Martha recorded the ratio of students and teachers at all the middle schools in the Timber Creek school district. The list is shown below.
20 students : 2 teachers
1 teacher per 15 students
1 teacher per 10 students
40 students : 4 teachers
5 teachers per 40 students
Part A: Identify the unit rates and ratios that are equivalent.
Part B: Give another unit rate or ratio that is not on the list but is equivalent to the ones from Part A. Explain why these are equivalent.
Part A:
To identify the unit rates and ratios that are equivalent, we need to find the ratios that represent the same value.
Let's list the ratios and unit rates:
1) 20 students : 2 teachers
2) 1 teacher per 15 students
3) 1 teacher per 10 students
4) 40 students : 4 teachers
5) 5 teachers per 40 students
To compare the ratios, we can simplify them by dividing both sides by the same number. Let's simplify each ratio:
1) 20 students : 2 teachers
Simplified ratio: 10 students : 1 teacher
2) 1 teacher per 15 students
Simplified ratio: 1 teacher : 15 students
3) 1 teacher per 10 students
Simplified ratio: 1 teacher : 10 students
4) 40 students : 4 teachers
Simplified ratio: 10 students : 1 teacher
5) 5 teachers per 40 students
Simplified ratio: 1 teacher : 8 students
Comparing the simplified ratios, we can see that ratio 1) and ratio 4) are equivalent, as they both represent the ratio of 10 students to 1 teacher. Therefore, the unit rates and ratios that are equivalent are:
- 10 students : 1 teacher
Part B:
A unit rate or ratio that is not on the list, but is equivalent to the ones from Part A, could be:
- 20 students : 2 teachers
This ratio is equivalent to ratio 1) from Part A. It represents the same value as the simplified ratio of 10 students : 1 teacher.
These ratios are equivalent because in both cases, there are 10 students per teacher. Whether it is expressed as 10 students : 1 teacher or 20 students : 2 teachers, the ratio remains the same and represents the same relationship between students and teachers.
Part A:
To find equivalent unit rates and ratios, we can simplify the given ratios.
Ratio 1: 20 students to 2 teachers
Simplified unit rate: 10 students per 1 teacher (dividing both sides by 2)
Ratio 2: 1 teacher per 15 students
Simplified unit rate: 1 teacher per 15 students (no further simplification is possible)
Ratio 3: 1 teacher per 10 students
Simplified unit rate: 1 teacher per 10 students (no further simplification is possible)
Ratio 4: 40 students to 4 teachers
Simplified unit rate: 10 students per 1 teacher (dividing both sides by 4)
Ratio 5: 5 teachers per 40 students
Simplified unit rate: 1 teacher per 8 students (dividing both sides by 5)
From the simplified unit rates, we can identify the following equivalent unit rates and ratios:
1) 10 students per 1 teacher (from Ratios 1 and 4)
2) 1 teacher per 15 students (from Ratio 2)
3) 1 teacher per 10 students (from Ratio 3)
4) 1 teacher per 8 students (from Ratio 5)
Part B:
To find another unit rate or ratio that is equivalent to the ones from Part A, we can use the concept of proportionality.
Let's say we have a ratio of x teachers per y students. To find the equivalent ratio, we can multiply or divide both the numerator and denominator by the same value.
For example, if we multiply both sides of Ratio 1 (20 students : 2 teachers) by 2, we get:
40 students : 4 teachers
This new ratio is equivalent to Ratio 4 (40 students : 4 teachers). The reason they are equivalent is that we multiplied both the numerator (number of students) and denominator (number of teachers) by the same value, which maintains the same ratio.
So, in general, to find another unit rate or ratio that is equivalent to the ones from Part A, we can multiply or divide both the numerator and denominator of any ratio by the same value. As long as the scaling factor is the same for both the numerator and denominator, the ratio remains equivalent.
20:2 = 40:4 since 20/2 = 40/4 = 10:1
Now check the others. As for equivalent ratios, just multiply both parts by any number. For example, 5:40 = 5*3:40*3 = 15:120