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Difference between revisions of "2018 AMC 10A Problems/Problem 14"

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==Solution 3==
 
==Solution 3==
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<math>\frac{3^{100}+2^{100}}{3^{96}+2^{96}}=\frac{2^{96}(\frac{3^{100}}{2^{96}})+2^{96}(2^{4})}{2^{96}(\frac{3}{2})^{96}+2^{96}(1)}=\frac{\frac{3^{100}}{2^{96}}+2^{4}}{(\frac{3}{2})^{96}+1}=\frac{\frac{3^{100}}{2^{100}}*2^{4}+2^{4}}{(\frac{3}{2})^{96}+1}=\frac{2^{4}(\frac{3^{100}}{2^{100}}+1)}{(\frac{3}{2})^{96}+1}</math>.
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<cmath>\frac{3^{100}+2^{100}}{3^{96}+2^{96}}=\frac{2^{96}\left(\frac{3^{100}}{2^{96}}\right)+2^{96}\left(2^{4}\right)}{2^{96}\left(\frac{3}{2}\right)^{96}+2^{96}(1)}=\frac{\frac{3^{100}}{2^{96}}+2^{4}}{\left(\frac{3}{2}\right)^{96}+1}=\frac{\frac{3^{100}}{2^{100}}\cdot2^{4}+2^{4}}{\left(\frac{3}{2}\right)^{96}+1}=\frac{2^{4}\left(\frac{3^{100}}{2^{100}}+1\right)}{\left(\frac{3}{2}\right)^{96}+1}.</cmath>
  
 
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.
 
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.
  
−
<math>\frac{2^{4}(\frac{3^{100}}{2^{100}}+1)}{(\frac{3}{2})^{96}+1}<\frac{2^{4}(\frac{3^{100}}{2^{100}})}{(\frac{3}{2})^{96}}</math>
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<cmath>\frac{2^{4}\left(\frac{3^{100}}{2^{100}}+1\right)}{\left(\frac{3}{2}\right)^{96}+1}<\frac{2^{4}\left(\frac{3^{100}}{2^{100}}\right)}{\left(\frac{3}{2}\right)^{96}},</cmath>
  
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<math>\frac{2^{4}(\frac{3^{100}}{2^{100}})}{(\frac{3}{2})^{96}}=\frac{3^{4}}{2^{4}}*2^{4}=3^{4}=81</math>
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<cmath>\frac{2^{4}\left(\frac{3^{100}}{2^{100}}\right)}{\left(\frac{3}{2}\right)^{96}}=\frac{3^{4}}{2^{4}}*2^{4}=3^{4}=81.</cmath>
  
 
So, our final answer is very close but not quite 81, and therefore the greatest integer less than the number is <math>\boxed{(A) 80}</math>
 
So, our final answer is very close but not quite 81, and therefore the greatest integer less than the number is <math>\boxed{(A) 80}</math>
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A faster solution. Recognize that for exponents of this size <math>3^{n}</math> will be enormously greater than <math>2^{n}</math>, so the terms involving <math>2</math> will actually have very little effect on the quotient. Now we know the answer will be very close to <math>81</math>.
 
A faster solution. Recognize that for exponents of this size <math>3^{n}</math> will be enormously greater than <math>2^{n}</math>, so the terms involving <math>2</math> will actually have very little effect on the quotient. Now we know the answer will be very close to <math>81</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>.
+
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{\text{\textbf{(A)}}}</math>.
  
 
==Solution 7==
 
==Solution 7==
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<cmath>3^{96} > 4\cdot 2^{100} = 64\cdot 2^{96}.</cmath>
 
<cmath>3^{96} > 4\cdot 2^{100} = 64\cdot 2^{96}.</cmath>
 
We now prove that <math>(3/2)^k > k</math> for all positive integers <math>k</math>.
 
We now prove that <math>(3/2)^k > k</math> for all positive integers <math>k</math>.
−
Clearly, <math>(3/2)^2 = 2.25 > 2</math>. Assume <math>(3/2)^k > k</math> where <math>k>=2</math>. Then <math>\left(\frac{3}{2}\right)^{k+1} > \frac{3k}{2} = k + \frac{k}{2}</math>. But since <math>k/2 >= 1</math>, we have that <math>(3/2)^{k+1} > k+1</math>. By induction (and <math>k=1</math> is trivial), the claim is proven.
+
Clearly, <math>(3/2)^2 = 2.25 > 2</math>. Assume <math>(3/2)^k > k</math> where <math>k\ge 2</math>. Then <math>\left(\frac{3}{2}\right)^{k+1} > \frac{3k}{2} = k + \frac{k}{2}</math>. But since <math>k/2 \ge 1</math>, we have that <math>(3/2)^{k+1} > k+1</math>. By induction (and <math>k=1</math> is trivial), the claim is proven.
  
−
Thus, <math>\left(\frac{3}{2}\right)^{96} > 96 > 64</math>. Writing this backwards and dividing both sides of the initial equation by <math>80</math> yields <math>\frac{3^{100}+2^{100}}{3^{96}+2^{96}} > 80</math>.
+
Thus, <math>\left(\frac{3}{2}\right)^{96} > 96 > 64</math>. Writing this proof backwards and dividing both sides of the initial equation by <math>80</math> yields <math>80 < \frac{3^{100}+2^{100}}{3^{96}+2^{96}} < 81</math>.
  
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==Solution 11==
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We know that in this problem, <math>3^{96}+2^{96}</math> times some number is equal to <math>3^{100}+2^{100}</math>. Multiplying answer <math>\boxed{\textbf{(B)}}</math> or 81 to <math>3^{96}+2^{96}</math> gives us <math>3^{100}+2^{96}\cdot3^4</math>. We know that <math>3^4\cdot2^{96}</math> is greater than <math>2^{100}</math>, so that means <math>\boxed{\textbf{(B)}}</math> or 81 is too big. That leaves us with only one solution: <math>80=\boxed{\textbf{(A) } 80}.</math>
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~ Terribleteeth
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==Solution 12==
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Dividing by <math>2^{96}</math> in both numerator and denominator, this fraction can be rewritten as <cmath>\frac{81 \times (1.5)^{96} + 16}{(1.5)^{96} + 1}.</cmath> Notice that the <math>+1</math> and the <math>+16</math> will be so insignificant compared to a number such as <math>(1.5)^{96},</math> and that thereby the fraction will be ever so slightly less than <math>81</math>. Thereby, we see that the answer is <math>\boxed{\text{(A)} \ 80}.</math>
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~ Professor-Mom [& wow there are now 12 sols to this problem :o :o :o this problem xDD]
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==Solution 13 (slightly similar to Solution 7)==
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If you multiply <math>(3^{96} + 2^{96})</math> by <math>(3^{4} + 2^{4})</math> (to get the exponent up to 100), you'll get <math>(3^{100} + 2^{100}) + 3^{96} \cdot 2^{4} + 2^{96} \cdot 3^{4}</math>. Thus, in the numerator, if you add and subtract by <math>3^{96} \cdot 2^{4}</math> and <math>2^{96} \cdot 3^{4}</math>, you'll get <math>\frac{(3^{4} + 2^{4})(3^{100} + 2^{100}) - 3^{96} \cdot 2^{4} - 2^{96} \cdot 3^{4}}{3^{96}+2^{96}}</math>. You can then take out out the first number to get <math>3^{4} + 2^{4} - \frac{3^{96} \cdot 2^{4} + 2^{96} \cdot 3^{4}}{3^{96}+2^{96}}</math>. This can then be written as <math>87 - \frac{16 \cdot 3^{96} + 16 \cdot 2^{96} + 75 \cdot 2^{96}}{3^{96}+2^{96}}</math>, factoring out the 16 and splitting the fraction will give you <math>87 - 16 - \frac{65 \cdot 2^{96}}{3^{96}+2^{96}}</math>, giving you <math>81 - \frac{65 \cdot 2^{96}}{3^{96}+2^{96}}</math>. While you can roughly say that <math>\frac{65 \cdot 2^{96}}{3^{96}+2^{96}} < 1</math> you can also notice that the only answer choice less than 81 is 80, thus the answer is <math>\boxed{\text{(A)} \ 80}.</math>
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~ Zeeshan12 [Now there's 13 :) ]
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==Solution 14 (Factoring)==
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If you factor out <math>3^{100}</math> from the numerator and <math>3^{96}</math> from the denominator, you will get <math>\frac{3^{100}\left(1+(\frac{2}{3}\right)^{100})}{3^{96}\left(1+(\frac{2}{3}\right)^{96})}</math>. Divide the numerator and denominator by <math>3^{96}</math> to get <math>\frac{81\left(1+(\frac{2}{3}\right)^{100})}{\left(1+(\frac{2}{3}\right)^{96})}</math>. We see that every time we multiply <math>\frac{2}{3}</math> by itself, it slightly decreases, so <math>1+(\frac{2}{3})^{100}</math> will be ever so slightly smaller than <math>1+(\frac{2}{3})^{96}</math>. Thus, the decimal representation of <math>\frac{\left(1+(\frac{2}{3}\right)^{100})}{\left(1+(\frac{2}{3}\right)^{96})}</math> will be extremely close to <math>1</math>, so our solution will be the largest integer that is less than <math>81</math>. Thus, the answer is <math>\boxed{\text{(A)} \ 80}.</math>
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~andy_lee
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==Video Solution (HOW TO THINK CREATIVELY!)==
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https://youtu.be/zb0AcwIDqdg
 +
 +
~Education, the Study of Everything
  
 
==See Also==
 
==See Also==

Latest revision as of 10:59, 27 October 2023

Problem

What is the greatest integer less than or equal to \[\frac{3^{100}+2^{100}}{3^{96}+2^{96}}?\]

$\textbf{(A) }80\qquad \textbf{(B) }81 \qquad \textbf{(C) }96 \qquad \textbf{(D) }97 \qquad \textbf{(E) }625\qquad$

Solution 1

We write \[\frac{3^{100}+2^{100}}{3^{96}+2^{96}}=\frac{3^{96}}{3^{96}+2^{96}}\cdot\frac{3^{100}}{3^{96}}+\frac{2^{96}}{3^{96}+2^{96}}\cdot\frac{2^{100}}{2^{96}}=\frac{3^{96}}{3^{96}+2^{96}}\cdot 81+\frac{2^{96}}{3^{96}+2^{96}}\cdot 16.\] 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 $\boxed{\textbf{(A) }80}$.

Solution 2

Let's set this value equal to $x$. We can write \[\frac{3^{100}+2^{100}}{3^{96}+2^{96}}=x.\] Multiplying by $3^{96}+2^{96}$ on both sides, we get \[3^{100}+2^{100}=x(3^{96}+2^{96}).\] Now let's take a look at the answer choices. We notice that $81$, choice $B$, can be written as $3^4$. Plugging this into our equation above, we get \[3^{100}+2^{100} \stackrel{?}{=} 3^4(3^{96}+2^{96}) \Rightarrow 3^{100}+2^{100} \stackrel{?}{=} 3^{100}+3^4\cdot 2^{96}.\] The right side is larger than the left side because \[2^{100} \leq 2^{96}\cdot 3^4.\] This means that our original value, $x$, must be less than $81$. The only answer that is less than $81$ is $80$ so our answer is $\boxed{A}$.

~Nivek

Solution 3

\[\frac{3^{100}+2^{100}}{3^{96}+2^{96}}=\frac{2^{96}\left(\frac{3^{100}}{2^{96}}\right)+2^{96}\left(2^{4}\right)}{2^{96}\left(\frac{3}{2}\right)^{96}+2^{96}(1)}=\frac{\frac{3^{100}}{2^{96}}+2^{4}}{\left(\frac{3}{2}\right)^{96}+1}=\frac{\frac{3^{100}}{2^{100}}\cdot2^{4}+2^{4}}{\left(\frac{3}{2}\right)^{96}+1}=\frac{2^{4}\left(\frac{3^{100}}{2^{100}}+1\right)}{\left(\frac{3}{2}\right)^{96}+1}.\]

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.

\[\frac{2^{4}\left(\frac{3^{100}}{2^{100}}+1\right)}{\left(\frac{3}{2}\right)^{96}+1}<\frac{2^{4}\left(\frac{3^{100}}{2^{100}}\right)}{\left(\frac{3}{2}\right)^{96}},\]

\[\frac{2^{4}\left(\frac{3^{100}}{2^{100}}\right)}{\left(\frac{3}{2}\right)^{96}}=\frac{3^{4}}{2^{4}}*2^{4}=3^{4}=81.\]

So, our final answer is very close but not quite 81, and therefore the greatest integer less than the number is $\boxed{(A) 80}$

Solution 4

Let $x=3^{96}$ and $y=2^{96}$. Then our fraction can be written as $\frac{81x+16y}{x+y}=\frac{16x+16y}{x+y}+\frac{65x}{x+y}=16+\frac{65x}{x+y}$. Notice that $\frac{65x}{x+y}<\frac{65x}{x}=65$. So , $16+\frac{65x}{x+y}<16+65=81$. And our only answer choice less than 81 is $\boxed{(A) 80}$ (RegularHexagon)

Solution 5

Let $x=\frac{3^{100}+2^{100}}{3^{96}+2^{96}}$. Multiply both sides by $(3^{96}+2^{96})$, and expand. Rearranging the terms, we get $3^{96}(3^4-x)+2^{96}(2^4-x)=0$. The left side is decreasing, and it is negative when $x=81$. This means that the answer must be less than $81$; therefore the answer is $\boxed{(A)}$.

Solution 6 (eyeball it)

A faster solution. Recognize that for exponents of this size $3^{n}$ will be enormously greater than $2^{n}$, so the terms involving $2$ will actually have very little effect on the quotient. Now we know the answer will be very close to $81$.

Notice that the terms being added on to the top and bottom are in the ratio $\frac{1}{16}$ 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: $\boxed{\text{\textbf{(A)}}}$.

Solution 7

Notice how $\frac{3^{100}+2^{100}}{3^{96}+2^{96}}$ can be rewritten as $\frac{81(3^{96})+16(2^{96})}{3^{96}+2^{96}}=\frac{81(3^{96})+81(2^{96})}{3^{96}+2^{96}}-\frac{65(2^{96})}{3^{96}+2^{96}}=81-\frac{65(2^{96})}{3^{96}+2^{96}}$. Note that $\frac{65(2^{96})}{3^{96}+2^{96}}<1$, so the greatest integer less than or equal to $\frac{3^{100}+2^{100}}{3^{96}+2^{96}}$ is $80$ or $\boxed{\textbf{(A)}}$ ~blitzkrieg21

Solution 8

For positive $a, b, c, d$, if $\frac{a}{b}<\frac{c}{d}$ then $\frac{c+a}{d+b}<\frac{c}{d}$. Let $a=2^{100}, b=2^{96}, c=3^{100}, d=3^{96}$. Then $\frac{c}{d}=3^4$. 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 $3^4=81$ as the denominator is too big and putting $3^4-1=80$ is too small. So we know that the answer is between $80$ and $81$, yielding $80$ 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 $80$ because otherwise there would have been a smaller answer, so we move onto $81$. We get:

$\frac{3^{100}+2^{100}}{3^{96}+2^{96}} \text{ ? } 3^4$

Cross multiply to get:

$3^{100}+2^{100} \text{ ? }3^{100}+(2^{96})(3^4)$

Cancel out $3^{100}$ and divide by $2^{96}$ to get $2^{4} \text{ ? }3^4$. We know that $2^4 < 3^4$, which means the expression is less than $81$ so the answer is $\boxed{(A)}$.

Solution 10.2

We know this will be between 16 and 81 because $\frac{3^{100}}{3^{96}} = 3^4 = 81$ and $\frac{2^{100}}{2^{96}} = 2^4 = 16$. $80=\boxed{(A)}$ is the only option choice in this range.


Explanation for why 80 is indeed the floor

We need $3^{100}+2^{100} > 80 \cdot 3^{96} + 5 \cdot 2^{100}$. Since $3^{100} = 81\cdot 3^{96}$, this translates to \[3^{96} > 4\cdot 2^{100} = 64\cdot 2^{96}.\] We now prove that $(3/2)^k > k$ for all positive integers $k$. Clearly, $(3/2)^2 = 2.25 > 2$. Assume $(3/2)^k > k$ where $k\ge 2$. Then $\left(\frac{3}{2}\right)^{k+1} > \frac{3k}{2} = k + \frac{k}{2}$. But since $k/2 \ge 1$, we have that $(3/2)^{k+1} > k+1$. By induction (and $k=1$ is trivial), the claim is proven.

Thus, $\left(\frac{3}{2}\right)^{96} > 96 > 64$. Writing this proof backwards and dividing both sides of the initial equation by $80$ yields $80 < \frac{3^{100}+2^{100}}{3^{96}+2^{96}} < 81$.


Solution 11

We know that in this problem, $3^{96}+2^{96}$ times some number is equal to $3^{100}+2^{100}$. Multiplying answer $\boxed{\textbf{(B)}}$ or 81 to $3^{96}+2^{96}$ gives us $3^{100}+2^{96}\cdot3^4$. We know that $3^4\cdot2^{96}$ is greater than $2^{100}$, so that means $\boxed{\textbf{(B)}}$ or 81 is too big. That leaves us with only one solution: $80=\boxed{\textbf{(A) } 80}.$

~ Terribleteeth

Solution 12

Dividing by $2^{96}$ in both numerator and denominator, this fraction can be rewritten as \[\frac{81 \times (1.5)^{96} + 16}{(1.5)^{96} + 1}.\] Notice that the $+1$ and the $+16$ will be so insignificant compared to a number such as $(1.5)^{96},$ and that thereby the fraction will be ever so slightly less than $81$. Thereby, we see that the answer is $\boxed{\text{(A)} \ 80}.$

~ 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 $(3^{96} + 2^{96})$ by $(3^{4} + 2^{4})$ (to get the exponent up to 100), you'll get $(3^{100} + 2^{100}) + 3^{96} \cdot 2^{4} + 2^{96} \cdot 3^{4}$. Thus, in the numerator, if you add and subtract by $3^{96} \cdot 2^{4}$ and $2^{96} \cdot 3^{4}$, you'll get $\frac{(3^{4} + 2^{4})(3^{100} + 2^{100}) - 3^{96} \cdot 2^{4} - 2^{96} \cdot 3^{4}}{3^{96}+2^{96}}$. You can then take out out the first number to get $3^{4} + 2^{4} - \frac{3^{96} \cdot 2^{4} + 2^{96} \cdot 3^{4}}{3^{96}+2^{96}}$. This can then be written as $87 - \frac{16 \cdot 3^{96} + 16 \cdot 2^{96} + 75 \cdot 2^{96}}{3^{96}+2^{96}}$, factoring out the 16 and splitting the fraction will give you $87 - 16 - \frac{65 \cdot 2^{96}}{3^{96}+2^{96}}$, giving you $81 - \frac{65 \cdot 2^{96}}{3^{96}+2^{96}}$. While you can roughly say that $\frac{65 \cdot 2^{96}}{3^{96}+2^{96}} < 1$ you can also notice that the only answer choice less than 81 is 80, thus the answer is $\boxed{\text{(A)} \ 80}.$

~ Zeeshan12 [Now there's 13 :) ]

Solution 14 (Factoring)

If you factor out $3^{100}$ from the numerator and $3^{96}$ from the denominator, you will get $\frac{3^{100}\left(1+(\frac{2}{3}\right)^{100})}{3^{96}\left(1+(\frac{2}{3}\right)^{96})}$. Divide the numerator and denominator by $3^{96}$ to get $\frac{81\left(1+(\frac{2}{3}\right)^{100})}{\left(1+(\frac{2}{3}\right)^{96})}$. We see that every time we multiply $\frac{2}{3}$ by itself, it slightly decreases, so $1+(\frac{2}{3})^{100}$ will be ever so slightly smaller than $1+(\frac{2}{3})^{96}$. Thus, the decimal representation of $\frac{\left(1+(\frac{2}{3}\right)^{100})}{\left(1+(\frac{2}{3}\right)^{96})}$ will be extremely close to $1$, so our solution will be the largest integer that is less than $81$. Thus, the answer is $\boxed{\text{(A)} \ 80}.$

~andy_lee

Video Solution (HOW TO THINK CREATIVELY!)

https://youtu.be/zb0AcwIDqdg

~Education, the Study of Everything

See Also

2018 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 13 		Followed by
Problem 15
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All AMC 10 Problems and Solutions

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