2005 CEMC Gauss (Grade 7) Problems/Problem 23
Contents
[hide]Problem
Using an equal-armed balance, if balances
and
balances
, which of the following would not balance
?
Solution 1
If balance
, then
would balance the equivalent of
.
Similarly,
would balance the equivalent of
.
If we take each of the answers and convert them to an equivalent number of
, we would have:
Therefore, and
do not balance the required. The answer is
.
Solution 2
Since balance
, then
would balance
.
Therefore,
would balance
, so since
balance
, then
would balance
, or
would balance
.
We can now express every combination in terms of only.
,
, and
equals
.
and
equals
.
and
equals
.
and
equals
.
and
equals
.
Therefore, since ,
, and
equals
, then it is
and
which will not balance with this combination. Thus, the answer is
.
Solution 3
We try assigning weights to the different shapes.
Since balance
, assume that each
weighs
and each
weighs
.
Therefore, since
balance
, which weigh
combined, then each
weighs
.
We then look at each of the remaining combinations.
,
, and
weigh
.
and
weigh
.
and
weigh
.
and
weigh
.
and
weigh
.
Therefore, it is the combination of and
which will not balance the other combinations. The answer is
.
See Also
2005 CEMC Gauss (Grade 7) (Problems • Answer Key • Resources) | ||
Preceded by Problem 22 |
Followed by Problem 24 | |
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CEMC Gauss (Grade 7) |