## NCERT Class VI Mathematics Chapter 12 Ratio and Proportion

National Council of Educational Research and Training (NCERT) Book for Class VI
Subject: Mathematics
Chapter: Chapter 12 – Ratio and Proportion

Class VI NCERT Mathematics Text Book Chapter 12 Ratio and Proportion is given below.

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’.

12.2 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 Rs 10 and cost of a pencil is Rs 2. How many times the cost of a pencil is the cost of a pen? 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,

6. Find the ratio of the following :

(a) 81 to 108      (b) 98 to 63

(c) 33 km to 121 km   (d) 30 minutes to 45 minutes

7. Find the ratio of the following:

(a) 30 minutes to 1.5 hours     (b) 40 cm to 1.5 m

(c) 55 paise to Re 1    (d) 500 ml to 2 litres

8. In a year, Seema earns Rs 1,50,000 and saves Rs 50,000. Find the ratio of

(a) Money that Seema earns to the money she saves.

(b) Money that she saves to the money she spends.

9. There are 102 teachers in a school of 3300 students. Find the ratio of the number of teachers to the number of students.

10. In a college, out of 4320 students, 2300 are girls. Find the ratio of

(a) Number of girls to the total number of students.

(b) Number of boys to the number of girls.

(c) Number of boys to the total number of students.

11. Out of 1800 students in a school, 750 opted basketball, 800 opted cricket and remaining opted table tennis. If a student can opt only one game, find the ratio of

(a) Number of students who opted basketball to the number of students who opted table tennis.

(b) Number of students who opted cricket to the number of students opting basketball.

(c) Number of students who opted basketball to the total number of students.

12. Cost of a dozen pens is Rs 180 and cost of 8 ball pens is Rs 56. Find the ratio of the cost of a pen to the cost of a ball pen.

13. Consider the statement: Ratio of breadth and length of a hall is 2 : 5. Complete the following table that shows some possible breadths and lengths of the hall.

12.3 Proportion

Consider this situation :

Raju went to the market to purchase tomatoes. One shopkeeper tells him that the cost of tomatoes is Rs 40 for 5 kg. Another shopkeeper gives the cost as 6 kg for Rs 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

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 Rs 30. Ashma gave Rs 12 and Pankhuri gave Rs 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

= Rs 12 : Rs 18 = 2 : 3

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 Rs 30. Ashma gave Rs 12 and Pankhuri gave Rs 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

= Rs 12 : Rs 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.

Sharing a ratio means something!

Consider the following examples :

• Raj purchased 3 pens for Rs 15 and Anu purchased 10 pens for Rs 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 Rs 60 and Roshan sells 4 kg of apples for Rs 120. Whose apples are more expensive? Ratio of the weight of apples = 2 kg : 4 kg = 1 : 2 Ratio of their cost = Rs 60 : Rs 120 = 6 : 12 = 1 : 2

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.

EXERCISE 12.2

1. Determine if the following are in proportion.

(a) 15, 45, 40, 120      (b) 33, 121, 9,96 (c) 24, 28, 36, 48

(d) 32, 48, 70, 210      (e) 4, 6, 8, 12 (f) 33, 44, 75, 100

2. Write True ( T ) or False ( F ) against each of the following statements :

(a) 16 : 24 :: 20 : 30     (b) 21: 6 :: 35 : 10          (c) 12 : 18 :: 28 : 12

(d) 8 : 9 :: 24 : 27           (e) 5.2 : 3.9 :: 3 : 4          (f) 0.9 : 0.36 :: 10 : 4

3. Are the following statements true?

(a) 40 persons : 200 persons = Rs 15 : Rs 75

(b) 7.5 litres : 15 litres = 5 kg : 10 kg

(c) 99 kg : 45 kg = Rs 44 : Rs 20

(d) 32 m : 64 m = 6 sec : 12 sec

(e) 45 km : 60 km = 12 hours : 15 hours

4. Determine if the following ratios form a proportion. Also, write the middle terms and extreme terms where the ratios form a proportion.

(a) 25 cm : 1 m and Rs 40 : Rs 160       (b) 39 litres : 65 litres and 6 bottles : 10 bottles

(c) 2 kg : 80 kg and 25 g : 625 g              (d) 200 ml : 2.5 litre and Rs 4 : Rs 50

EXERCISE 12.3

1. If the cost of 7 m of cloth is Rs 294, find the cost of 5 m of cloth.

2. Ekta earns Rs 1500 in 10 days. How much will she earn in 30 days?

3. If it has rained 276 mm in the last 3 days, how many cm of rain will fall in one full week (7 days)? Assume that the rain continues to fall at the same rate.

4. Cost of 5 kg of wheat is Rs 30.50.

(a) What will be the cost of 8 kg of wheat?

(b) What quantity of wheat can be purchased in Rs 61?

5. The temperature dropped 15 degree celsius in the last 30 days. If the rate of temperature drop remains the same, how many degrees will the temperature drop in the next ten days?

6. Shaina pays Rs 7500 as rent for 3 months. How much does she has to pay for a whole year, if the rent per month remains same?

7. Cost of 4 dozens bananas is Rs 60. How many bananas can be purchased for Rs 12.50?

8. The weight of 72 books is 9 kg. What is the weight of 40 such books?

9. A truck requires 108 litres of diesel for covering a distance of 594 km. How much diesel will be required by the truck to cover a distance of 1650 km?

10. Raju purchases 10 pens for Rs 150 and Manish buys 7 pens for Rs 84. Can you say who got the pens cheaper?

11. Anish made 42 runs in 6 overs and Anup made 63 runs in 7 overs. Who made more runs per over?

What have we discussed?

1. For comparing quantities of the same type, we commonly use the method of taking difference between the quantities.

2. In many situations, a more meaningful comparison between quantities is made by using division, i.e. by seeing how many times one quantity is to the other quantity. This method is known as comparison by ratio.

For example, Isha’s weight is 25 kg and her father’s weight is 75 kg. We say that Isha’s father’s weight and Isha’s weight are in the ratio 3 : 1.

3. For comparison by ratio, the two quantities must be in the same unit. If they are not, they should be expressed in the same unit before the ratio is taken.

4. The same ratio may occur in different situations.

5. Note that the ratio 3 : 2 is different from 2 : 3. Thus, the order in which quantities are taken to express their ratio is important.

6. A ratio may be treated as a fraction, thus the ratio 10 : 3 may be treated as 10/3 .

7. Two ratios are equivalent, if the fractions corresponding to them are equivalent. Thus, 3 : 2 is equivalent to 6 : 4 or 12 : 8.

8. A ratio can be expressed in its lowest form. For example, ratio 50 : 15 is treated as 50/15 ; in its lowest form 50/15  = 10/3.  Hence, the lowest form of the ratio 50 : 15 is 10 : 3.

9. Four quantities are said to be in proportion, if the ratio of the first and the second quantities is equal to the ratio of the third and the fourth quantities.

Thus, 3, 10, 15, 50 are in proportion, since 3/10 = 15/50.  We indicate the proportion by 3 : 10 :: 15 : 50, it is read as 3 is to 10 as 15 is to 50. In the above proportion, 3 and 50 are the extreme terms and 10 and 15 are the middle terms.

10. The order of terms in the proportion is important. 3, 10, 15 and 50 are in

proportion, but 3, 10, 50 and 15 are not, since 3/10 is not equal to 50/15.

11. The method in which we first find the value of one unit and then the value of the required number of units is known as the unitary method. Suppose the cost of 6 cans is Rs 210. To find the cost of 4 cans, using the unitary method,

we first find the cost of 1 can. It is Rs 210/6 or Rs 35. From this, we find the price of 4 cans as Rs 35 × 4 or Rs 140.