A group of students were asked some questions about how they travelled to school. 1/2 of them said they walked. 1/3 of them said they travelled by bus. In the group there were more than 20 students and less than 30.
a) How many students were there in the group
b) what fraction of students came by car? give answer in simplest form.
The answer will be an integer that is evenly divisible by both 2 and 3. There is only one such number between 19 and 29, inclusive. Try them all yourself to see which it is.
To find the number of students in the group, we can start by considering the fractions mentioned in the question.
It states that 1/2 of the students said they walked and 1/3 of the students said they traveled by bus. We can express these fractions as ratios:
1/2 of the students = (1/2) * N
1/3 of the students = (1/3) * N
Where N represents the total number of students in the group.
Now, we need to find the value of N that satisfies the given conditions.
The question mentions that there were more than 20 students and less than 30 students. This narrows down our range of possible values for N to the interval [21, 29].
We can try each value in this interval to see which value of N gives us a valid fraction for the number of students who walked and traveled by bus.
Let's start by trying N = 21:
(1/2) * 21 = 10.5
(1/3) * 21 = 7
Since both fractions are not whole numbers, this value of N doesn't work.
Now, let's try N = 22:
(1/2) * 22 = 11
(1/3) * 22 = 7.33
Again, the fractions are not whole numbers, so 22 doesn't work.
Continuing this process, we can try each number in the range [21, 29] until we find the value of N that satisfies the conditions.
Let's try N = 23:
(1/2) * 23 = 11.5
(1/3) * 23 = 7.67
Once again, the fractions are not whole numbers, so 23 doesn't work.
We continue this process until we reach N = 26:
(1/2) * 26 = 13
(1/3) * 26 = 8.67
Here, the fraction (1/2) * N = 13 is a whole number, but (1/3) * N = 8.67 is not. Therefore, 26 is not the correct value either.
Finally, we reach N = 27:
(1/2) * 27 = 13.5
(1/3) * 27 = 9
Now, both fractions are whole numbers, so N = 27 satisfies the conditions.
a) Therefore, there were 27 students in the group.
To find the fraction of students who came by car, we need to subtract the sum of the fractions for walking and traveling by bus from 1 (since these three options represent all the possibilities).
Let's calculate the fraction of students who came by car:
Fraction of students who came by car = 1 - (Fraction of students who walked + Fraction of students who traveled by bus)
Fraction of students who came by car = 1 - (1/2 + 1/3)
To add these fractions, we need a common denominator, which is 6:
Fraction of students who came by car = 1 - (3/6 + 2/6)
Fraction of students who came by car = 1 - (5/6)
Simplifying this expression:
Fraction of students who came by car = 1 - 5/6
= (6/6) - (5/6)
= 1/6
b) Therefore, the fraction of students who came by car is 1/6.