2018 AMC 10A Problems/Problem 14

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

Solution 1

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 out 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 2

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 $\boxed{(A)}$

~RH

Solution 3

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 strictly 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 4

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: $\boxed{(A)}$ .

Solution 5 (Using the answer choices)

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 6(logic)

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

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