# The Problem

A gymnast received these scores from five judges
in the state competition. Her score for each event is found by
computing the average (mean) after the high and low scores are
thrown out. Find *a* if the vault was her worst event. Find
a if the vault was her best event.

### Floor: a, 9.3, 8.1, 8.9, 9.5

Bars: 7.6, 8.2, 8.5, 8.2, 8.9

Vault: 9.5, 8.9, 9.4, *a*, *a*

Beam: 8.4, 8.5, 8.4, *a*, 8.7

### (Source: Mathematics Teaching in the Middle School,
Mar-Apr 1996).

### Submit your idea for an investigation to InterMath

# The Solution

### In order to determine the values for a if the vault is to
be the worst or the best event, one must first determine the totals
for each event, independent of a. So we can use the following
values for each event:

### Floor: 35.8 + a

### Bars: 41.4

### Vault: 27.8 + 2a

### Beam: 34 + a

###

### First case: Vault as the worst event

### In order for the vault to be the worst event, we already know
that a must be less than the difference between the given bars
score and the base score for the vault. So we know that a must
be less than half of 41.4 - 27.8.

### 41.4-27.8=13.6

### 13.6/2 = 6.8

### Now we can pick a reasonable value for a, and decide if it
is the solution we are looking for, or if it needs to be adjusted.
Lets look at 6.5 first. If the gymnast recieved a score of 6.5
for each a, we would be given the following values:

### Floor: 42.3

### Bars: 41.4

### Vault: 40.8

### Beam: 40.5

### We can now see that the value we tried for a is too large,
since the vault recieved a higher score than the beam. So, lets
pick a smaller number. Using a = 6.1 we the following values are
produced:

### Floor: 41.9

### Bars: 41.4

### Vault: 40

### Beam 40.1

### In this instance we can see that 6.1 is a solution for a if
the vault is the worst event. This means that any score below
6.1 would produce the lowest score on the vault.

###

### Second case: Vault as the best event

### For this case we can use the basic information we found in
the first part of the problem. We know that for the vault to be
the best event, it must get a higher score than the bars, so we
know that a is going to be greater thn 6.8.

###

### Lets look at a=7.5 first, which gives us the following values
for each event:

### Floor: 43.3

### Bars: 41.4

### Vault: 42.8

### Beam: 41.5

### This value for a is too small, so we need to choose a larger
value for a. Lets look at a=8.1, which produces the following
values for each event:

### Floor: 43.9

### Bars: 41.4

### Vault: 44

### Floor: 43.9

### So we can see that in order for the vault to be the best event,
the gymnast must score at least 8.1 for each a value.