40 children were selling tickets for a charity. The ratio of the number of tickets each boy sold to the number of tickets each girl sold was 3:2.
a)If 106 tickets were sold altogether, how many boys were there?
b)What is the ratio of the number of boys to the number of girls who sold the tickets?(Give your answer in its simplest form)
There is no way that 106 can be divided into the ratio of 3:2.
However, 40 can be so divided, since 40 = 8*5, so 24:16 = 3:2
I suspect a typo.
To find the answers to these questions, we can set up a system of equations based on the given information.
Let's assume that there are b boys and g girls selling tickets. According to the given ratio, the number of tickets sold by each boy to each girl is 3:2. This means that for every 3 tickets sold by a boy, 2 tickets are sold by a girl.
a) If 106 tickets were sold altogether, we can determine the number of tickets sold by the boys and girls using the ratios.
The total number of tickets sold by the boys is 3 multiplied by the number of boys, which gives us 3b.
The total number of tickets sold by the girls is 2 multiplied by the number of girls, which gives us 2g.
So, we have the equation: 3b + 2g = 106.
b) To find the ratio of the number of boys to the number of girls who sold the tickets, we need to simplify the ratio using the total number of boys and girls.
To do this, we divide both sides of the equation by the greatest common divisor (GCD) of 3 and 2, which is 1.
Dividing each term of the equation by 1 doesn't change its value, so our equation becomes: (3b + 2g) / 1 = 106 / 1.
Since the GCD of 3 and 2 is 1, the ratio of the number of boys to the number of girls is simply the coefficient of b to the coefficient of g.
Thus, the ratio of boys to girls is 3:2.
Now we have a system of equations:
1) 3b + 2g = 106
2) Ratio of boys to girls = 3:2
To solve for the number of boys, we can use either substitution or elimination method. Let's use the elimination method.
From equation 1, we can isolate b by subtracting 2g from both sides: 3b = 106 - 2g.
Next, we can divide both sides by 3: b = (106 - 2g) / 3.
Now we substitute this value of b into equation 2 to find the number of boys. The ratio of boys to girls is 3:2, so:
(106 - 2g) / 3 = 3/2.
To solve this equation, we can cross-multiply:
2(106 - 2g) = 3(3).
212 - 4g = 9.
Simplifying, we get -4g = -203.
Dividing both sides by -4, we have:
g = 203/4 = 50.75 (rounded to two decimal places).
Since the number of children must be a whole number, we need to round this to the nearest whole number. As we cannot have a fractional number of girls, we cannot have 0.75 of a girl. Therefore, we round down to 50 girls.
Now we substitute this value of g back into equation 1 to find the number of boys:
3b + 2(50) = 106,
3b + 100 = 106,
3b = 6,
b = 6/3 = 2.
Therefore, there were 2 boys and 50 girls selling the tickets.
To summarize:
a) There were 2 boys selling tickets.
b) The ratio of boys to girls selling the tickets is 2:50, which simplifies to 1:25.