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MATHS SOLUTION GAME – 3 rd of ESO –PROPORTIONALIT YNOTE: The problems below are not in order of difficulty.1. An old man bought a packet containing one hundred and twenty candies. He wantsto offer them to his four grandchildren. He decides to distribute candies in directlyproportional way, according to the children’s ages. They are aged respectively 4, 6, 6and 8. How many candies does every child get?2. Anna loves to play tennis, and she wants to buy a new racket. At the shoppingcenter, they sell one for €180 plus a 16% in taxes. Which is the article’s final price?3. To get to Almeria from Seville, the morning train takes five hours, travelling at thespeed of 100 km/h. The evening train is slower: it takes six hours and fifteen minutesto cover the same route. What is the speed (in kilometers per hour) of the Seville-Almeria evening train?4. A recipe for a cake says: “Ingredients (for 4 persons): -Two eggs -Six spoonfuls of sugar -A quarter litre of milk…”Calculate the quantity of ingredients (eggs, sugar, milk) needed to prepare the samecake for six persons.5. A can contains 400ml of fruit juice. Its label reads: Nutritional content of this can Proteins 0.2 g Carbohydrates 7.6 gHow much (in grams) proteins and carbohydrates are there in an one-liter bottle ofthe same juice?6. The price of a box of mushrooms, in September, is 15€. In October it increasesby 6%, in November it decreases by 1.90€ and on the first day of December itincreases again by 9%. How much does a box of mushrooms cost in December?7. Robert’s uncle weighs 101 kilos, so he decides he has to go on a diet. The doctortells him that his ideal weight would be 89 kilos. Thanks to this diet, he loses 3kilograms in six weeks. If Robert’s uncle continues with the same diet, how manydays will take him to reach his ideal weight?8. Five printers take six minutes to print six hundred pages. If we decide to use seven
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printers, how much time will it take to print 1,400 pages?9. A newspaper publishes an article containing some statistics about sports. It saysthat 80 out of 1,500 persons like extreme sports. Express this quantity (that is, thenumber of people who like extreme sports) in percentage. Write the result with anapproximation of at least two decimal digits.10. Seven workers take seven hours to fix some holes in a street. How many time (inminutes) does it take to five workers to do the same job?11. I want to distribute €620 to my three nephews in inversely proportional parts,according to their ages (so that the youngest gets more money and the oldest getsless). My nephews are: Alex, aged 1; Bob, aged 3, and Carl, aged 7. How manymoney does everyone of them get?12. To build the Pyramid of Cheops, the Egyptian pharaoh hired 20,000 people. Itsconstruction took twenty years of work. Let’s imagine that Egypt’s pharaoh decidedto hire 10,000 workers more. In this case, how much time would it have taken tocomplete the Pyramid? (Approximate the result to one decimal digit only)VOCABULARYCan = lata de bebidaHire (to) = contratar, alquilarNephew = sobrino(If there is a word you don’t know, just raise your hand and ask)
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SolutionsP1. (OLD MAN, CANDIES) The 4, 6, 6, 8 years old grandchildren receive respectively20,30,30,40 sweets.P2. (TENNIS RACKET) The racket’s final price is €208.80P3. (TRAINS) The train’s speed is 80 km/hP4. (CAKE’S RECIPE) 3 eggs, 9 spoonfuls of sugar, 0.375 l of milkP5. (PROTEINS) A one-litre bottle contains 0.5g of proteins and 19g ofcarbohydrates.P6. (BOX OF MUSHROOMS) In December a box costs €15.26P7. (DIET) It will take him 168 days (=24 weeks) to reach his ideal weightP8. (PRINTERS) It will take 10 minutesP9. (SPORTS PERCENTAGE) The percentage of people who like extremesports is 5.33%P10. (FIXING A HOLE) It will take 588 minutes (a little less than 10 h)P11. (MONEY TO NEPHEWS) Nephew A: 420 Nephew B: 140 Nephew C: 60P12. (PYRAMID) A little more than 13 years (exactly 13.33333333333….)
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NOTES ON SOLUTIONSOld man , candies:The proportionality constant N is 120/(4+6+6+8) = 5.The 4-years old grandchild receives 4x5=20 sweets.The 6-years old children receive 6x5=30 sweets each.The 8-years old child receives 8x5=40 sweetsBox of mushroomsSeptember: 15€October: 15.90€November: 14€December: 15.26€DietIt will take 168 days (=24 weeks) to reach his ideal (89kgs) weightPrinters (prop. Compuesta)The more printers, the less minutes (inverse)The more pages, the more minutes (direct) Inv dir7/5 times 600/1400 = 6/x x=6 times 5 times 1400 / 4200 x = 10 minutesMoney to nephews (inversely proportional)k/1 + k/3 + k/7 = 6201 + 1/3 + 1/7 = (21+7+3)/21 = 31/21k = 620 / (31/21) = 420So : Nephew A: 420/1=420 Nephew B: 420/3 = 140 Nephew C: 420/7 = 60Alex, Bob and Carl get respectively 420, 140 and 60 euros.Tennis racketThe racket’s final price is €208.80Pyramid (simple inverse prop)A little more than 13 years (exactly 13.33333333333….)
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