1. MATHS SOLUTION GAME – 3 rd of ESO –
PROPORTIONALIT Y
NOTE: The problems below are not in order of difficulty.
1. An old man bought a packet containing one hundred and twenty candies. He wants
to offer them to his four grandchildren. He decides to distribute candies in directly
proportional way, according to the children’s ages. They are aged respectively 4, 6, 6
and 8. How many candies does every child get?
2. Anna loves to play tennis, and she wants to buy a new racket. At the shopping
center, 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 the
speed of 100 km/h. The evening train is slower: it takes six hours and fifteen minutes
to 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 same
cake 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 g
How much (in grams) proteins and carbohydrates are there in an one-liter bottle of
the same juice?
6. The price of a box of mushrooms, in September, is 15€. In October it increases
by 6%, in November it decreases by 1.90€ and on the first day of December it
increases 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 doctor
tells him that his ideal weight would be 89 kilos. Thanks to this diet, he loses 3
kilograms in six weeks. If Robert’s uncle continues with the same diet, how many
days 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

2. printers, how much time will it take to print 1,400 pages?
9. A newspaper publishes an article containing some statistics about sports. It says
that 80 out of 1,500 persons like extreme sports. Express this quantity (that is, the
number of people who like extreme sports) in percentage. Write the result with an
approximation of at least two decimal digits.
10. Seven workers take seven hours to fix some holes in a street. How many time (in
minutes) 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 gets
less). My nephews are: Alex, aged 1; Bob, aged 3, and Carl, aged 7. How many
money does everyone of them get?
12. To build the Pyramid of Cheops, the Egyptian pharaoh hired 20,000 people. Its
construction took twenty years of work. Let’s imagine that Egypt’s pharaoh decided
to hire 10,000 workers more. In this case, how much time would it have taken to
complete the Pyramid? (Approximate the result to one decimal digit only)
VOCABULARY
Can = lata de bebida
Hire (to) = contratar, alquilar
Nephew = sobrino
(If there is a word you don’t know, just raise your hand and ask)

3. Solutions
P1. (OLD MAN, CANDIES) The 4, 6, 6, 8 years old grandchildren receive respectively
20,30,30,40 sweets.
P2. (TENNIS RACKET) The racket’s final price is €208.80
P3. (TRAINS) The train’s speed is 80 km/h
P4. (CAKE’S RECIPE) 3 eggs, 9 spoonfuls of sugar, 0.375 l of milk
P5. (PROTEINS) A one-litre bottle contains 0.5g of proteins and 19g of
carbohydrates.
P6. (BOX OF MUSHROOMS) In December a box costs €15.26
P7. (DIET) It will take him 168 days (=24 weeks) to reach his ideal weight
P8. (PRINTERS) It will take 10 minutes
P9. (SPORTS PERCENTAGE) The percentage of people who like extreme
sports 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: 60
P12. (PYRAMID) A little more than 13 years (exactly 13.33333333333….)

4. NOTES ON SOLUTIONS
Old 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 sweets
Box of mushrooms
September: 15€
October: 15.90€
November: 14€
December: 15.26€
Diet
It will take 168 days (=24 weeks) to reach his ideal (89kgs) weight
Printers (prop. Compuesta)
The more printers, the less minutes (inverse)
The more pages, the more minutes (direct)
Inv dir
7/5 times 600/1400 = 6/x  x=6 times 5 times 1400 / 4200  x = 10 minutes
Money to nephews (inversely proportional)
k/1 + k/3 + k/7 = 620
1 + 1/3 + 1/7 = (21+7+3)/21 = 31/21
k = 620 / (31/21) = 420
So : Nephew A: 420/1=420
Nephew B: 420/3 = 140
Nephew C: 420/7 = 60
Alex, Bob and Carl get respectively 420, 140 and 60 euros.
Tennis racket
The racket’s final price is €208.80
Pyramid (simple inverse prop)
A little more than 13 years (exactly 13.33333333333….)