MATH 6

STUDY AND LEARNLet’s study how to divide fractions in mixed forms.Have you joined a girl scout or boy scout jamboree? Did you enjoy the hikingactivity?Read the problem.Example 1: A hiking trail in a camping site is 6 1 kilometers long. If a boy scout 4averages 2 1 kilometers per hour, how many hours will it take him to reach the end of 2the trail?Divide to get the number of hours needed to reach the end of the trail.61  21 42What kind of fractions are 6 1 and 2 1 ? 42Can we divide these at once?Change 6 1 and 2 1 to improper fractions. How will you change them? 426 1  2 1  25  5 4 2 42Recall how you divide a fraction by another fraction.Multiply the dividend by the reciprocal of the divisor.The reciprocal of 5 is _____. 26 1  2 1 = 25  5 = 51 4 2 42 25 x 2 = 5 or 2 1 45 2 2 21 2

Can you use cancellation? If yes, do so.What is the answer? Is it simplified already?Therefore, it will take 2 1 hours to reach the end of the trail. 2Example 2: as you follow the steps in dividing.Find the value of n in 3 1  2 2  n . 46Supply the missing number in each31 22  13  14 = 13 x 3 39 or 1 11 46 4 64 6 2= 28 2 28 14 2Simplify your answer.Did you get 39 or 1 11 ? 28 28If yes, very good! If not, review your computations.Example 3:This time, solve for the answer of this exercise as fast as you can.What is the quotient of 4 2 1 2 ? 56Is your answer 33 or 3 3 ? 10 10 3

TRY THESEDivide. Match the quotient in Column A with the mixed number in Column B. Writeonly the letter of the correct answer in the blank.______ 1) Column A Column B______ 2) 42 24 A. 2 6______ 3)______ 4) 68 13______ 5) 71 32 B. 117 27 28 10 2  6 1 C. 1 35 38 39 82 42 D. 2 13 96 46 26 31 E. 3 4 79 7 F. 111 15 WRAP UPIn dividing a mixed number by another mixed number:  Change the mixed numbers into improper fractions.  Multiply the dividend by the reciprocal of the divisor.  Reduce the quotient to its lowest term. 4

ON YOUR OWNA. Find the quotient. Write the letter of the correct answer in the box that corresponds to the quotient.A. 4 1 13 M. 2 1 11 38 10 5B. 3 1  2 1 N. 5 1  2 1 23 42D. 4 3  2 1 S. 8 1 11 55 36I. 6 1 11 W. 3 3 1 1 48 84G. 4 1  2 7 K. 1 2 11 2 10 33Decode the favorite sports of the children.71 2 7 55 13 13 55 21 12 35 21 21 7 10 9 4 4 9 10 3 33 10 1111 55 11 55 2 1 12 2 9 4 9 10 3 If you are able to name the sport, you have a perfect score of 17!!! 5

GRADE VI WORD PROBLEMS USING DIRECT PROPORTIONObjective: Solve word problems using direct proportion REVIEWGoing Up the LadderFind the missing term in each proportion as you go up the ladder. END x5 12 10 4:9 = 8:x x  20 5 25 5  15 9x 3:10=9:x x:20 = 4:54:x = 20:15 1

STUDY AND LEARNThere are problems involving proportion where two quantities in a proportion aredirectly related. A proportion is direct, if an increase in one quantity produces thesame rate of increase in the other quantity and vice versa.Example 1: Lola Ester uses 3 tablespoons of baking powder to make 16 pancakes. How manytablespoons of baking powder does she need to make 48 pancakes?This is a direct proportion. An increase in the amount of baking powder produces acorresponding increase in the number of pancakes and vice-versa.Let’s look at the table.No. of tablespoons of baking powder 3NNo. of pancakes 16 48The proportion is: 3:16 = N:48 or 3N 16 48 16N = 144 N = 144 16 N = 9 tablespoons of baking powderCheck: extremes 16 x 9 = 144 3 x 48 = 144 3:16 = 9:48 The product of the means is equal to the means product of the extremesExample 2:A store has t-shirts for sale at “2 for Php390.” At this rate, how much will6 t-shirts cost?Proportion: 2 : 380 = 6 : N or 2 =6Example 3: 380 N 2N = 380 x 6 N = 2280 = 1 140 2 N = Php1 140 – cost of 6 t-shirts 2

The ratio of two numbers is 3:5. If the larger number is 30, what is the smallernumber?Smaller number 3NBigger number 5 30Proportion: 3: 5 = N:30 5N = 90 N = 90 5 N = 18 smaller number TRY THESEWrite the proportion and solve for the answer.1. If 12 ball pens are bought for Php75, how much will you pay for 24 ball pens at the same rate? Proportion:2. The ratio of men to women in a certain college is 7 to 5. How many women are there if there are 350 men? Proportion:3. A car travels 275 kilometers in 5 hours. How far will it travel in 9 hours at the same speed? Proportion:4. The ratio of math books to science books is 8:5. How many science books are there if there are 152 math books? Proportion:5. Five men can produce 20 armchairs in one day. How many armchairs can be made by 7 men in a day? Proportion: 3

WRAP UP A proportion is direct if as one quantity increases, the other quantity also increases or if one quantity decreases, the other quantity also decreases by the same ratio. Two quantities, A and B, are in direct proportion if by whatever factor A changes, B changes by the same factor. To solve a problem, transform the problem into a proportion, then find the missing term. ON YOUR OWNSolve each problem.1) In a zoo, there are 2 lions for every 5 monkeys. How many lions are there if the zoo has 45 monkeys?2) Three bananas cost Php5.00. How many bananas can be bought with Php20.00?3) A typist can finish a 300-word article in about 6 minutes. At the same rate, how long will it take her to type a 4500-word report?4) The ratio of Pete’s money to George’s is 3:4. If George has Php360, how much does Pete have?5) The ratio of two numbers is 4:7. If the smaller number is 32, find the bigger number. 4

GRADE VI WORD PROBLEMS USING PARTITIVE PROPORTIONObjective: Solve word problems using partitive proportion.REVIEWWhat is N in each proportion? Write the answer in your notebook.START 3 N 2 N 4 N N  15 N  20 3  18 4 28 3 12 7 35 5 25 9 45 6N 18  N N  28 48  N 56  N 72  8 20 10 5 35 54 9 63 9 81 N 36  9 N  12 3  12 5  20 6  42 40 N 8 32 10 N 12 N 7N 3 N 15  5 7 14 18 N 7 N 9 27 IT’S 8N GOLD! 11 33 9N 13 39If you get all the values of N correctly, then you’re ready to solve word problemsinvolving proportion. 1

STUDY AND LEARNSome problems involve partitive proportions.You will study how to solve this type of problem.Study the examples to know more about this type of proportion.Example 1: Two friends invested a capital of Php60 000 in a small store. If their initialcontributions are in the ratio 2:3, how much did each one invest? Think of the sum as divided into 2 + 3 = 5 equal parts. So, one number is 2 5 of the sum and the other is 3 . The invested capital Php60 000 is made up of 5 5 equal parts. Write and solve the proportion for each quantity.1) 2 = N 2) 3 = N 5 Php60 000 (use cross-multiplication) 5 Php60 0005N = 2 x Php60 000 5N = 3 x Php60 000N = 120000 N = 180000 5 5N = Php24 000 N = Php36 000In this problem, you will find the relative value of each quantity given in theproportion.Example 2: An 80-meter rope is cut into 2 pieces given in the ratio of 5:15. How long is theshorter rope? the longer rope?If the ratio of 2 pieces of rope is 5 : 15, consider the 80-meter rope as made up of 20equal parts.To find the length of the shorter rope, you can use this proportion: 2

5  N , where N is the length of the shorter rope20 80Using cross multiplication, 20N = 5 x 80 20N = 400 N = 400 20 N = 20So the length of the shorter rope is 20 meter or 20 m.The length of the longer rope is 80 m – 20 m = 60 m.You can also use proportion to find the length of the longer rope.The proportion for the longer rope is:15  N , where N is the length of the longer rope. Then use cross-multiplication.20 8020N = 15 x 8020N = 1200 N = 60 m – longer piece of ropeN = 1200 20So, the length of the shorter rope is 80 m – 60 m = 20 mExample 3: Rebecca filled three containers, A, B and C with orange juice in the ratio 2:3:4.If the total amount of orange juice is 36 l, find the capacity of the 3 containers. The sum of the ratio is 2 + 3 + 4 = 9 equal parts. Each quantity is equal to 2 , 3 and 4 , respectively. The capacity of the 99 9following containers is:A. 2  N B. 3  N C. 4  N 9 36 9 36 9 36 9N = 2 x 36 9N = 3 x 36 9N = 4 x 36 N = 144 N = 72 N = 108 9 9 9 N = 16L N = 8L N = 12LExample 4: 3

A 75-meter log is to be cut into 3 pieces in the ratio 4:5:6. How long is theshortest piece? longest piece? Sum of the ratio = 15 equal parts Can you w rite the proportion of each piece?Shortest piece = =N Longest piece = NIs the shortest piece 20 m? Is the longest piece 30 m?TRY THESESolve the following problems.1. Two numbers have the ratio of 4:7. If their sum is 143, what are the numbers?2. The ratio of math books to Filipino books in the library is 8 to 5. How many math books are there if there are 247 books in all?3. The ratio of Carl’s money to that of Ramon is 3:4. If together they have Php6 300, how much does each boy have?4. The ratio of the angles of a triangle is 2:3:4. What is the measure of each angle of the triangle?5. The ratio of the sides of a triangle is 4:6:8. If the perimeter of the triangle is 90 centimeters, what is the length of each side?WRAP UP Partitive proportion shows the relative value of each quantity in a proportion. In a partitive proportion, find the total number of equal parts, then express each quantity as a proportion. 4

ON YOUR OWNRead and answer the following problems.1. In a class of 40 students, the ratio of boys to girls is 3:5. How many are girls? How many are boys?2. The ratio of the three sides of a triangle is 1:2:3. What is the measurement of each side if the perimeter of the triangle is 120 cm?3. A sum of Php500 is divided among 3 girls, Ana, Beth and Cynthia in a ratio 2:3:5. What is the share of each girl? Who gets the biggest share?4. Apple, carrot and celery juice are mixed in the ratio 3:5:4. The volume of these mixed juices is 60 l. How many liters of carrot juice are found in the mixture?5. The total weight of Cris, Bong and David is 112 kg. Their weights are in the ratio 3:1:4. What is Cris’ weight? How much heavier is David than Bong?6. Mr. Lee cut a roll of ribbon into three pieces X, Y and Z in the ratio 4:2:1. The length of the longest piece is 28 m. Find the lengths of Y and Z. How long is the roll of ribbon?7. Ronald draws three lines in different colors red, yellow and green. Their lengths are in the ratio 1:3:5. The yellow line is 18 cm long. How long is the green line? How long is the red line? 5

GRADE VI WORD PROBLEMS USING INVERSE PROPORTION Objective: Solve word problems using inverse proportion. REVIEWSolve the following problems using direct proportion. 1. The ratio of Anne’s age to her mother is 1:3. If Anne is 10 years old now, how old is her mother? 2. A narra tree casts a shadow of 9 meters at the time a 2-metre acacia tree casts a shadow of 6 meters. How tall is the narra tree? 3. If 4 packs of milk cost Php90, how much will 10 packs of milk cost? 4. In a mixture, the ratio of sugar to cocoa is 1:3. If 20 cups of sugar are used, how many cups of cocoa are used? 5. Seven grams of iron combined with 4 grams of sulfur form iron sulfide. How much sulfur should be combined with 56 grams of iron to form iron sulfide? Proportion: ________ STUDY AND LEARN At this point, you know when two quantities are in direct proportion. This time, you will explore when two quantities are inversely proportional or indirectly proportional. A proportion is indirect when an increase in one quantity produces a corresponding decrease in another quantity or vice-versa. This means a change in one produces an opposite change in the other. 1

Example 1: If 8 men can finish a work in 6 days, how many men can finish the same work in 4 days? If the work be finished by 8 men in 6 days, will the same amount of work be finished by more or less than 8 men in 4 days? If you decrease the number of days to finish the work, there should be a corresponding increase in the number of men. How will you form the proportion of this kind of problem? Solution:To form this proportion, write the number of days as one ratio and the number of men asthe other ratio. Place the higher value of each ratio on the same side of the quantity. No. of days no. of men6 : 4 = N : 8 or 4 : 6 = 8 : NHigher value higher value4N = 48N = 48 so, 12 men are needed to finish the work in 4 days 4N = 12Note that the higher number in each ratio is on the same side of the quantity.Example 2: Twelve boys can consume the food in 5 days, how many days will they food last if there are only 8 boys? 12 boys → food will last for 5 days 8 boys → will the food last for more or less than 5 days? Can you write now the proportion? 12:8 = : Is your answer 7 1 days? 2 2

 Let’s solve. 12 : 8 = N : 5 8N = 60 N = 60 = 7 1 days 82 TRY THESEWrite the proportion for each problem and solve.1. If a can of dog food can feed 5 dogs for 4 days, how long will it last if you have 8 dogs to feed? Proportion: ___________2. Ten mountain climbers brought enough food for 14 days. If 4 more joined them on the way, for how long will the food last? Proportion: ___________3. If 4 men can do a job in 6 days, how many men are needed to do the job in 8 days? Proportion: ___________ WRAP UP Remember:  Two quantities are indirectly or inversely proportional if an increase in one quantity results in a decrease in the other quantity or vice versa.  To write the inverse or indirect proportion, write the higher number in each ratio on the same side. 3

ON YOUR OWNEvaluate yourself now. Do you think you have already mastered the skill of solvingproblems using indirect proportion? Solve the following problems.1. If 3 farmers can plow a field in 4 days, how long will it take 6 farmers to plow the same field? Proportion: ___________2. Four pumps can fill a tank in 42 minutes. How long can 6 pumps of the same kind fill the tank? Proportion: ___________3. I have enough money to have a vacation for 12 days if I spend Php500 a day. How many days will my money last if I decided to spend only Php400 a day? Proportion: ___________4. A contractor has enough money to pay 8 workers for 15 days. If he adds 8 more workers, for how many days of work can he pay the workers at the same rate? Proportion: ___________5. Five people can clean the whole house in 8 hours. If the owner gets 3 additional cleaners, how long will it take to clean the whole house? Proportion: ___________ 4

GRADE VI FINDING THE PERCENTAGEObjective: Find the percentage when the rate and base are given.REVIEWLet us find out first if you still recall other topics in Mathematics. Answer the exercisesbelow in your notebook. Read the directions for each one carefully.A. Complete the table by writing the equivalent of the given number in decimal, fraction or percent. The first one is done for you. Decimal Fraction Percent 0.5 50% 0.75 50 or 1 75% 100 2 36%2) 1) 36 or 9 3) 1.4 100 25 120%4) 140 or 7 100 5 5)B. Multiply. Simplify the answer if possible.1) 45 x 1 4) 12.4 x 0.65 2 5) 5 8 x 62) 63 x 0.3 10 23) 1.2 x 2 3Let us first recall base, rate and percentage. 1

STUDY AND LEARNIn the equation below, identify the rate, base and percentage.Example 1: 20% of 8 = 1.6a) 20% is the ____.b) 8 is the ____.c) 1.6 is the ____.If your answers are a) rate, b) base and c) percentage, then you are correct.What is 20% x 8?How will you solve for the percentage?We need to change or rewrite 20% first as decimal or fraction. 20% = 20 = 0.20 100 20% of 8 means 0.20 x 8 = 1.6 Therefore, 20% of 8 = 1.6Example 2: Let us now solve for 12% of 60. 2

LESSON 1In the Mathematics expression 12% of 60 is ______, the word “of” stands for the“multiplication operation” and the word “is” means “equal to”. The blank can be anyletter. Therefore, “12% of 60 is ____” can be written as 12% x 60 = Q.We now solve for the percentage following the steps above. 0.12 x 60 = Q 7.2 = QHence, 12% x 60 = 7.2.Let us go back to the examples. Compare the numbers given by writing <, > or = insidethe . Answer in your notebook.20% x 8 = 1.6 12% x 60 = 7.220%  100% 12%  100%8  1.6 60  7.2If your answer to the first  is <, you are right. Twenty percent and 12% are both lessthan 100%.If your answer to the second  is >, you are correct. The base 8 is bigger or more thanthe percentage 1.6. Similarly, the base 60 is bigger or more than the percentage 7.2.It simply means that IF THE RATE IS LESS THAN 100%, THE PERCENTAGE ISLESS THAN THE BASE.R < 100%P<BLet us move on to another way of finding the percentage, that is, by changing the rate tofraction.Let us again use 20% x 8.1. Write the expression as an equation. 20% x 8 = N2. Change 20% to a fraction. 20% = 20 or 13. Multiply 20 by 8. 100 5 100 20 x 8 = N 100 1.6 = NNote that we got the same answer: 20% x 8 = 1.6This means that whether we change the rate to a decimal or a fraction we get the sameanswer. 3

Let us try another example. 4% of 900 4% = 0.04 900 x 0.04 = TLet us follow the steps.1. Write the expression as an equation. 4 x 900 = T 36 = T2. Change the rate to a fraction. 100 13. Multiply. 36 4 x 900 = 3600 = 100 100 1 So 4% x 900 = 36Now try this one on your own. Just follow the steps and you won’t go wrong. 15% of 400 is ____.What answer did you get?LESSON 2 Here are the equations with the answer.Now let us answer some more. 1) 100% of 65 is 65 2) 100% of 120 is 120 Get the percentage of the following 3) 100% of 2 415 is 2 415 equations.1) 100% of 65 is ___2) 100% x 120 = N3) 100% of 2 415 = CWhat do you notice with the base and percentage? You will notice that the base and thepercentage are the same.This means that: IF THE RATE IS 100%, THE PERCENTAGE AND BASE AREEQUAL OR THE SAME.R = 100% P=BLESSON 3 4

Using the same steps in Lesson 1, solve for the percentage of the following expressions:1) 120% of 82) 150% of 503) 200% of 355Your answer should be 9.6, 75 and 710, respectively. Confused? Don’t be. It’s reallyvery simple.Let us look at these three equations.1) 120% of 8 = 9.62) 150% of 50 = 753) 200% of 355 = 710What do you notice with the percentage and the base?Are the percentages more than the bases? Yes, they are.Now look at the rates.Are the rates more than 100%? Yes, they are.This means that: IF THE RATE IS MORE THAN 100%, THE PERCENTAGE ISMORE THAN THE BASE.R > 100% P>BTRY THESEA. Find the percentage by changing the rate to: Decimal Fraction1) 6% of 7 4) 90% of 70 5) 41% of 8802) 40% of 935 6) 4.2% of 1 4503) 87 1 % of 32 2B. Find the percentage. Answer in your notebook. Look closely at the rate.1) 101% of 50.5 is ____ 5

2) 132% of 983) 140% x 2 3604) 233% x 6505) 200 1 % x 10 4 WRAP UP  To find the percentage, change the rate to a decimal or a fraction, then multiply with the base.  If the rate is less than 100%, the percentage is less than the base.  If the rate is 100%, the percentage is equal to the base.  If the rate is more than 100%, the percentage is more than the base. ON YOUR OWN 6) 100% of 16 7) 200% of 59A. Find the percentage. 8) 1 % x 50 1) 36% of 75 2 2) 104% of 90 3) 15 2 % x 175 5 4) 72 % x 250 5) 100.1% x 304 6

GRADE VI FINDING THE RATEObjectives: Find the rate or percent when the percentage and base are given. Derive a formula for finding the rate or percent.REVIEWA. Rewrite each of the following as a division sentence. 1) 4 x A = 12 2) 3 x B = 63B. Divide. 3) 20  25 4) 792  22C. Identify the rate, base and percentage. Find the missing number. 5) 36% x __ = 75 6) __ x 8 = 9D. Complete the table. Decimal Fraction Percent 1.5 7) 8) 10)9) 1 3 1

STUDY AND LEARNLet us work on two equations given in the review exercise.1) __ x 8 = 92) 64 x __ = 100What are missing in each of these equations?We can write the two equations as:1) N% x 8 = 92) 64 x E% = 100Here, we are given a multiplication sentence with one of the factors missing. You havelearned that in order to get the missing factor, we need to use the inverse operation-division.Let us try and solve equation number 1 step by step.1. Divide both sides by 8 so only N% will be left N% x 8 = 9on the left side. N% x 8 = 9 882. Remember, % is the same as (x 0.01) or N% = 1.125(x 1 ). We now rewrite the equation. N x 0.01 = 1.125 1003 Divide both sides by 0.01 N x 0.01 = 1.125 0.01 0.01 N = 112.5We now complete the equation as 112.5% x 8 = 9.Therefore, the missing rate or percent is 112.5%. Take note that it is not 1.125%.Try answering equation number 2 on your own. Do it step by step.Look at the solution below and see if you got the same answer.64 x E% = 10064 x E% = 100 2

64 64E% = 1.5625E x 0.01 = 1.5625E x 0.01 = 1.5625 0.01 0.01E = 156.25%Now, we know that the missing rate or percent is 156.25%.The complete equation now is 64 x 156.25% = 100.Try this on your own: G% of 6 is 3.Remember “of” means multiplication and “is” represents the equal sign.What value of G did you get?If you got 50 = G, then you have the correct answer.We now have 50% of 6 is 3, or 50% x 6 = 3.Let us go back to the last example you solved. G% x 6 = 3Let us identify the rate, base and percentage. G% x 6 = 3Rate (R) Base (B) Percentage (P)Let us replace the given with the letters R, B and P.G% x 6 = 3R% x B = PLet us do the steps which were illustrated earlier.1. Divide both sides by B. R% x B = P R% x B = P2. % can be written as (x 0.01) or (x 1 ). In this case let us 100 BB R% = P use (x 1 ) to replace % sign. 100 B Rx 1 = P3. Divide both sides by 1 . 100 100 B Rx 1 = P 100 B 3

R= P  1 B 1004. Using the rule in dividing fractions, we get the reciprocal of 1 and change  to x. 100We have succeeded to derive the formula in finding the rate: Rate = percentage x 100 baseLet us now try this formula to find the rate of 6 is R% of 30. Let us compare the answerwe will get to the answer obtained using the steps we used earlier.Step-by-Step Formula6 = R% x 30 6 = R% x 30 6  R%  x  30 6 x 100 = R30 30 300.2 = R x 0.010.2 = R x 0.01 0.2 x 10 = R0.01 0.01 20 = ROr6 = R% x 30 Notice that these two are the same. This6 = R% x 30 means that the formula is a30 30 simplification of the step-by-step way of solving for the rate. 10.2 = R x 1000.2  100 = R20 = RNow use the formula in answering these questions.1) 7 = R% x 562) 20 x R% = 4The answers are 1) R = 12.5 and 2) R = 20.Therefore, the rates are 12.5% and 20% respectively. 4

You are now ready to answer the practice exercises. Take your time in answering. Readthe directions carefully. TRY THESEFind the missing rate or percent by finding the value of the letter first, then give thecomplete equation.1) H% of 10 is 82) K% of 24 = 63) 9 is M% x 994) 45 = B% x 1205) 540 x F% = 90 WRAP UPTo find the rate or percent:  rewrite the equation properly  divide the percentage by the base ( P ) B  rewrite % as (x 0.01) or (x 1 ) 100  divide both sides by 0.01 or 1 100  rewrite the answer with the % sign.The formula in finding the rate is: R = P x 100 B 5

ON YOUR OWN Read the direction carefully. Answer in your notebook. Find the rate or percent. Then fill in the blanks to complete the sentences. Remember to rewrite the equation first. It is better to use fractions for some answers.1) A% of 50 is 10. a. The value of A is ____. b. The missing rate is ____. c. The complete equation is ____.2) 20 is C% of 150. a. The value of C is ____. b. The missing rate is ____. c. The complete equation is ____.3) 12 is 108 x J%. a. The value of J is ____. b. The missing rate is ____. c. The complete equation is ____.4) L% of 75 is 15. a. The value of L is ____. b. The missing rate is ____. c. The complete equation is ____.5) 450 x Q% = 75 a. The value of Q is ____. b. The missing rate is ____. c. The complete equation is ____. 6

GRADE VI FINDING THE BASEObjectives: Find the base when the percentage and rate are given. Derive the formula in solving for the base. REVIEWRead the directions carefully. Write your answers in your notebook.A. Identify the missing part. Write rate, base or percentage. 1) N% x 8 = 7 2) 45% x 50 = P 3) 36 x A% = 3.6 4) z = 16 x 100% 5) s x 9% = 120B. Solve for the value of the letters. 1) W% x 95 = 23.75 2) 65% x 365 = H 3) 495 x R% = 249.975 4) 40 x 120% = T 5) U% x 95 = 87.4 1

STUDY AND LEARNYou have already learned how to find the percentage as well as how to find the rate. Atthis point, you are going to study how to find the base.Look at the sample below. 9% of a number is 120. Find the base.Let’s solve for the base step-by-step.1. Change 9% to decimal. 9% = 0.92. Divide the percentage by the rate. B x 0.09 = 120 B = 120  0.09 1 B = 1 333.33The complete equation is 1333 x 9% = 120. 3Let us solve another example:80% of a number is 40. Find the number. 80% x B = 40 0.80 x B = 40 B = 40  0.80 B = 50Let us check if the answer is correct using the proportion method.80% x B = 40 80 x B = 40100 80 = 40100 B80B = 40 x 10080B  4000 80 80 B = 50 2

Therefore, whether you use the equation method or the proportion method you will getthe same answer.Let’s have other examples.Equation Method Proportion Method1) C x 25% = 4 C x 25% = 4 C x 25 = 4 C = 4  0.25 C = 16 100 25 = 4 100 C 25C = 4 x 100 25C  400 25 25 C = 16 16 x 25% = 42) 30 = 15% x A 30 = 15% x A 30 = 0.15 x A 30 = 15 x A A = 30  0.15 A = 200 100 30  15 A 100 3000  15A 15 15 200 = A 30 = 15% x 200Let us go back to the equation method of solving for the base. Let us answer thisexample: 75% x B = Php450Step 1 75% x B = Php450 0.75 x B = Php450Step 2 B = Php450  0.75 B = Php60075% x Php600 = Php450Let us go back and identify the parts of the equation. 75% x B = 450Rate (R) Base (B) Percentage (P) 3

Using these parts, let us derive a formula for finding the base.Step 1 R x B = PStep 2 B= P RDon’t forget that the rate is with % sign.Try answering the two equations using this formula. Again, take your time in answering.1) 25% x B = 1102) 32.4 = 45% x BLet us see if you got the same solution and answer.1) 25% x B = 110 2. 32.4 = 45% x B B P B P R RB  110 B  32.4 25% 45%B  110 B  32.4 0.25 0.45B = 440 B = 72If you get both correct, very good! You can now proceed to the exercises under “TryThese.” Good luck! TRY THESERead each direction carefully. Answer in your notebook. Take your time in answering.A. Find the base. Use the equation method. 1) 81 = 80% x F 2) 27 = 12% x G 4

3) 36% x H = 216 4) 62.5% x I = 45B. Find the base. Use the proportion method. 5) 45% x J = 37.8 6) 315 = 87.5% x K 7) 65% x L = 1040C. Use the formula in finding the value of B, or the base. Answer in your notebook. 1) 24 = 60% x B 2) 70 = B x 14% 3) 27 = 9% x B 4) 20% x B = 80 5) 150% x B = 1800 WRAP UP To find the base:  Change rate to a decimal or a fraction.  Divide the percentage by the rate.  The quotient is the value of the base. There are two methods in solving for the base:  Equation method  Proportion method The formula in finding the base is B  P . R 5

ON YOUR OWNRead the directions carefully. Answer in your notebook.A. Find the base then write the complete equation. Use the proportion method.1) 45% x B = 90 3) B x 33 1 %  352) 12% x B = 48 3 4) 63 = B x 18%B. Find the base then rewrite the complete equation. Use the proportion method.5) 1 % x B = 4 7) B x 10% = 3 26) 15% x B = 3C. Find the base then rewrite the complete equation. Use the formula B  P . R8) 10% x B = 98 10) 46 = 4% x B9) B x 17.5% = 175 6

GRADE VISOLVING WORD PROBLEMS INVOLVING FINDING THE PERCENTAGE Objective: Solve word problems involving finding the percentage.REVIEWA. Identify the base, rate and percentage in each equation. The first one has been done for you. Write the answers in your notebook. Equation Base Rate Percentage 8 36% N1) 36% of 8 = N2) 65 x 95% = N3) 46% x A = 234) D x 10% = 115) 6 x 9% = M6)  x 500 = 250B. Find the percentage. The first one has been done for you. 1) 16 2 % x 15 = 2.5 3 2) 5.5% x 68.7 3) 20% x 8 1 3 4) 37 1 % x 24 2 5) 18 1 x 5% 4 6) 95 x 250% 1

STUDY AND LEARNCan you tell at once how many correct answer you get in a test if the rate and base aregiven in the problem?Read the problem. William got 90% correct answers in a 50-item test in Math VI. How many itemsdid he answer correctly?Questions:1. What will you find in the problem? Are you going to look for the number of correct items?2. Which data are given in the problem? Is 90% the rate? Is 50-item test the base?3. If the base and the rate are given in the problem, how will you transform it to an equation?Since you will be finding the percentage, the equation will be:90% of 50 = P90% x 50 = P4. You can now solve for the percentage using the different methods shown.Using an EquationThe basic equation for finding the percentage:Rate (R) x Base (B) = Percentage (P)From here, you now have 90% x 50 = PSolve now for P. 90% x 50 = P or 90% x 50 = P 0.9 x 50 = P 90 x 50 = P 45 = P 100 45 = PUsing Proportion 2

We have 90% is to 100% and P is to 50.Setting the Proportion90:100 = P:50Solving for P: 90:100 = P:50 90 = P 90 x 50 = 100 x P 100 50 4500 = 100P 90 x 50 = 100 x P 100 100 45 = P 4500 = 100P 100 100Using the Shortcut Method 45 = P  90 = P 100 50 90 x 50 = P 100 4500 = P 100 45 = P100% = 50 10% = 5 x9=x9 90% 45So, William got 45 items correct.Let’s have another example: Mang Juan was able to gather 320 kilograms of atis to be sold the next day.During the market day, he was able to sell 75% of the fruits. How many kilograms of atisdid Mang Juan sell?1. What is the total number of kilograms of atis? _____. This is the base.2. How much was he able to sell? ______. This is the rate.3. If the base is 320 and the rate is 75%, what is the equation for solving the percentage?4. Did you have 75% of 320 = N? Solve for N using any method you used for Problem A. 3

5. How many kilograms of atis did Mang Juan sell? The answer is _____ kilograms. This is the percentage.This time, try to work on this problem. Berto is given a weekly allowance of Php500. He sets aside 20% of this to put inhis coin bank. How much does he put in his coin bank?Base = _____ Rate = _____ Percentage = _____Equation: _____________Answer: Berto puts _____ in his bank.TRY THESEA. Identify the base, rate in each problem. Solve for the percentage. Fill the boxes to complete the table. Word Problems Base Rate Percentage1. Marian earns 300 a day tending the store. Shegives 25% of it to her parents. How much moneydoes Marian give to her parents?2. There are 55 mangoes in the basket. Forty percentof them are ripe. How many ripe mangoes arethere in the basket?3. Mang Nic has 3 hectares of land to plant. He hasplanted 60% of the land with palay. How manyhectares were planted with palay?4. Every payday Maryl budgets 70% of her Php1,500earning for bill payments. How much does shebudget for paying all her bills?5. Pedro was tasked to fill 5 big bottles with grated papaya for the “atchara.” He was able to fill 150%of the bottles. How many bottles was he able tofill with grated papaya?B. Let us answer two problems. You may use any of the methods. Write the answers to the blanks in your notebook. 4

1) Joyce is helping her mother pack “kutsinta” to be sold in the market. Mother gave her 20 plastic bags to fill. Joyce was able to fill 80% of the plastic bags with “kutsinta.” How many plastic bags did Joyce fill?a) R = ____ B = ____ P = ____b) How many plastic bags did Joyce fill?c) What kind of child is Joyce? Why?2) Mico is helping his father harvest corn in their field. He was able to harvest corn from 85% of the 1 000 square meters of the field. How much of the field was Mico able to harvest corn from?a) R = ____ B = ____ P = ____b) How much of the field was he able to harvest corn from?c) What kind of child is Mico? Why?d) If you were Mico, would you do the same? Why?WRAP UPIn solving problems involving finding the percentage: Find what the problem asks for. Identify which the base and the rate are in the given data. Transform the problem in equation form. Solve for the percentage using any of the following methods: - using equation - using proportion - using the shortcut method Check if the answer is correct. 5

On Your OwnRead the problems carefully. Use the guide given in answering them.1) There are 50 pupils in class. In a Math test, only 16% got a score of 95. How many pupils got 95?a) R = ____ B = ____ P = ____b) Proportion:c) Solution:d) Complete Answer2) A family spent 4 650 for food this month. 32% of this was used to buy a sack of rice. How much was the sack of rice?a) R = ____ B = ____ P = ____b) Solution:c) Complete Answer 6

GRADE VI SURFACE AREA OF A CUBEObjective: Find the surface area of a cube. REVIEWFind the area enclosed by of each of the following squares. Write your answer in your notebook.1) A = ____ 4) A = ____ A = ____ 31 m A = ____ 15 mm A = ____ 22) 5) 3 32 cm dm 53) 9.7 dm 1

STUDY AND LEARNThe surface area of a figure is the sum of the areas of all its faces.A cube is a three-dimensional figure which has length, width and height. It has squarefaces and the sides are called edges.Example 1:Look at the cube on the right. How many faces does a cube have? 6 A B C Can you name all of them? plane ABCD, faces D H plane ADEF, plane ABGF, plane BCHG, F G vertex plane CHED, plane EFGH E What kind of a plane figure does the shape of a face look like? square How many edges does a cube have? (12) How many vertices does a cube have? (8)How can we get the area of a square?Do you still remember? Area = side x side or in symbols s x s or s2 = 4 cm x 4 cm Area = 16 cm2s = 4 cmSince, the six faces of a cube are all squares, then the surface area of a cube is 6 times thearea of one face.In symbols,SA = 6(side x side) or 6(s2) where s is the side of a face, which is a square 2

Example 2:Take a look at this example: SA = 6(s x s) = 6(12 cm x 12 cm)12 cm = 6(144 cm2) SA = 864 cm2Take a look at this word problem as another example.Find the surface area of a cube whose edge is 8.3 cm.How long is one edge of the side? 8.3 cmSA = 6(side x side) = 6(8.3 cm x 8.3 cm) = 6(68.89 cm2)SA = 413.34 cm2 TRY THESEA. Find the surface areas of the following cubes. Write your answer in your notebook. 1) 2) 3) 9 dm 13 dm 20 cmSA = ____ SA = ____ SA = ____ 3

B. Copy the table in your notebook. Then complete it by finding the surface areas of the 5 given cubes.Cube Edge Surface Area A 7.3 m B 15.2 cm C 9.4 dm D 12 m 3m E 4C. Find the surface area of each of the cubes in Column A. Then, match your answer with Column B. Write only the letter in your notebook. AB1. A. e = 11 cm SA = 530.16 cm22. B. SA = e = 9.4 cm 2,747.76 cm23. C. e = 13.2 cm SA =4. 3,750 cm2 e = 21.4 cm D. SA =5. 1,045.44 cm2 e = 25 dm E. SA = 726 cm2 4

WRAP UP  The surface area of a figure is the sum of the areas of all its faces.  The formula in finding the surface area of a cube is 6 times the area of one face. In symbols, SA = 6(side x side) or 6(s2) ON YOUR OWNA. Find the surface area of each of the following cubes. Write your answers in your notebook. 1) 2) 5 cm 5.8 dm3) 4) 17 cm 14.7 mm5) 5 dm 8 5

B. Find the surface area of the following cubes. Write your answer in your notebook. 1) 2) 17 cm 17 cm 21.3 cm 21.3 cm 21.3 cm 17 cm3) 4) 7 1 dm 7 1 dm 7 dm 7 dm 5 5 10 10 7 dm 7 1 dm 10 5 SA = ____ SA = ____5) 36 cm 36 cm 36 cm SA = ____ 6

GRADE VISURFACE AREA OF A RECTANGULAR PRISMObjective: Find the surface area of a rectangular prism. REVIEWFind the area of each of the following rectangles. Write your answers in your notebook.1. A = _____ 5 dm 12 dm2. A = _____ 4 dm 6 dm A = _____3. 17 cm 7 cm A = _____4. 8.2 mm 4.3 mm A = _____5. 5m 1

7m STUDY AND LEARNRead the problem. The surface area of a solid figure is the sum of the areas of all its faces.The surface area of a rectangular solid which is also called rectangular prism is the sumof the areas of all the faces. height = 3 cm width = 5 cm length = 10 cm  How many faces does the figure have? (6)  How many vertices are there? (8)  How many edges are there? (12)  What can you say about the faces? Are they all the same? (No)To visualize clearly the faces of the rectangular prism, we will unfold the figure. SIDE (E) FB R TBO O OA T 10 cm N P C T 10 cm T KO M (A) (B) (C) (D) 3 cm 5 cm 3 cm SIDE (F) 3 cm 5 cm  Which pairs of faces are the same in size and shape? (Front and Back; Top and Bottom; right side and left side)  What plane figure do the faces of a rectangular prism represent? (rectangle)  How do you get the area of a rectangle? (A = base x height or length x width) 2

 To find the surface area of the figure, we need to find the sum of the areas of the 6 faces of the rectangular prism. Face Area of Face (cm2) A 10 x 3 = 30 B 10 x 5 = 50 C 10 x 3 = 30 D 10 x 5 = 50 E 5 x 3 = 15 F 5 x 3 = 15 190TOTALThe surface area of the figure is 190 cm2.We can also solve the same problem this way:We know that:1. the bottom and top faces have the same dimensions, 10 cm by 5 cm2. the left and right faces (sides) have the same dimensions, 5 cm by 3 cm3. the front and back faces have the same dimensions, 10 cm by 3 cmUsing the area formula for a rectangle, A = bh, we have:1. for bottom and top faces A = (10)(5) = 50 cm250 x 2 = 100 cm22. for left and right faces (sides) A = (5)(3) = 15 cm215 x 2 = 30 cm23. for front and back faces A = (10)(3) = 30 cm230 x 2 = 60 cm2Adding these areas to find the surface area, we have100 cm2 – bottom and top30 cm2 – left and right 3