika. Therefore, ratio is 90 : 180 = 1 : 2. Since both the ratios are the same, so the distribution is fair

Two friends Ashma and Pankhuri went to market to purchase hair clips. They purchased 20 hair clips for ₹ 30. Ashma gave ₹ 12 and Pankhuri gave ` 18. After they came back home, Ashma asked Pankhuri to give 10 hair clips to her. But Pankhuri said, “since I have given more money so I should get more clips. You should get 8 hair clips and I should get 12”.

Can you tell who is correct, Ashma or Pankhuri? Why? Ratio of money given by Ashma to the money given by Pankhuri

= ₹ 12 : ₹ 18 = 2 : 3

According to Ashma’s suggestion, the ratio of the number of hair clips for Ashma to the number of hair clips for Pankhuri

= 10 : 10 = 1 : 1

According to Pankhuri’s suggestion, the ratio of the number of hair clips for Ashma to the number of hair clips for Pankhuri

= 8 : 12 = 2 : 3

Now, notice that according to Ashma’s distribution, ratio of hair clips and the ratio of money given by them is not the same.

But according to the Pankhuri’s distribution the two ratios are the same.

Hence, we can say that Pankhuri’s distribution is correct.

Ratio and Proportion

Unitary Method

Consider the following situations:

● Two friends Reshma and Seema went to market to purchase notebooks. Reshma purchased 2 notebooks for ₹24. What is the price of one notebook?

● A scooter requires 2 litres of petrol to cover 80 km. How many litres of petrol is required to cover 1 km?

These are examples of the kind of situations that we face in our daily life. How would you solve these?

Reconsider the first example: Cost of 2 notebooks is ₹24.

Therefore, cost of 1 notebook = ₹24 ÷ 2 = ₹12.

Now, if you were asked to find cost of 5 such notebooks. It would be = ₹12 × 5 = ₹60

Reconsider the second example: We want to know how many litres are needed to travel 1 km.

For 80 km, petrol needed = 2 litres.

Therefore, to travel 1 km, petrol needed = \(\frac{2}{80}=\frac{1}{40}\) litres.

     Now, if you are asked to find how many litres of petrol are required to cover 120 km?

     Then petrol needed = \(\frac{1}{40}\)×120 litres = 3 litres.

     The method in which first we find the value of one unit and then the value of required number of units is known as Unitary Method.

We see that,

Distance travelled by Karan in 2 hours = 8 km

Distance travelled by Karan in 1 hour = \[\frac{8}{2}\] km = 4 km

Therefore, distance travelled by Karan in 4 hours = 4 × 4 = 16 km

Similarly, to find the distance travelled by Kriti in 4 hours, first find the distance travelled by her in 1 hour.

Example: If the cost of 6 cans of juice is ₹210, then what will be the cost of 4 cans of juice?

Solution: Cost of 6 cans of juice = ₹210

Therefore, cost of one can of juice = 210 6 = ₹35

Therefore, cost of 4 cans of juice = ₹35 × 4 = ₹ 140.

Thus, cost of 4 cans of juice is ₹140.

Example: A motorbike travels 220 km in 5 litres of petrol. How much distance will it cover in 1.5 litres of petrol?

Solution: In 5 litres of petrol, motorbike can travel 220 km. Therefore, in 1 litre of petrol, motor bike travels = \[\frac{220}{5}\] km

Therefore, in 1.5 litres, motorbike travels = \(\frac{220}{5}\times 1.5\)km

                                                           = \(\frac{220}{5}\times \frac{15}{10}\)km

                                                           = 66 km.

Thus, the motorbike can travel 66 km in 1.5 litres of petrol.

Example: If the cost of a dozen soaps is ₹153.60, what will be the cost of 15 such soaps?

Solution: We know that 1 dozen = 12

Since, cost of 12 soaps = ₹ 153.60

Therefore, cost of 1 soap = \(\frac{153.60}{12}\) = ₹12.80

Therefore, cost of 15 soaps = ₹12.80 × 15 = ₹192

Thus, cost of 15 soaps is ₹192.

Example : Cost of 105 envelopes is ₹350. How many envelopes can be purchased for ₹100?

Solution: In ₹350, the number of envelopes that can be purchased = 105

Therefore, in ₹1, number of envelopes that can be purchased =\(\frac{105}{350}\)

Therefore, in ₹100, the number of envelopes that can be purchased = \(\frac{105}{350}\)× 100 = 30

Thus, 30 envelopes can be purchased for ₹100.

Example: A car travels 90 km in 2 \(\frac{1}{2}\) hours.

(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.

Solution: (a) In this case, time is unknown and distance is known.

Therefore, we proceed as follows : 2 \(\frac{1}{2}\)hours = \(\frac{5}{2}\) hours = \(\frac{5}{2}\) × 60 minutes = 150 minutes.

90 km is covered in 150 minutes

Therefore, 1 km can be covered in \(\frac{150}{90}\) minutes

Therefore, 30 km can be covered in \(\frac{150}{90}\) × 30 minutes i.e. 50 minutes

Thus, 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 \(\frac{1}{2}\) hours (i.e. \(\frac{5}{2}\) hours) = 90 km

Therefore, distance covered in 1 hour = 90 ÷ \(\frac{5}{2}\) km = 90 × \(\frac{2}{5}\) = 36 km

Therefore, distance covered in 2 hours = 36 × 2 = 72 km.

Thus, in 2 hours, distance covered is 72 km.