Exercises:

MODULE 3 LINKING RATIO, RATE, AND PROPORTION Lessons Target: Write ratios in different ways; Express ratio in simplest form; Determine two ratios as a proportion; Find the missing the term in a proportion; Solve problems involving ratio, direct, partitive, and inverse proportion; and Solve word problems involving percentage, base, and rate. LESSON 1: Ratio and Fraction Pre-Assessment: Express the ratio as fraction in their lowest term. 1. 54:36 2. 2:75 3. 21:63 4. 4:7 5. 24:9 Study and Learn Ratio The newspaper headline indicates that 80 out of every 100 Filipinos read newspapers. Therefore, 20 out of 100 Filipinos do not read newspapers at all. The comparison of the numbers of people who do not read newspapers with those who do can be written as a ratio, 20:80. This read as, "twenty is to eighty". The numbers 20 and 80 are the terms of the ratio. When each term of a ratio is multiplied or divided by the same nonzero number, an equivalent ratio is produced. For example: 20:80= 200:800 = 2:8 = 1:4 The ratio 1:4 is in lowest terms because the only factor that the terms have in common is 1.The order in which a comparison or ratio is expressed is important. The quantity which is mentioned first in the comparison must be written first in the ration. Let us consider the following example. Here we have 2 cars and 1 bicycle. The ratio of the number of cars to the number of bicycle is 2:1. However, the ratio of the number of bicycle to the numbers of cars is 1:2. Both ratios tell us that are there are 2 cars for every 1 bicycle. + Writing Ratios as Fractions We can also express ratio as a fraction. Suppose we have three candles and 5 matchsticks. The ratio of the number of candles to the number of matchsticks is 3:5. Written as a fraction, the ratio of the number of candles to the number of matchsticks is Number of candles Number of matchsticks , So, 3:5=3 Remember that the order in which a ratio is expressed is important. The quantity which is mentioned first in the ratio must be written in the numerator of the fraction. Therefore, the ratio of the number of matchsticks to the number of candies is . Example 1: Express the following statement as (i) a ratio, (ii) a fraction in the order given. The number of engines to the number of carriages Solution: There are 1 engine and 4 carriages. (i) The ratio of the number of engines to the number of carriages is 1:4. (ii) Written as a fraction, 1:4=- Example 2: Given that Aisah's height is = of Farouks's height, what is the ratio of Aisah's height to Farouk's height? 1 unit Draw models to represent the Solution: Aisah's height information. Farouk's height 3 units to 5 units. The ratio of Aisah's height and Farouk's height is 3:5 We compare the quantities of 2 or more sets of objects using ratio. The actual amount in each quantity may not be used. + Ratios in the Simplest Form Let's take a look at the ratio 12:8. How do we write it in simplest form? First find all the equivalent ratios of 12:8Divide 12:8 by a common factor 2 to get 6:4 +2 12:8 : 2 Divide 6:4 by a common factor 2 to get 3:2 6:4 6:4 and 3:2 are equivalent ratios of 12:8 : 2 3:2 + 2 The ratio 3:2 cannot be divided exactly by a common factor to get another equivalent ratio. Thus, 3:2 is the ratio in simplest form. Fractions should always be expressed in simplest form. Example 3: Write the following ratio in its simplest form. a.30:6 b. 1 2 : 1 - Solution: 30:6 a. : 6 + 6 Divide both sides of the ratio by the greatest 5:1 common factor. 30:6= 5:1 30:6 and 5:1 are equivalent ratios. 5:1 is the simplest form of 30:6. b. Since 15 = = and 1} = 10 . Then the ratio of 12 to 12 = > : = X - = > = 3:2 = 15 x 39 = = = 3:2 37 70 2 Equivalent Ratios Multiplying or dividing the first and second terms of the ratio by the same nonzero number generate or form equivalent ratios. For example, consider the ratio 6:4 We have - = 6+2 6 : 6X3 18 4 + 2 4 X 3 12 Therefore, 3:2 and 18:12 are equivalent ratios of 6:4+ Ratios of Three Quantities Example 4: Aina made 6 sandwiches, 3 pies, and 9 tarts. The ratio of the number of sandwiches to the number of pies to the number of tarts is 6:3:9. 1 unit We can put them in groups of 3 1 unit 2 units to 1 unit to 3 units Iunit The ratio of the number of sandwiches to the number of pies to the number of tarts is 2:1:3. 6:3:9 = 2:1:3. They are equivalent ratios. 2:1:3 is a ratio in the simplest form. Rates In our previous discussion, we learned that a ratio is a comparison of two like quantities. Another term that we will need to learn is rate. For example, when a person buys a car, one of the considerations is usually how much gasoline the car consumes. The fuel consumption, in liters, is compared to the distance the car travels, in hundreds of kilometres. Such a comparison is called a rate because the two quantities that are being compared have different units. Here are some more example: 70 Km per hour (per one hour) 0.4 per Km (per one Km) P 35 per Kg (per one Kg) 10 centavos to the peso (per one peso)Example 5: Express 180 Km in 3 hours as a rate. Solution: 180 Km in 3 hours or - Km in hour is 60 Km per hour. Thus, rate = 60 Km per hour or 60 Kph. LESSON 2: Direct, Partitive, and Inverse Proportions Pre-Assessment: Match the proportion with the related equation. Write the letter in each blank. 15 n a. 4 X n = 8 x 15 2. n 12 b. 3 x 4 = 12 x n 3. Na A 18 3 100 A/M P IW c. 12 X n = 6x15 4. = d. 4 x 15 = 3xn 5 . e. 12 x 15 = 3 x n Study and Learn Proportion Mohammad had 4 seedlings of squash and 6 seedlings of tomatoes, while Ali had 8 seedlings of squash and 12 seedlings of tomatoes. For Mohammad, the ratio of squash to tomato seedlings is 4:6. For Ali, the ratio of squash to tomato is 8:12. Express each ratio in simplest form: WIN WIN 12 Since both ratios have the simplest form - , then we say that the ratios are equal. An equation stating that two ratios are equal is called proportion. Therefore, * = is a proportion. Another way to write this proportion is 4:6 = 8:12. The numbers 4,6,8, and 12 are the terms of proportion. The first and fourth terms, 4 and 12 are called extremes. The second and third terms, 6 and 8 are the means. Means 4:6 = 8:12 Extremes Suppose we take the product of the meuns and the product of the extremes. 8 6 X 8 = 48 6 12 6 X 8 = 48 4 x 12 = 48 4:6 = 8:12 4 x 12 = 48The above illustrates the law of proportion, which states that the product of the means is equal to the product of the extremes. We can use this law to solve for the missing term in a proportion. Example 1: What is the value of n in 3:7 = 12: n Solution: 1 = 12 4 16 16 X n = 4 x 12 16 X n = 48 n = 48 + 16 n = 3 Thus, the missing value of n is 3. Check by substituting the value for n. Then, determine if the ratios are equal. 1 = 12 3 12 16 41 16 12 16 = 12 +4 - A 100 16 + 4 Therefore, the ratios are equal. Solving Problems on Proportion by Using the Model Approach Solving problems by using an approach can make the given problem clearer and easier to solve. Example: Armaden mixed apple juice and pear juice in the ratio 3:7 to make the 6L of mixed fruit juice. How many liters of apple juice did he use? Solution: Apple juice 61 Pear juice He used 1 = L of apple juice. 10 units = 6 L lunit = L 10 3 units = ~ x 3 = 18 18 10 10 1 7 Direct Proportion If we multiply the terms of a ratio by the same number, then we generate ratios that form a proportion. From _ we can generate the following equal ratios: 3 3X 1 = 3 4X 1 3X 2 6 4 X2 3 x 3 9 4 X 3 12 3 X 4 12 4 X 4 16 3 x 5 = 15 4 X 5 20> The first terms are 3, 6,9,12, and 15. Their corresponding terms are 4, 8, 12, 16, and 2 If the terms of one ratio are increased the same number of times to get another ratio then the two ratios form a direct proportion. The concept of direct proportion can be used to solve word problems study the following examples: 1. Omar's father took the family on a trip. At the average rate of 35 Km/h, how long did it take them to travel a distance of 350 Km? Solution: Since the relationship is between the number of kilometres and the number of hours. We write a proportion in which both ratios are in the form . T Km The given ratio ; the second ratio is . We write the two ratios as a proportion. Km on a trip 35 350 n Number of hours 35 x n = 1x 35 35n = 350 35n _ 350 35n 35 n = 10 h It took them 10 h to travel a distance of 350 Km. + Indirect Proportion Assume that a man has to pay a salary loan of P12,000. Below is the table showing the options he can choose to pay his salary loan on installment Amount Paid per Month Time taken in months P 1,000 12 Increasing P 2,000 6 Decreasing amount number of P 3,000 4 months P 4,000 W P 6,000 N We notice that the time taken for payments decreases as the amount paid per month increases. Therefore, if the amount paid per month is doubled, the time taken for payment is halved. For example, Amount Paid per Month Time taken in monthsd 20 ratio 2,000 6 halved doubled P 4,000 3 More examples: 1. If 8 men can do a certain job in 12 days, how many men will be required to do the same job in 16 days? If we analyze the given problem carefully, we can deduce that fewer men will be required to finish the same job for a greater number of days. Thus, there are quantities that may not be directly proportional to one another. In this case, an increase in one quantity results in a proportional decrease in the other, and vice versa. Such a proportion is called INDIRECT PROPORTION. LET US SOLVE THE PROBLEM USING INDIRECT PROPORTION Let n = number of men required to do the work in 16 days. Consider: In 12 days, it takes 8 men to do the work. In 16 days, it will take less than 8 men to do the work. Therefore, n will be less than 8 Solution: n 12 Checking: 8 16 6 n = 8 x 12 12 16 8 16 n = 6 six men can finish the work 3 3 4 4 2. If a car traveling at the rate of 40km/h takes 10 hours to travel a certain distance, how long would it take the same car to travel the same distance at the rate of 50km/h? Solution: Lesser rate: lesser time = greater rate: greater time 40: n = 50: 10 40 (10) = 50(n) 400 50 n 50 n = 8 It takes 8 hours to cover the same distance at the rate of 50 km/h. Partitive Proportion A quantity or a whole can be divided into two or more equal or unequal parts calledPARTITIVE PROPRTION. 2 parts 1 Part 3 parts 2 parts 1 Part 3 parts 1:1:2 2:3:3 We use ratios to illustrate partitive proportion. Study these examples, 1. A can of cookies contains 100 pieces in all. If 3 children share the cookies in the ratio 1:1:2, how many cookies will each child get? Solution: In 1:1:2, we add the numbers 1 + 1 +2 = 4, therefore, 1:1:2 1+ 1+2= 4 n: n:2n n+n+2n = 100 4n = 100 * 2n = 2 (25) Cn = 25 we substitute the value of n. 1:1:2 = n: n:2n = 25:25:50 Of the children, one will have 25 cookies, the second will have 25 pieces, and the third will get 50 cookies. 2. The ratio of the length of a playground to its width is 5:2. Find its length if the width is 20 m. Solution: When we say that the ratio of the length to the width is 5:2, it means that if its length is 5m then its width is 2m. In other words, the length is - times its width and its width is - times its length. Now in this case, we can solve for the length. nnanenty Width = 20 m Length = = x 20 m =50'm Therefore, the length is 50 m. LESSON 3: Percent, Fraction, and Decimals Pre-Assessment: Write the shaded part of the figures as (a) fraction, (b) decimal, and (c) percent of the whole figure.2. b. Study and Learn Since percents are alternative representations of fractions and decimals, it is important to be able to convert among all three forms, as suggested in the diagram. Fractions Decimals Percents + Changing Percent to Fraction Every percent can be expressed as a fraction, with 100 as its denominator, since percent means per hundred. When the fraction is not in its simplest form, we need to reduce it to its lowest terms. Let us study the diagram below. Predicted Price for a Kilogram of Rice in 2017 More than P60 Do not know 1% 1% P40 to P50 58% RICE P30 to P40 34% P50 to P60 6% 58% of the people predict that a kilogram of rice would cost between P40 and P50 in 2017. 58% means "58 out of 100" or tor . 100 . This fraction can be simplified further that is: 1+ 2471 58 29 100 + 2 Hence, 58% is the same as Below are more examples:Example 1: Change 40% to a fraction in its lowest terms. Solution: + 20 Write the denominator as 100 40% = - and simplify the fraction. 100 $20 Example 2: Express 135% as a fraction in lowest terms. Solution: 5 Write the number over 100. Divide 135% = 135 100 20 both the numerator and the denominator by the GCF. Simplify the fraction . Example 3: Convert to 10- % to a fraction. Solution:10 - % = -2 = 21 + 100 21 x 1 21 2 100 200 Changing Fraction to Percent In renaming a fraction as a percent, multiply the given fraction with a fraction equal to 1 to give a product of 100 in the denominator. This will change it to its equivalent percent. Let us study the example below. Ronnie makes 16 out of 20 shots in the basketball game. What percent of his shots does he make? From the given information, the ratio of the shots made with the total number of shots is 16:20 or . Let us express to to 20 to an equivalent fraction whose denominator is 100. 16 16X5 80 20 20*5 100 = 80% Thus , 16 20 - = 80% and Ronnie makes 80% of his shots. Here are some more example: 1. = 3x25 75 4 4x25 100 = 75% 1X100 100 100+6 16- 2. 6 6X100 600 600+6 100 = 16- % Study these examples showing another way of changing fractions to percents. 1. = = = X 100% = 300% = 75% 2. 2- = x 100% = 500% 2 - = 250% Changing Fraction to DecimalsMother prepare pizza pie for snacks. She is taking away 2 out of 10 slices. Two out of 10 slices may be written as: In fraction form In decimal form 2 10 0.2 Both read as "two tenths" Look at these examples. 1. What part of the whole figure is shaded? Five out of 10 equal parts Fraction 10 or 0.5 Decimal Both read as "five tenths" 2. What part of 1 peso is 1 centavo? One peso equals 100 centavos, so 1 centavo is 1 out of 100 parts. Fraction 100 0.01 Decimal Both read as "one hundredth" Here are more example: 8 10 = 0.8 = 1.2 24 10 100 = 0.24 8 = 0.08 12 100 = 1.02 24 1000 = 0.024 100 8 = 0.008 1 -4 = 1.002 24 = 0.002 1000 1000 10000 Do you see any relationship between the zeros in the denominator and the number of decimal places? We can also convert decimal numbers into fractions or mixed numbers. Study these examples: 1. 0.04 = 100 25 2. 3.005 = 3-5 1000 = 3-1 200LESSON 4: Percentage, Base, and Rate Pre-Assessment: Solve for N. 1. N% of 70 is 35? 2. 5% of N is 30? Study and Learn: Finding the Percent of a Number (Percentage) Earlier, we showed that to find a fractional part of a number, we multiply the fraction by the number. Similarly, to find the percent of a number, change the percent to a decimal or fraction and then multiply. Example 1: On a 20-item test, Omar answered = of the problems correctly. How many problems did he answer correctly? Solution: of 20 = = x = 16 problems of " means to multiply Therefore, Omar answered 16 problems correctly. Example 2: Find 75% of 20. Solution: Since 75% = 0.75 Then, 75% of 20 = 0.75 x 20 = 15.00 Therefore, 75% of 20 is 15. Alternative Solution: Since 75% = 75 3 100 4 Then 75% of 20 = 3 x 20 = 15. Therefore, 75% of 20 is 15. Percent Ending in Fraction When a percent contains a fraction, we need to change the fraction to a decimal first. Thus, 6-% may be written as 6.5%. moving the decimal point to the left and removing the % symbol, 6.5% = 0.065. Example 3: Find 6-% of P550. Since 6=% = 6.5% = 0.065. Then, 6-% of P550 = 0.065 x P550 = P35.75.Thus, 6-% of P550 is P35.75. When the numbers are replaced by the corresponding letters, A, Band R, we have: PERCENT PROPORTION: A B = R 100 Where A = amount, B= base, R = percent. Example 1: What is 5 percent of 40? Solution: 5 represents R, because it is followed by the word percent. 40 represents B, because it follows the words "percent of". What number is 5 percent of /40)? R B This represents A, because it is the remaining number after R and B have been identified. A = 5% x 40 A = 0.05 x 40 A = 2 Example 2: 6 is what percent of 30? Solution: Identify the numbers A, B, and R. 6 is what percent of 30)? R B 5 = Rx 20 R =6 + 30 or , R =6 30 R = 20% Therefore, 6 is 20 percent of 30. Example 3: 10% of what number is 50? Solution: 10% of what number is 50? BLINKING RATIO, RATE, AND PROPORTION RATIO AND FRACTION Exercise 1 A. Express the ratio as fraction in their lowest term. Show your solution. Example: 53:36 = 54 NIN 36 = NIH NIW 54 18 NEW 36 18 or 1 a) 4: UIN b) 36:9# c) 45 min:1 hour =B. Solve the ratio problem. Farouk has 50 red marbles, 70 green marbles, and 42 yellow marbles. What is the ratio of the number of green marbles to the number of red marbles to the total number of marbles he has? DIRECT, INDIRECT, AND PARTITIVE PROPORTIONS Exercise 2 Answer the following problems. 1. There 5 teachers for every 250 pupils in a barangay school. How many teachers are there if there are 2000 pupils? (USE DIRECT PROPORTION)2. If an emergency pack of food cans lasts 5 days for 10 men, how long will the food last if there are 20 men? (USE INDIRECT PROPORTION) 3. The three angles of a triangle have a total degree measure of 180. If the angles of the triangle have measures in the ratio 2:3:3, what is the measure of each angle? (USE PARTITIVE PROPORTIONS)PERCENT, FRACTION, AND DECIMALS Exercise 3 Express the following as a percent, as a decimal, and as a fraction in its simplest form. ( PERCENT DECIMAL | FRACTION SIMPLEST FORM 1. 30 out of 100 2. 90 out of 100 3. 62 out of 100 PERCENTAGE, BASE, AND RATE Exercise 4 A. Complete the following. 1. What percent of 70 is 18? Px70=18 P=18+70 P= %2. P30 is what percent of P600? Px600= 30 P= 30+600 P = % 3. 60% of 90 is N. N= 60% x 90 N = 0.60 x 90 N = % B. Solve the problem. On a Math test, Aiman answered 80% of the questions correctly. If he answered 40 questions correctly, how many questions were there on the test