Respuesta :
Answer
- There are 39 children in each group
- The number of groups that can be formed is 51 groups
SOLUTION
Problem Statement
The question tells us to form teams from 3 groups of children with 390, 234, and 1365 children respectively. We are to arrange them such that they have the same number of children a group. We are asked to find:
1. The number of children in each group.
2. The largest number of groups
Solution
Question 1:
- The number of children in each group must be the same number. That is, if there are x number of children in group 1, there must be x children in all the other groups as well.
- This demands that each group must contain the same number of children from each grade. That is, if we take x children from the third graders to fill up group 1, then, we must also take x children from the third graders for all the other groups.
This is true for all the other grades.
- This means that we are looking for a number common to all the 3 grades. If we can find the largest number common to the number all the grades, then, that number represents the total number of children in each group. Thus, if we find that y is the number common to 390, 234, and 1365, then, the number of children in a group must be y.
- This value of y is, by definition, the Highest common factor of the 3 numbers, 390, 234, and 1365. This is because the Highest Common Factor is simply the largest number that is common to a set of numbers.
- Thus, to find the number of children in each group, we simply find the Highest Common Factor of the numbers 390, 234, and 1365.
[tex]\begin{gathered} 390=2\times5\times3\times13 \\ 234=2\times3\times3\times13 \\ 1365=5\times7\times3\times13 \\ \\ \text{Notice that the number common to all 3 numbers is:} \\ 3\times13=39 \\ \\ \text{Thus, the HCF is 39} \end{gathered}[/tex]- This means that there are 39 children in each group
Question 2:
- If each group has 39 children from the 3 grades and each group takes an equal number of children from the grades, we can easily calculate the number groups can be formed.
[tex]\begin{gathered} \text{There are 39 children in each group.} \\ \text{But we are taking equal number of kids from each grade.} \\ \\ \text{Thus, the number of groups formed from the third graders are:} \\ \frac{390}{39}=10groups \\ \\ \text{The number of groups formed from the fourth graders are:} \\ \frac{234}{39}=6\text{groups} \\ \\ \text{The number of groups formed from the fifth graders are:} \\ \frac{1365}{39}=35\text{groups} \\ \\ \text{Thus, the total number of groups formed is:} \\ (10+6+35)=51groups \end{gathered}[/tex]- The number of groups that can be formed is 51 groups
Final Answer
- There are 39 children in each group
- The number of groups that can be formed is 51 groups
