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Number # Number
of Throws # of people
1 # 7
2 # 6
3 # 6
4 # 4
5 # 2

in a certain game , each person threw beanbag at a target until the person missed the target .The table shows the results for 25 people who played the game. for example , 4 people hit the target on their first 3 throws and missed on their 4th throw. Based on the information in the table, which of the following must be true?

I.More than half the people hit the target on their first throw.
II. For all of the throws attempted, more hit the target than missed the target.
III. o one hit the target 5 times.

A)I only.
B)II only.
C)I and III only.
D)II and III only.
E)I , II, and III .

I. The number of times the target is hit is n-1 where n is the number of throws (because a person stops throwing when he/she misses the target) so a new chart of the number of times the target was actually hit vs. the number of people (derived from this chart would be)

0 # 7
1 # 6
2 # 6
3 # 4
4 # 2

Clearly, 7 is less than half of 25, so I is true.

That eliminates choices B and D and choices C and E both included III. so let's test III.

If "o one hit the target 5 times" means "only one hit the target 5 times" than III is false because the most times anyone hit the target is 5-1 = 4 times = A,

If "o one hit the target 5 times" means "0 hit the target 5 times" than III is true.

If III is true, than that leaves choices C and E. Testing II, the number of throws attempted is 7(1)+6(2)+6(3)+4(4)+2(5)=7+12+18+16+10= 63. By the new chart above, the sum of the number of times the target was hit is 7(0)+6(1)+6(2)+4(3)+2(4)= 0+6+12+12+8= 38. 38 is more than half of 63, so II is true. Since there is no "I and II" choice, E must be the correct answer.

It's quite easy to see that
I. more than half of people hit the target on their first throw -> I is true,
and
II. there are exactly 25 misses and many more hits -> II is true.

E is the correct answer.

Just as a bonus: no one hit the target on their 5-th throw -> III is true too.

## Replies to: Math (sat) problem!

I. The number of times the target is hit is n-1 where n is the number of throws (because a person stops throwing when he/she misses the target) so a new chart of the number of times the target was actually hit vs. the number of people (derived from this chart would be)

0 # 7

1 # 6

2 # 6

3 # 4

4 # 2

Clearly, 7 is less than half of 25, so I is true.

That eliminates choices B and D and choices C and E both included III. so let's test III.

If "o one hit the target 5 times" means "only one hit the target 5 times" than III is false because the most times anyone hit the target is 5-1 = 4 times = A,

If "o one hit the target 5 times" means "0 hit the target 5 times" than III is true.

If III is true, than that leaves choices C and E. Testing II, the number of throws attempted is 7(1)+6(2)+6(3)+4(4)+2(5)=7+12+18+16+10= 63. By the new chart above, the sum of the number of times the target was hit is 7(0)+6(1)+6(2)+4(3)+2(4)= 0+6+12+12+8= 38. 38 is more than half of 63, so II is true. Since there is no "I and II" choice, E must be the correct answer.

# of misses equals # of people eliminated

and

# of hits equals # of people left.

Let's expand the table.

Throw #...throws...misses...hits

1...25...7...18

2...18...6...12

3...12...6...6

4...6...4...2

5...2...2...0

I.

18 people - more than half - hit the target on their first throws.

I is true.

II.

Total misses 7+6+6+4+2=25

Total hits 18+12+6+2+0=38

More people hit the target than missed it.

II is true.

Out of 5 answers only E includes both I and II.

E is the answer.

The results of the game may be represented by the following diagram:

1 ..................!!!!!!!

2 ............!!!!!!

3 ......!!!!!!

4 ..!!!!

5 !!

It's quite easy to see that

I. more than half of people hit the target on their first throw -> I is true,

and

II. there are exactly 25 misses and many more hits -> II is true.

E is the correct answer.

Just as a bonus: no one hit the target on their 5-th throw -> III is true too.