Math  /  Data & Statistics

QuestionProblem \# 2: (Finding the missing value) Lucy took three tests. If her median score was 82 , her mean score was 87 , and the range was 17 , what were her three test scores? Use two different strategies (the balance method and the equation).
Problem \# 3: (Finding average speed) For the first 3 hours, we drove 72 miles per hour on average. For the next 1 and a half hour, we drove 55 miles per hour. What was our average speed?
Problem \# 4: (Finding overall average) On a certain exam, Mr. Smith graded 10 papers of section A and found that the mean for this group of students was 70 . Then he graded 20 papers of section B and found that the mean for the second group was 80 . Then he graded 25 papers of section C and found that the mean for the third group was 82 . What was the overall mean of the combined group of students?

Studdy Solution

STEP 1

1. Lucy took three tests, and we need to find the scores.
2. The median score is 82.
3. The mean score is 87.
4. The range of the scores is 17.

STEP 2

1. Use the balance method to find the scores.
2. Use the equation method to find the scores.

#### Balance Method
3. 1. Calculate the total distance traveled.
2. Calculate the total time traveled.
4. Find the average speed.

STEP_1: High_Level_Step: 1 Calculate the distance for the first segment:
Distance1=72×3=216 miles \text{Distance}_1 = 72 \times 3 = 216 \text{ miles}
High_Level_Step_Completed: FALSE
STEP_2: High_Level_Step: 1 Calculate the distance for the second segment:
Distance2=55×1.5=82.5 miles \text{Distance}_2 = 55 \times 1.5 = 82.5 \text{ miles}
High_Level_Step_Completed: TRUE
STEP_3: High_Level_Step: 2 Calculate the total distance:
Total Distance=216+82.5=298.5 miles \text{Total Distance} = 216 + 82.5 = 298.5 \text{ miles}
High_Level_Step_Completed: TRUE
STEP_4: High_Level_Step: 2 Calculate the total time:
Total Time=3+1.5=4.5 hours \text{Total Time} = 3 + 1.5 = 4.5 \text{ hours}
High_Level_Step_Completed: TRUE

STEP 3

Assume the scores are abc a \leq b \leq c . Given that the median score is 82, we have:
b=82 b = 82

STEP 4

The mean score is 87, so:
a+b+c3=87 \frac{a + b + c}{3} = 87
Substitute b=82 b = 82 :
a+82+c3=87 \frac{a + 82 + c}{3} = 87
a+82+c=261 a + 82 + c = 261
a+c=179 a + c = 179

STEP 5

The range is 17, so:
ca=17 c - a = 17

STEP 6

Solve the system of equations:
a+c=179 a + c = 179 ca=17 c - a = 17
Add the equations:
2c=196 2c = 196 c=98 c = 98
Substitute back to find a a :
a+98=179 a + 98 = 179 a=81 a = 81
Thus, the scores are 81,82,98 81, 82, 98 .
#### Equation Method
STEP_1: Assume the scores are abc a \leq b \leq c . Given that the median score is 82, we have:
b=82 b = 82
STEP_2: The mean score is 87, so:
a+b+c3=87 \frac{a + b + c}{3} = 87
Substitute b=82 b = 82 :
a+82+c3=87 \frac{a + 82 + c}{3} = 87
a+c=179 a + c = 179
STEP_3: The range is 17, so:
ca=17 c - a = 17
STEP_4: Solve the system of equations:
a+c=179 a + c = 179 ca=17 c - a = 17
Add the equations:
2c=196 2c = 196 c=98 c = 98
Substitute back to find a a :
a+98=179 a + 98 = 179 a=81 a = 81
Thus, the scores are 81,82,98 81, 82, 98 .
The scores are: 81,82,98 81, 82, 98 .
### Problem #3: Finding Average Speed
_ASSUMPTIONS_:
1. The first segment of the trip was 3 hours at 72 miles per hour.
2. The second segment was 1.5 hours at 55 miles per hour.

STEP 7

Calculate the average speed:
Average Speed=Total DistanceTotal Time=298.54.5=66.33 miles per hour \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{298.5}{4.5} = 66.33 \text{ miles per hour}
The average speed is 66.33 66.33 miles per hour.
### Problem #4: Finding Overall Average
_ASSUMPTIONS_:
1. Section A has 10 papers with a mean of 70.
2. Section B has 20 papers with a mean of 80.
3. Section C has 25 papers with a mean of 82.

_HIGH_LEVEL_APPROACH_:
1. Calculate the total score for each section.
2. Calculate the total number of papers.
3. Find the overall mean.

STEP_1: Calculate the total score for section A:
Total ScoreA=10×70=700 \text{Total Score}_A = 10 \times 70 = 700
STEP_2: Calculate the total score for section B:
Total ScoreB=20×80=1600 \text{Total Score}_B = 20 \times 80 = 1600
STEP_3: Calculate the total score for section C:
Total ScoreC=25×82=2050 \text{Total Score}_C = 25 \times 82 = 2050
STEP_4: Calculate the total number of papers:
Total Papers=10+20+25=55 \text{Total Papers} = 10 + 20 + 25 = 55
STEP_5: Calculate the overall mean:
Overall Mean=Total ScoreA+Total ScoreB+Total ScoreCTotal Papers=700+1600+205055=78.18 \text{Overall Mean} = \frac{\text{Total Score}_A + \text{Total Score}_B + \text{Total Score}_C}{\text{Total Papers}} = \frac{700 + 1600 + 2050}{55} = 78.18
The overall mean is 78.18 78.18 .

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