Two boxes contain pictures of hockey and basketball players.
In one box, the ratio of hockey players to basketball players is 4:3.
In the other box, the ratio is 3:2.
The boxes contain the same number of pictures.
a) What could the total number of pictures be?
b) Which box contains more pictures of hockey players?
The minimum number in one box would be 4 + 3 = 7, and in the other it would be 3 + 2 = 5
Now, find the least common multiple of those two, which is 7 ∙ 5 = 35
So in each box, there are 35 pictures.
H = hockey players
B = basketball players
In one box, the ratio of hockey players to basketball players is 4:3 mean:
H / B = 4 / 3
Multiply both sides by B
H = 4 B / 3
H + B = 35
4 B / 3 + B = 35
4 B / 3 + 3 B / 3 = 35
7 B / 3 = 35
Multiply both sides by 3
7 B = 105
Divide both sides by 7
B = 105 / 7
B = 15
H = 35 - 15
H = 20
In the first box, there would then be 20 hockey players and 15 basketball players.
In the other box, the ratio is 3:2 mean:
H / B = 3 / 2
Multiply both sides by B
H = 3 B / 2
H + B = 35
3 B / 2 + B = 35
3 B / 2 + 2 B / 2 = 35
5 B / 2 = 35
Multiply both sides by 2
5 B = 70
Divide both sides by 5
B = 70 / 5
B = 14
H = 35 - 14
H = 21
In the other box, there would then be 21 hockey players and 14 basketball players.
box1:
Hockey cards --- 4x
basketball cards --- 3x
box2
hockey cards --- 3y
basketball cards -- 2y
7x = 5y
y = 7x/5
so x must be a multiple of 5 in order for y to be a whole number
x -- y
5 7
10 14
15 21
....
notice that as x increases by 5 , the y increases by 7
and the slope of y = 7x/5 is 7/5 ?? , mmmhhhh
can't it be 7, 10 too?
You asked: "What could the total number of pictures be?"
Bosnian found the smallest such total, but there there are an infinite number of solutions
e.g. take one of my x and y pairs
x = 15, y = 21
so in Box1, there would be 60 hockey cards and 45 basketball cards
notice 60:45 = 4:3
the total would be 105 cards in box1
in box2 there would be 63 hockey cards and 42 basketball cards
for a total of 63+42 = 105 , the same as box1
You said: "can't it be 7, 10 too?"
The answer is no
First of all what are the 7 and 10 supposed to be??
7:10 ≠ 4:3 nor 3:2
a) Well, let's start by assuming that the total number of pictures in both boxes is "x". In the first box, the ratio of hockey players to basketball players is 4:3. So, out of the "x" pictures in this box, we can say that 4/7 of them are hockey players, and 3/7 are basketball players. In the second box, the ratio is 3:2. So, out of the "x" pictures in this box, we can say that 3/5 of them are hockey players, and 2/5 are basketball players. Since both boxes contain the same number of pictures, we can equate these ratios: 4/7x = 3/5x. Cross-multiplying, we get 20x = 21x. Dividing both sides by x, we get 20 = 21. Oh no, this math doesn't make any sense! It looks like there was an error along the way. I'm afraid I can't give you a specific answer for the total number of pictures. But hey, at least we had a good laugh with this mathematical mishap!
b) Since we couldn't determine the total number of pictures, it's impossible to say for sure which box contains more pictures of hockey players. But hey, let's imagine a scenario where one box contains only hockey pictures, and the other box contains pictures of clowns instead. In this case, the box with the hockey pictures would definitely have more hockey players. But wait, we're talking about basketball and hockey players, not clowns! Silly me, mixing up my scenarios. But hey, in the absence of concrete information, let's just say both boxes have an equal number of hockey players. That way, no box feels left out, right?
To find the total number of pictures in the boxes and determine which box contains more hockey players, we need to set up equations based on the given ratios.
Let's assume the total number of pictures in each box is x.
In the first box, the ratio of hockey players to basketball players is 4:3. This means that in the first box, the number of hockey players can be represented by (4x/7), and the number of basketball players can be represented by (3x/7).
In the second box, the ratio of hockey players to basketball players is 3:2. This means that in the second box, the number of hockey players can be represented by (3x/5), and the number of basketball players can be represented by (2x/5).
Now, since both boxes contain the same number of pictures, the total number of pictures in the first box should be equal to the total number of pictures in the second box.
So, (4x/7) + (3x/7) = (3x/5) + (2x/5)
Now we can solve this equation to find the value of x, which represents the total number of pictures in each box.
Multiplying both sides of the equation by 35 (the least common multiple of 7 and 5) to eliminate the denominators, we get:
20x + 15x = 21x + 14x
35x = 35x
Since both sides of the equation are equal, we can see that the value of x can be any positive number.
a) Therefore, the total number of pictures in the boxes could be any positive number.
b) To determine which box contains more pictures of hockey players, we can compare the expressions for the number of hockey players in both boxes.
In the first box, the number of hockey players is (4x/7), and in the second box, the number of hockey players is (3x/5).
Since we know that x can be any positive number, we cannot determine which box contains more pictures of hockey players without knowing the actual value of x.