Grade 8 Inverse Proportions

Transcription

1 ID : ww-8-inverse-proportions [1] Grade 8 Inverse Proportions For more such worksheets visit Answer t he quest ions (1) There are f ew taps and holes in tank. Each tap can f ill empty tank in 4 minutes, while each hole can empty a f ull tank in 8 minutes. If there are 3 taps and 2 holes, how long would it take to f ill empty tank? (2) If 4 boys can f inish a work in 24 hours, how long will it take if 16 boys do it. (3) 21 women can f inish a work in 12 hours. If 7 women f ell ill bef ore the work started and did not work, how long would it take to f inish the work? (4) Tap A can f ill a tank in 6 hours, while tap B can f ill the same tank in 3 hours. If both taps are open together, how long will it take to f ill an empty tank? (5) 3 workers can build a 48 meter long wall in 4 days. At the same rate, how long will it take f or 6 workers to build a 120 meter long wall? Choose correct answer(s) f rom given choice (6) Elizabeth and Deborah together can do a work in 42 hours, but if Paul also helps them they can f inish it in 6 hours only. How long would it take if Paul has to do it alone? a. 10 hours b. 7 hours c. 5 hours d. 8 hours (7) Two taps A and B can f ill a water tank in 20 and 5 hours respectively. If they are turned up alternatively f or one hour each (staring with tap A). Find the time taken to f ill the tank. a. 8 hours b. 7.5 hours c. 6 hours d. 9 hours (8) Tap A can f ill a tank in 8 hours, while tap B can empty the f ull tank in 10 hours. If both taps are open together, how long will it take to f ill an empty tank? a. 43 hours b. 40 hours c. 41 hours d. 38 hours (9) It takes machine M, x hours to manuf acture a deck of cards that machine N can manuf acture in y hours. If machine M operates alone f or z hours and is then joined by machine N until 40 decks are f inished, f or how long will the two machines operate simultaneously? a. 40(x - z) b. 40x - z c. (40x - z)y d. 40x - z

2 ID : ww-8-inverse-proportions [2] Fill in the blanks (10) Pump A can f ill a tank in 20 minutes, while pump B can f ill the same tank in 5 minutes. Both pumps are turned on together to f ill the empty tank, but when tank became half f ull, pump B stops working and rest of the tank is f illed by pump A only. It took minutes to f ill the tank. (11) Train X takes 28 hours to travel f rom station A to station B, while train Y takes 21 hours to travel f rom station B to station A. If train X and Y starts at same time f rom stations A and B respectively, trains will cross each other af ter hours. (Assume trains travel with unif orm speed) (12) Jennif er can complete a task in 24 minutes, while Michael can complete the same task in 12 minutes. If both Jennif er and Michael work together, then the task will be completed in minutes. (13) Michael works twice as f ast as William. If Michael can complete a work in 3 minutes, it will take minutes to complete the work when both Michael and William works together. (14) If a man can do a work in 7 days, and woman can do the same work in 14 days, it will take days if 2 men and 3 women do the same work. (15) Laura can complete some work in 36 days, while Michelle can complete the same work in 45 days. They started the work together, and when the work was half complete, Michelle f ell ill and Laura had to f inish the remaining work. The work got compled in days Edugain ( All Rights Reserved Many more such worksheets can be generated at

3 Answers ID : ww-8-inverse-proportions [3] (1) 2 minutes It takes one tap 4 minutes to f ill 1 tank. So in 'x' minutes it f ills x/4 of the tank So with 3 taps, it will f ill (3 * x )/4 of the tank. It takes one hole 8 minutes to f ill 1 tank. So in 'x' minutes it empties x/8 of the tank So with 2 holes, it will empty (2 * x )/8 of the tank. So the equation f ormed is (3 * x )/4 - (2 * x )/8 = 1 (subtraction is being done because we have to remove the quantity that is being emptied by the holes ) Solving this equation we get x = 2 minutes (2) 6 hours We know 4 boys can f inish a work in 24 hours. So 1 person will take 24*4 hours to f inish a work So 16 boys will take 24*4/16 hours to f inish a work So total time taken is 6 hours. (3) 18 hours We know 21 women can f inish a work in 12 hours. So 1 person will take 12*21 hours to f inish a work Since 7 women f ell ill, so the remainig workers are now 21-7 = 14. So 14 women will take 12*21/14 hours to f inish a work So total time taken is 18 hours.

4 (4) 2 hours ID : ww-8-inverse-proportions [4] We know that f or A, it takes 6 hours to f ill 1 tank. So in 1 hour it will f ill 1/6 of the tank. So in 'x' hours it will f ill x/6 of the tank We know that f or B, it takes 3 hours to f ill 1 tank. So in 1 hour it will f ill 1/3 of the tank. So in 'x' hours it will f ill x/3 of the tank Since both taps are used to f ill the tank so x/6 + x/3 = 1 (1 ref erring to 1 f ull tank ) Solving this we get x = 2 hours (5) 5 days Since work done by 3 workers in 4 days = 48 meter Work done by 3 workers in one day = 48/4 = 12 meter Work done by one worker in one day = 12/3 = 4 meter Work done by 6 workers in one day = 4 6 = 24 meter Since 24 meter long wall is done by 6 workers in one day 120 meter long wall will be done by 6 workers in 120/24 = 5 days (6) b. 7 hours Elizabeth and Deborah together can do a work in 42 hours. So we consider it as one unit. So if they can complete a work in 42 hours,then in 'x' hours they can complete x/42 of the work. Let say Paul can complete a work in 'y' hours. So in 'x' hours, x/y of the work can be done. If Elizabeth, Deborah and Paul do the work together they complete it in 6 hours. So the equation f ormed is x/42 + x/y = 1. Here x = 6. Now the equation is 6/42 + 6/y = 1. Solving f or y we get y = 7. So Paul completes the work individually in 7 hours.

5 (7) a. 8 hours ID : ww-8-inverse-proportions [5] Tap A takes 20 hours to f ill a water tank. So in 1 hour it f ills 1/20 of the tank. Tap B takes 5 hours to f ill a water tank. So in 1 hour it f ills 1/5 of the tank. Since the tank is f illed by each tap in alternate hours, so in one hour 1/20 of the tank is f illed, and the second hour 1/5 of the tank is f illed. We have to f ind out the number of hours required to f ill the whole tank. So 1/20 + 1/5 + 1/20 + 1/ = 1 When this value equals to one, We get the number of hours the tank was f illed in. Step 6 So here the tank was f ull in 8 hours

6 (8) b. 40 hours ID : ww-8-inverse-proportions [6] If you look at the question caref ully, you will notice that Tap A can f ill a tank in 8 hours, while tap B can empty the f ull tank in 10 hours. If both taps are open together, then the time it will take to f ill an empty tank is equal to the LCM of the time in which Tap A can f ill a tank and tap B can empty the f ull tank. Calculating the LCM of 8 and 10. All prime f actors of 8: is a factor of is a factor of is a factor of = All prime f actors of 10: is a factor of is a factor of = 2 5 Now the LCM of 8 and 10 is = = 40 Step 6 Theref ore the time it will take to f ill an empty tank = 40 hours

7 ID : ww-8-inverse-proportions [7] (9) c. (40x - z)y Machine M takes x hours to manuf acture a deck of cards. So in 1 hour it manuf actures 1/x of the deck and in z hour it manuf actures z/x of the deck. Machine N takes y hours to manuf acture a deck of cards. So in 1 hour it manuf actures 1/y of the deck. Both machines have to manuf acture 40 decks and machine M operates alone f or z hours. Since, machine M operates z/x decks in z hours. Theref ore, the remaining decks = 40x - z x In 1 hour, the number of decks manuf actured by both machines = 1 x + 1 y = y + x = or we can say that, the time taken by both machines to manuf acture decks = 1 hour The time taken by both machines to manuf acture 1 deck = 1 = hours The time taken by both machines to manuf acture 40x - z x decks = 40x - z x = (40x - z)y hours Thus, the two machines will operate (40x - z)y hours simultaneously.

8 (10) 12 ID : ww-8-inverse-proportions [8] Pump A can f ill the tank in 20 minutes. So in 1 minute it would f ill 1/20 of the tank. So in 'x' minutes it would f ill x/20 of the tank. Similarly, Pump B can f ill the tank in 5 minutes. So in 1 minute it would f ill 1/5 of the tank. So in 'x' minutes it would f ill x/20 of the tank. Here when both the pumps are open, they together f ill only half of the tank. So the equation f ormed would be x/20 + x/5 = 1/2 Solving this we get x = 2 We know that the remaining half tank was f illed by pump A. So if pump A takes 20 minutes to f ill the whole tank, then it would take 20/2 minutes to f ill half of the tank Step 6 So the total time taken to f ill the tank is /2 = 12 minutes. (11) 12 Train X takes 28 hours to reach f rom A to B that is 1 side. So in 1 hour it will reach 1/28 of the distance. So in 'x' hours it will reach x/28 of the distance. Train Y takes 21 hours to reach f rom B to A that is 1 side. So in 1 hour it will reach 1/21 of the distance. So in 'x' hours it will reach x/21 of the distance. So the equation f ormed is x/28 + x/21 = 1 (When both trains will meet at a point, so the total distance between the two stations is covered ) Solving this we get x = 12 hours

9 (12) 8 ID : ww-8-inverse-proportions [9] We know that f or Jennif er, it takes 24 minutes to complete a task. So in 1 minute it will complete 1/24 of the task. So in 'x' minutes it will complete x/24 of the task. We know that f or Michael, it takes 12 minutes to complete a task. So in 1 minute it will complete 1/12 of the task. So in 'x' minutes it will complete x/12 of the task Since both Jennif er and Michael together complete the task so x/24 + x/12 = 1 (1 ref erring to 1 complete task) Solving this we get x = 8 minutes (13) 2 Let Michael take 'y' minutes to complete the task. Michael is 2 times f aster than that of William. It takes Michael to complete work in 3 minutes. So it takes William 2*3 = 6 minutes to complete the task. We know that f or Michael, it takes 3 minutes to complete a task. So in 1 minute it will complete 1/3 of the task. So in 'x' minutes it will complete x/3 of the task. We know that f or William, it takes 6 minutes to complete a task. So in 1 minute it will complete 1/6 of the task. So in 'x' minutes it will f ill x/6 of the task Step 6 Since both Michael and William together complete the task so x/3 + x/6 = 1 (1 ref erring to 1 complete task) Step 7 Solving this we get x = 2 minutes

10 (14) 2 ID : ww-8-inverse-proportions [10] A man can do a work in 7 days. So in 1 day, he could do 1/7 of the work. So 2 men in 1 day could do 2/7 of the work. So in 'x' days x/7 of the work would be done. So 2 men in 'x'days could do (2*x)/7 of the work. A woman can do a work in 14 days. So in 1 day, she could do 1/14 of the work. So in 'x' days x/14 of the work would be done. So 3 women in 'x'days could do (3*x)/14 of the work. So the equation f ormed would be (2*x)/7 + (3*x)/14 = 1 Solving this we get x = 2 So the work can be completed in 2 days (15) 28 Laura can complete a work in 36 days. So in 1 day it would complete 1/36 of the work. So in 'x' days it would complete x/36 of the work. Michelle can complete a work in 45 days. So in 1 day it would complete 1/45 of the work. So in 'x' days it would complete x/45 of the work. Here when both Laura and Michelle work together, they complete only half of the work. So the equation f ormed would be x/36 + x/45 = 1/2 Solving this we get x = 10 We know that the remaining work was completed by Laura. So if Laura takes 36 days to complete the whole work, then it would take 36/2 days to complete half load of work lef t Step 6 So the total time required to complete the work is /2 = 28 days.