Difference between revisions of "2018 AMC 10A Problems/Problem 14"
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Notice that the terms being added on to the top and bottom are in the ratio <math>\frac{1}{16}</math> with each other, so they must pull the ratio down from 81 very slightly. (In the same way that a new test score lower than your current cumulative grade always must pull that grade downward.) Answer: <math>\boxed{(A)}</math>. | Notice that the terms being added on to the top and bottom are in the ratio <math>\frac{1}{16}</math> with each other, so they must pull the ratio down from 81 very slightly. (In the same way that a new test score lower than your current cumulative grade always must pull that grade downward.) Answer: <math>\boxed{(A)}</math>. | ||
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==Solution 7== | ==Solution 7== |
Revision as of 17:24, 5 August 2022
Contents
- 1 Problem
- 2 Solution 1
- 3 Solution 2
- 4 Solution 3
- 5 Solution 4
- 6 Solution 5
- 7 Solution 6 (eyeball it)
- 8 Solution 7
- 9 Solution 8
- 10 Solution 9
- 11 Solution 10 (Using the answer choices)
- 12 Explanation for why 80 is indeed the floor
- 13 Solution 11
- 14 Solution 12
- 15 Solution 13 (slightly similar to Solution 7)
- 16 See Also
Problem
What is the greatest integer less than or equal to
Solution 1
We write
Hence we see that our number is a weighted average of 81 and 16, extremely heavily weighted toward 81. Hence the number is ever so slightly less than 81, so the answer is
.
Solution 2
Let's set this value equal to . We can write
Multiplying by
on both sides, we get
Now let's take a look at the answer choices. We notice that
, choice
, can be written as
. Plugging this into our equation above, we get
The right side is larger than the left side because
This means that our original value,
, must be less than
. The only answer that is less than
is
so our answer is
.
~Nivek
Solution 3
We can ignore the 1's on the end because they won't really affect the fraction. So, the answer is very very very close but less than the new fraction.
So, our final answer is very close but not quite 81, and therefore the greatest integer less than the number is
Solution 4
Let and
. Then our fraction can be written as
.
Notice that
.
So ,
.
And our only answer choice less than 81 is
(RegularHexagon)
Solution 5
Let . Multiply both sides by
, and expand. Rearranging the terms, we get
. The left side is decreasing, and it is negative when
. This means that the answer must be less than
; therefore the answer is
.
Solution 6 (eyeball it)
A faster solution. Recognize that for exponents of this size will be enormously greater than
, so the terms involving
will actually have very little effect on the quotient. Now we know the answer will be very close to
.
Notice that the terms being added on to the top and bottom are in the ratio with each other, so they must pull the ratio down from 81 very slightly. (In the same way that a new test score lower than your current cumulative grade always must pull that grade downward.) Answer:
.
Solution 7
Notice how can be rewritten as
. Note that
, so the greatest integer less than or equal to
is
or
~blitzkrieg21
Solution 8
For positive , if
then
. Let
. Then
. So answer is less than 81, which leaves only one choice, 80.
- Note that the algebra here is synonymous to the explanation given in Solution 6. This is the algebraic reason to the logic of if you get a test score with a lower percentage than your average (no matter how many points/percentage of your total grade it was worth), it will pull your overall grade down.
~ ccx09
Solution 9
Try long division, and notice putting as the denominator is too big and putting
is too small. So we know that the answer is between
and
, yielding
as our answer.
Solution 10 (Using the answer choices)
Solution 10.1
We can compare the given value to each of our answer choices. We already know that it is greater than because otherwise there would have been a smaller answer, so we move onto
. We get:
Cross multiply to get:
Cancel out and divide by
to get
. We know that
, which means the expression is less than
so the answer is
.
Solution 10.2
We know this will be between 16 and 81 because and
.
is the only option choice in this range.
Explanation for why 80 is indeed the floor
We need . Since
, this translates to
We now prove that
for all positive integers
.
Clearly,
. Assume
where
. Then
. But since
, we have that
. By induction (and
is trivial), the claim is proven.
Thus, . Writing this proof backwards and dividing both sides of the initial equation by
yields
.
Solution 11
We know that in this problem, times some number is equal to
. Multiplying answer
or 81 to
gives us
. We know that
is greater than
, so that means
or 81 is too big. That leaves us with only one solution:
~ Terribleteeth
Solution 12
Dividing by in both numerator and denominator, this fraction can be rewritten as
Notice that the
and the
will be so insignificant compared to a number such as
and that thereby the fraction will be ever so slightly less than
. Thereby, we see that the answer is
~ Professor-Mom [& wow there are now 12 sols to this problem :o :o :o this problem xDD]
Solution 13 (slightly similar to Solution 7)
If you multiply by
(to get the exponent up to 100), you'll get
. Thus, in the numerator, if you add and subtract by
and
, you'll get
. You can then take out out the first number to get
. This can then be written as
, factoring out the 16 and splitting the fraction will give you
, giving you
. While you can roughly say that
you can also notice that the only answer choice less than 81 is 80, thus the answer is
~ Zeeshan12 [Now there's 13 :) ]
See Also
2018 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.