2. How to compare objects? We can compare two objects by subtraction or division. We can compare these objects with their lengths. Which is big? Which of the trees is taller?

3. Which is liked by all? In class 6 D, there are 30 students. 18 of them are cricket players, whereas 12 of them are interested in basketball. So there are (18-12)= 6 players more in cricket. Hence cricket has majority & therefore it is the one liked by all

4. We can consider another example, A Recipe for pancakes uses 3 cups of flour and 2 cups of milk. If we compare these two quantities by division, It is : 3 = 1.5 2 1 This means that the amount of flour used is 1.5 times the amount of milk used. This comparison by ‘Ratio’ division is the ‘Ratio’.

6. What do you mean by ‘Ratio’? A ratio compares values. A ratio says how much of one thing there is compared to another thing. In your Physical Education class, you have learned that a basketball team is made up of 12 players. But not all of them are allowed to play inside the court at the same time. In a basketball team, only 5 of them must be inside the court at a time, others remain seated at the bench. In symbols, it is 5:7 – meaning, THE RATIO OF THE IN-COURT PLAYERS TO THE BENCHED PLAYERS IS 5 TO 7. 5:7 is an example of a RATIO- the quotient of x is divided by y, where y is not equal to zero. A RATIO can be written as  x to y  x:y  x/y

7. Here the first quantity (or x) is called the first term or antecedent The second quantity(or y) is called the second term or consequent. We can compare two quantities only if they are of the same units. We cannot compare say 5 hours and 10 kg. We cannot compare 4 m and 3 km as the units are different. In the second case, however we can compare the two quantities by making the units uniform (i.e. By converting km to meters) There are 3 blue squares to 1 yellow square Ratios can be shown in different ways: Using the ":“(colon) to separate the values = 3 : 1 Instead of the ":" you can use the word "to" = 3 to 1 Or write it like a fraction = 3 1

8. How can we write ratios?  Examples of writing the ratio:  There are 9 shapes, out of which 3 are squares and 6 are circles. ratio of squares to circles is 3/6 ratio of squares to circles is 3 to 6 ratio of squares to circles is 3:6  What is the ratio of cats to mice? Number of Cats:3 Number of Mice:6 Express the ratio as a fraction: Express the ratio in words: Express the ratio with a colon: 3 to 6 3:6

9. If the numerator and denominator do not have the same units it may be easier to convert to the smaller unit so we don’t have to work with decimals… 3cm/12m = 3cm/1200cm = 1/400 (1 m = 100 cm) 2kg/15g = 2000g/15g = 400/3 (1 kg = 1000g) 2g/8g = 1/4 Of course, if they are already in the same units, we don’t have to worry about converting. Good deal

10. Equivalent Ratios If two ratios have the same value when simplified, then they are called Equivalent Ratios. Equivalent ratios can be obtained by multiplying or dividing both sides by the same non-zero number. The two ratios 8 : 24 and 4 : 12 are equivalent. There are 10 dolls for every 40 children in a preschool. Then the ratio of the number of children to that of the dolls = 40:10 = 4:1.

11. Express the ratio in simplest form: 10 = 10/5 = 2 45 45/5 9 6 = 6/6 = 1 18 18/6 3 15 = 15/5 = 3 50 50/5 10 8 = 8/4 = 2 12 12/4 3 6 = 6/3 = 2 15 15/3 5

12. "Part-to-Part" and "Part-to- Whole" Ratios Here are some examples to illustrate this idea:-  There are 5 pups, 2 are boys, and 3 are girls Part-to-Part: The ratio of boys to girls is 2:3 or 2 / 3 The ratio of girls to boys is 3:2 or 3 / 2 Part-to-Whole: The ratio of boys to all pups is 2:5 or 2 / 5 The ratio of girls to all pups is 3:5 or 3 / 5  Let’s look at a classroom: In a classroom there are 16 boys and 15 girls. Ratios can be part-to-part – 16 boys 15 girls Ratios can be part-to-whole – 16 boys 31 students

14. What do you mean by proportion? If two ratios are equal, we say that they are in proportion and use the symbol ‘::’ or ‘=’ to equate the two ratios. 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 know as respective terms. First and fourth terms are known as extreme terms. Second and third terms are known are middle terms. Consider, x:6 = 10:12 Here the first term x and the last term 12 are called the extreme terms or extremes. 6 and 10 are called the middle terms or means. When two ratios are equal, the product of the extremes = the product of the means.

15. Types of proportions There are two types of proportions: Direct Proportion o If two varying quantities are always in the same ratio(are equal), they are said to be directly proportional to one another. Another name for direct proportion is direct variation. o Two quantities, A and B, are in direct proportion if by whatever factor A changes, B changes by the same factor. For example, o the cost of buying oranges is proportional to the number of oranges bought, i.e. the more oranges we buy, the more we have to pay.

16. Inverse proportion If increased in one quantity causes decrease in other quantity or decrease in one quantity, then we say that both quantities are inversely related. Suppose that 20 men build a house in 6-days. If men are increased to 30 then they take 4-days to build the same house. If men become 40, they take 2-days to build the house. i.e. It can be seen that as the no. of men is increased, the time taken to build the house is decreased in the same ratio.

17. Unitary Method The unitary method is the process of finding the value of something for one item(unit) an then using this value for calculating the value for several such items(units). Consider 16 mangoes cost Rs.64.how much would 22 mangoes cost? The cost of 16 mangoes = Rs.64 :. Cost of one mango = Rs. 64/16 Cost of 22 mangoes = Rs.[64 22] = Rs.88 16

18. Finding the missing quantity when the ratios are equal.. Find the missing number in each of the ratios given below: 3:4=6:__ Let x be the missing number Therefore, 3:4=6:x ¾ = 6/x hence, by cross multiplication: 3x= 6 4 x= 24/3 = 8 3:4= 6:8

19. Fill IN THE Blanks Complete the following statements The we get when we compare two quantities by division is called the ratio of two quantities. If the two terms of a ratio have no common factor other than 1, the ratio is in form. Its. Usually the ratio of two numbers is written in its simplest form. In a ratio, the first term is called the antecedent and the second term is called the t. In a ratio, if we change the order of the numbers, the value of the ratio changes. To compare two quantities with different units, we must take the units of the two ratios the same. A ratio is a pure number and has no units. If two ratios are equal, they are said to be in proportion. In the proportion 2:3=6:9, 2 and 9 are called the extremes and 3 and 6 are called the mean s. If two ratios are equal, the product of the extremes, is equal to the product of the means. Quotient simplest antecedent consequent value units equal extremes means

20. Using ratios In Maps: The Scale The scale of the map is the ratio of a length on the map to the length it represents on the ground. The scale is given in most of the maps. The scale is often given in the form of the Representative or RF in short. Here in the map the ratio was 1:20000000 i.e. 1cm= 200 km

21. If two villages are 5 km apart and on the map the distance is represented by 5 cm, then the ratio is 5cm: 5 km = 5cm:500000cm = 1: 100000 So the representative fraction(RF) is 1:100000 or 1/100000 Any length on the ground is 100000 times the corresponding length on this map. When we talk of the scale of the map we mean its Representative fraction.

22. Questions To Practice 1.A workshop has 24 employers of which 12 are skilled workers, 2are supervisors and the rest are unskilled workers. Find the ratio of the number of skilled workers to supervisors. Number of skilled workers = 12 Number of supervisors = 2 Ratio of number of skilled workers: supervisors = 12:2 = 6:1

23. (2)6 bars of chocolate cost Rs.96. How much would 15 bars of chocolate cost? Cost of 6 bars of chocolate = Rs.96 Cost of 1 bar of chocolate = Rs.96 = Rs.16 6 Therefore, Cost of 16 bars of chocolate = Rs.16 15 = Rs.240

24. (3) A motorbike travels 220 km in 5 litres of petrol. How much distance will it cover in 1.5 litres of petrol? In 5 litres of petrol, motorbike can travel 220km. Therefore, in 1 litre of petrol, motor bike travels = 220 km 5 Therefore, in 1.5 litres, motorbike travels= 220 = 1.5 km 5 = 220 15 km = 66 km 5 10 Thus the motorbike can travel 66 km in 1.5 litres of petrol