1. Copyright © Ed2Net Learning, Inc. 1 Ratios Grade 6

2. Copyright © Ed2Net Learning, Inc. 2 1.Solve: k - 38 = 50 2. Solve: 4r + 80 = 100 3. Solve: 3(n + 4) = 18 4.Solve: 78 > 12y 5. Write as an expression: Twice Sandra’s age Warm up

3. Copyright © Ed2Net Learning, Inc. 3 To solve an equation means to find the values of the variable that make the equation true. The values of the variable are called the SOLUTIONS of the equation. 18 is a solution of: a + 2 = 20 because: a = 18 Review: Solve Equations by Adding & Subtracting

4. Copyright © Ed2Net Learning, Inc. 4 1. Isolate the variable you wish to solve for 2. Substitute your answer into the original equation and check that it works Solve Equations by Multiplication and Division 8x = 48 Example: 8x = 48 x = 6 8(6) = 48 48 = 48

5. Copyright © Ed2Net Learning, Inc. 5 1.Combine like terms 2. Isolate the variable you wish to solve for 3.Substitute your answer into the original equation and check that it works Solving Two Step Equations

6. Copyright © Ed2Net Learning, Inc. 6 Example of Two step Equations 4x + 6 = 6x - 5 4x + 6 - 6 = 6x - 5 – 6 4x = 6x -11 4x -6x = 6x -6x -11 -2x = -11 x = 11 2

7. Copyright © Ed2Net Learning, Inc. 7 In this lesson we will see how to translate verbal phrases in to algebraic expressions Examples: The number is twice as much The weight is 5 less than half of it Expressions & Equations

8. Copyright © Ed2Net Learning, Inc. 8 An inequality is a statement with a symbol of or ≥ between numerical or variable expressions. Example: x + 7 ≤ 11. x is a variable. x x is a very common variable that is used in algebra, but you can use any letter (a, b, c, d,....) to be a variable Defining Inequality Inequalities with Add and Subtract

9. Copyright © Ed2Net Learning, Inc. 9 Inequality Symbols SYMBOL MEANING < less than > greater than ≤ less than or equal to ≥ greater than or equal to

10. Copyright © Ed2Net Learning, Inc. 10 Rule No : 1 - Addition If a < b, then a + c < b + c Rules to Solve Linear Inequalities Rules to Solve Linear Inequalities Rule No : 2 - Subtraction If a < b, then a - c < b - c

11. Copyright © Ed2Net Learning, Inc. 11 For all numbers a, b and c, the following rules are true 1.If c is positive and a < b, then ac < bc and a b, c = 0 c c 3. If c is positive and a > b, then ac > bc and a b, c = 0 c c < > Inequalities with Multiply and Divide Inequalities with Multiply and Divide

12. Copyright © Ed2Net Learning, Inc. 12 When solving Inequalities, if you multiply and divide each side by the same negative number, you must reverse the direction of the inequality symbol. -4h > 12 4 4 When we simplify, because we're dividing by a negative number, h < -3 Example

13. Copyright © Ed2Net Learning, Inc. 13 Ratios A way of comparing two quantities of the same kind Tells how one number is related to another Ratio of two numbers X & Y(<>0) is X/Y (proper or improper fraction) written as X:Y Each ratio is read as ‘ Ratio of X to Y’

14. Copyright © Ed2Net Learning, Inc. 14 The numbers in a ratio are called ‘Terms’. The first term is called as ‘Antecedent’ and the second term is called as ‘Consequent’. The colon (:) is generally used to represent a ratio. Example: 2 : 5, 5/3, 4 to 7 In the above example Antecedents are 2, 5 & 4 and Consequents are 5, 3 & 7 respectively. Terms in a Ratio

15. Copyright © Ed2Net Learning, Inc. 15 Ratio in its lowest form A ratio should always be expressed in its lowest form A ratio can be expressed in its simplest form by dividing both the terms of the ratio by their highest common factor (HCF) or by expressing the terms as its factors first and then simplifying them

16. Copyright © Ed2Net Learning, Inc. 16 1) 10:25 HCF of 10 and 25 is 5 10:25 = (10/5) : (25/5) 10:25 = 2:5 2) 9: 24 = 9/24 9:24 = (3*3)/(3*8) = 3/8 9:24 = 3:8 Examples

17. Copyright © Ed2Net Learning, Inc. 17 Equivalent Ratios Two ratios are said to be equivalent when they can be expressed as a same ratio in their lowest form. Example: 4:8, 8:16, 32:64 are equivalent ratios Because, all these ratios can be expressed as 1:2 in their lowest form

18. Copyright © Ed2Net Learning, Inc. 18 3/9, 4/12, 5/15 are equivalent fractions. The above fractions can be written in their lowest form as 1/3, 1/3 & 1/3 respectively. 1/3 = 1:3 So, all the above fractions are equivalent ratios Examples Equivalent fractions are always equivalent ratios as they can be expressed as a same fraction in their lowest form.

19. Copyright © Ed2Net Learning, Inc. 19 Evaluating the ratio of two quantities Evaluating the ratio of two quantities Quantities should be expressed in the same unit of measurement while calculating their ratio

20. Copyright © Ed2Net Learning, Inc. 20 15cm, 1.2 m = 15 cm : 120cm = 15:120 50 days, 1 yr = 50 days : 365 days = 50:365 5 Km, 100m = 5000m: 100m = 5000:100 45 minutes, 2 hrs = 45minutes:120minutes= 45:120 3.2 km, 3miles = 3.2 km :4.8 km (Hint 1 mile = 1.6 km) Examples

21. Copyright © Ed2Net Learning, Inc. 21 Proportion Ratios are said to be in proportion when they are equivalent If p:q and r:s are equivalent ratios, then they are written as p:q = r:s or p: q :: r :s Here p,q,r,s are said to be in proportion The terms p & s are called extremes and q & r are called means

22. Copyright © Ed2Net Learning, Inc. 22 Rule of proportion says, product of extremes is equal to product of means If p:q = r:s then, p*s = q*r If p:q = q:r then q q^2 = p*r, here q is called the mean proportion Mean Proportion

23. Copyright © Ed2Net Learning, Inc. 23 Two quantities are said to be in direct variation if they are related in such a way that when one increases(or decreases), the other also increases (or decreases) in such a manner that the ratio of the two remains the same. Direct Variation

24. Copyright © Ed2Net Learning, Inc. 24 Suppose a car travels a distance of 50km in half an hour, 150 km in 3 hrs and 250 km in 5hrs. Let us find the speed's of the car S = Distance traveled/ time taken S = 50/1, 150/3, 250/5 S = 50 km/hr, 50km/hr, 50km/hr The ratio of distance traveled to the time taken remains the same in the above case. The distance traveled and time taken are said to be in direct variation Example

25. Copyright © Ed2Net Learning, Inc. 25 The method of finding values by, first finding the value of one unit by division and then the value of the required number of units by multiplication is called the unitary method Unitary method

26. Copyright © Ed2Net Learning, Inc. 26 Suppose the cost of 30 pens is $120. Find the cost of 45 pens. Step 1(Division): First find the cost a single pen The cost of 30 pens = $120 The cost of one pen = $120/$30=$4 Step2(Multiplication): Find the cost of 45 pens The cost of one pen= $4 The cost of 45 pens = 4*45 = $180 This way of solving the problem is called unitary method Example

27. Copyright © Ed2Net Learning, Inc. 27 Let’s take a break!

28. Copyright © Ed2Net Learning, Inc. 28

29. Copyright © Ed2Net Learning, Inc. 29 1.Express the following ratios in their simplest form. a)24 : 18 b) 1/4 : 1 2.Fill in the gaps in each ratio, so that it becomes 2 : 3 when simplified. a) 6 : __ b) __ : 12 3.Express the following ratios as fractions. 1) 4 : 10 2) 0.5 : 1 ½ 4.In a test containing 30 questions of 1 mark each and 20 questions of 2 mark each, John got 56 marks. Compare his marks with the total marks. Your Turn

30. Copyright © Ed2Net Learning, Inc. 30 5. Mark and July were running an industry. In a year each of them earned a profit of $1800 and $1200. What is the ratio of individual’s share compared to total profit? 6. From 2 liters of milk, 10 cups of coffee can be made. To make 6 cups of coffee, how many liters of milk is needed? 7. Fill in the blanks to make each of the following a true proportion. a) 1 / __ = 7 / 42. b) 4 : __ = 36 : 27 Your Turn

31. Copyright © Ed2Net Learning, Inc. 31 8. Insert = or for the following 1) 54 / 25 __ 9 / 4 2) 12 : 15 __ 36 : 45 9. Find the value of X in the following proportions. 1) X : 6 = 55 : 11 2) X : 3 / 4 = 40 : 30 10. Find the mean proportion between 4 and 9. Your Turn

32. Copyright © Ed2Net Learning, Inc. 32 1. A school team won 9 matches out of the 20 basketball matches played. What is the ratio of the games lost to the total games played?

33. Copyright © Ed2Net Learning, Inc. 33 2. The ratio of the length of a play ground to its width is 5 : 2. Find its length if the width is 40 meters?

34. Copyright © Ed2Net Learning, Inc. 34 3. A worker is paid $160 for 5 days. What should be paid to him for 28 days?

35. Copyright © Ed2Net Learning, Inc. 35 1. Ratio of two numbers X & Y( 0) is X/Y (proper or improper fraction) written as X:Y. A ratio should be expressed in its lowest form 2. In the ratio X : Y, the term X is called the antecedent and the term Y is called the consequent 3. All equivalent ratios can be expressed in one common ratio in its lowest form 4. Ratios are said to be in proportion when they are equivalent Lets review what we have learned today

36. Copyright © Ed2Net Learning, Inc. 36 5. In the proportion p:q =r:s, p & s are extremes and q & r are means and p*s=q*r 6. Two quantities are said to be in direct variation if they are related in such a way that when one increases(or decreases), the other also increases (or decreases) in such a manner that the ratio of the two remains the same. 7. The method of finding values by, first finding the value of one unit by division and then the value of the required number of units by multiplication is called the unitary method. Recap

37. Copyright © Ed2Net Learning, Inc. 37 Great Job! Keep practicing!!