inclusionAI/Ring-lite-rl-data huggingface dataset:
A frog starts climbing up a 12-meter deep well at 8 AM. For every 3 meters it climbs up, it slips down 1 meter. The time it takes to slip 1 meter is one-third of the time it takes to climb 3 meters. At 8:17 AM, the frog reaches 3 meters from the top of the well for the second time. How many minutes does it take for the frog to climb from the bottom of the well to the top?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Determine the number of field extensions between \( \mathbb{Q}(\zeta^3) \) and \( \mathbb{Q}(\zeta) \), where \( \zeta \) is a primitive 12th root of unity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
How many ways are there to arrange the letters $A, A, A, H, H$ in a row so that the sequence $HA$ appears at least once?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Two people $A$ and $B$ start from the same place at the same time to travel around a circular track of length $100$ m in opposite directions. Initially, $B$ moves more slowly than $A$ until they meet. After meeting, $B$ doubles his speed and next meets $A$ at the starting point. Let $d$ m be the distance traveled by $B$ before he met $A$ for the first time after leaving the starting point. Find the integer closest to $d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41}
For positive integer $n$ let $a_n$ be the integer consisting of $n$ digits of $9$ followed by the digits $488$. For example, $a_3 = 999,488$ and $a_7 = 9,999,999,488$. Find the value of $n$ so that an is divisible by the highest power of $2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
In the figure, the triangle $\triangle ABC$ is isosceles, with $\angle BAC = 20^\circ$.

Given that $BC = BD = BE$, determine the measure of the angle $\angle BDE$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
In the given diagram, each Chinese character inside a circle represents a number between 1 and 9. Different characters correspond to different numbers. The sum of the numbers at the three vertices of each small triangle is equal. If the sum of the 7 numbers is 12, what number does "杯" (cup) represent?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Olya drew $N$ different lines on a plane, any two of which intersect. It turned out that among any 15 lines, there are always two that form an angle of $60^{\circ}$ between them. What is the largest possible value of $N$ for which this is possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
A right regular hexagonal prism has bases $ABCDEF$, $A'B'C'D'E'F'$ and edges $AA'$, $BB'$, $CC'$, $DD'$, $EE'$, $FF'$, each of which is perpendicular to both hexagons. The height of the prism is $5$ and the side length of the hexagons is $6$. The plane $P$ passes through points $A$, $C'$, and $E$. The area of the portion of $P$ contained in the prism can be expressed as $m\sqrt{n}$, where $n$ is not divisible by the square of any prime. Find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{323}
Determine the number of distinct ways to arrange 8 indistinguishable red beads and 32 indistinguishable blue beads into a necklace such that there are at least 2 blue beads between any two red beads.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30667}
Let  $A_1B_1C_1$  be a triangle with  $A_1B_1 = 16, B_1C_1 = 14,$  and  $C_1A_1 = 10$ . Given a positive integer  $i$  and a triangle  $A_iB_iC_i$  with circumcenter  $O_i$ , define triangle  $A_{i+1}B_{i+1}C_{i+1}$  in the following way:

(a)  $A_{i+1}$  is on side  $B_iC_i$  such that  $C_iA_{i+1}=2B_iA_{i+1}$ .
(b)  $B_{i+1}\neq C_i$  is the intersection of line  $A_iC_i$  with the circumcircle of  $O_iA_{i+1}C_i$ .
(c)  $C_{i+1}\neq B_i$  is the intersection of line  $A_iB_i$  with the circumcircle of  $O_iA_{i+1}B_i$ .

Find \[ \left(\sum_{i = 1}^\infty [A_iB_iC_i] \right)^2. \]

Note:  $[K]$  denotes the area of  $K$ .

*Proposed by Yang Liu*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10800}
An archipelago consists of $N \geqslant 7$ islands. Any two islands are connected by at most one bridge. It is known that from each island no more than 5 bridges lead out, and among any 7 islands there must be two connected by a bridge. What is the maximum value that $N$ can take?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
if the product of the integers w, x, y, and z is 720, and if 0 < w < x < y < z < 20, what is the value of w + z?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
求具有下述性质的最小正数 c：对任意整数 n\geq4，以及集合 A\subseteq\{1,2,\cdots,n\}，若 |A|>cn，则存在函数 f:A\rightarrow\{-1,1\}，满足\left|\ds{a\in A}{}f(a)\cdot a\right|\leq1.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
2. We consider sports tournaments with $n \geqslant 4$ teams where each pair of teams plays against each other at most once. We call such a tournament balanced if every quartet of teams plays exactly three matches among themselves. Not all teams play against each other.

Determine the largest value of $n$ for which a balanced tournament with $n$ teams exists.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given a rectangular grid measuring 8 by 6, there are $48$ grid points, including those on the edges. Point $P$ is placed at the center of the rectangle. Find the probability that the line $PQ$ is a line of symmetry of the rectangle, given that point $Q$ is randomly selected from the other $47$ points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{12}{47}}
The roots of a monic cubic polynomial $p$ are positive real numbers forming a geometric sequence. Suppose that the sum of the roots is equal to $10$. Under these conditions, the largest possible value of $|p(-1)|$ can be written as $\frac{m}{n}$, where $m$, $n$ are relatively prime integers. Find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2224}
Let  $ \mathcal{R}$  be the region consisting of the set of points in the coordinate plane that satisfy both  $ |8 \minus{} x| \plus{} y \le 10$  and  $ 3y \minus{} x \ge 15$ . When  $ \mathcal{R}$  is revolved around the line whose equation is  $ 3y \minus{} x \equal{} 15$ , the volume of the resulting solid is  $ \frac {m\pi}{n\sqrt {p}}$ , where  $ m$ ,  $ n$ , and  $ p$  are positive integers,  $ m$  and  $ n$  are relatively prime, and  $ p$  is not divisible by the square of any prime. Find  $ m \plus{} n \plus{} p$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{365}
在平面直角坐标系中，设$\vec{e}$是一个单位向量，向量$\vec{a}$满足$\vec{a} \cdot \vec{e}=2$，并且$|\vec{a}|^{2} \leqslant 5|\vec{a}+t \vec{e}|$对任意实数$t$成立。请提供$|\vec{a}|$的取值范围左端点的两倍与右端点之差。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In a college's independent admission test for minor languages, the school received 5 recommendation spots, of which there are 2 for Korean, 2 for Japanese, and 1 for Russian. Both the Japanese and Korean spots require female participants. After selection, the school nominated 3 females and 2 males for a total of 5 recommended candidates. Find the number of different recommendation methods.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
The rank of a rational number \( q \) is the unique \( k \) for which \( q = \frac{1}{a_1} + \cdots + \frac{1}{a_k} \), where each \( a_i \) is the smallest positive integer such that \( q \geq \frac{1}{a_1} + \cdots + \frac{1}{a_i} \). Let \( q \) be the largest rational number less than \( \frac{1}{4} \) with rank 3, and suppose the expression for \( q \) is \( \frac{1}{a_1} + \frac{1}{a_2} + \frac{1}{a_3} \). Find the ordered triple \( \left(a_1, a_2, a_3\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(5, 21, 421)}
In triangle $ABC$, $AB = 28$, $AC = 36$, and $BC = 32$. Let $D$ be the point on segment $BC$ such that $\angle BAD = \angle DAC$. Let $E$ be the unique point such that $DE \parallel AB$ and line $AE$ is tangent to the circumcircle of $\triangle ABC$. Find the length of segment $AE$. \( \text{(Express your answer as a number.)} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Given ten 0's and ten 1's, how many 0-1 binary sequences can be formed such that no three or more consecutive 0's are together? For example, 01001001010011101011 is such a sequence, but the sequence 01001000101001110111 does not satisfy this condition.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24068}
\(a, b, c,\) and \(d\) are four non-zero numbers such that the quotients

$$
\frac{a}{5}, \quad \frac{-b}{7a}, \quad \frac{11}{abc}, \quad \text{and} \quad \frac{-18}{abcd}
$$

are all positive. Determine the signs of \(a, b, c,\) and \(d\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a > 0, b < 0, c < 0, d < 0}
A regular 201-sided polygon is inscribed inside a circle with center $C$. Triangles are drawn by connecting any three of the 201 vertices of the polygon. How many of these triangles have the point $C$ lying inside the triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["338350"]}
A scientist walks through a forest and records the heights of 5 trees in integers. She notices that the height of each tree is either double or half the height of the tree to its right. Unfortunately, some data was lost due to rain. Based on her observations, she is able to determine the missing values. The height of the second tree is 11 meters. Calculate the average height of these trees in meters, rounded to one decimal place.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24.2}
15. Given the set $A=\{1,2,3,4\}$, every second, Andel randomly picks a number from $A$ (repetition is allowed), and stops picking when the sum of the last two numbers picked is a prime number. What is the probability that the last number picked is “1”?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{15}{44}}
Find the smallest positive integer $n$ such that there exists a sequence of all rational numbers $a_1, a_2, \dots$ where no subsequence of length $n$ forms an arithmetic progression.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $A_{10}$ denote the answer to problem 10. Two circles lie in the plane; denote the lengths of the internal and external tangents between these two circles by $x$ and $y$, respectively. Given that the product of the radii of these two circles is $15 / 2$, and that the distance between their centers is $A_{10}$, determine $y^{2}-x^{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Determine the number of ring homomorphisms from the ring of real numbers to itself that satisfy the condition $f(0) = 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let a sequence $\left\{a_{n}\right\}_{n=0}^{\infty}$ be defined by $a_{0}=\sqrt{2}, a_{1}=2$, and $a_{n+1}=a_{n} a_{n-1}^{2}$ for $n \geq 1$. The sequence of remainders when $a_{0}, a_{1}, a_{2}, \cdots$ are divided by 2014 is eventually periodic with some minimal period $p$ (meaning that $a_{m}=a_{m+p}$ for all sufficiently large integers $m$, and $p$ is the smallest such positive integer). Find $p$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Find the inverse Laplace transform of the function: $$\mathcal{L}^{-1}\left\{\dfrac{1}{(s^2+a^2)^2}\right\}$$

Given the identity: $tf(t)\longleftrightarrow -\frac{\mathrm d}{\mathrm ds}F(s)$ and $\int_{0}^{t}g(\tau)d\tau\longleftrightarrow \frac{1}{s}G(s)$, where $f(t)$ and $g(t)$ are the inverse Laplace transforms of $F(s)$ and $G(s)$, respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sin (a t)-a t \cos (a t)}{2 a^3}}
On a rectangular sheet of paper, a picture in the shape of a "cross" is drawn using two rectangles $ABCD$ and $EFGH$, whose sides are parallel to the edges of the sheet. It is given that $AB=9$, $BC=5$, $EF=3$, $FG=10$. Find the area of the quadrilateral $AFCH$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52.5}
One day the Beverage Barn sold $252$ cans of soda to $100$ customers, and every customer bought at least one can of soda. If the maximum possible median number of cans of soda bought per customer on that day is $x$, what is the value of $10x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
If $x_1, x_2, x_3, x_4$ are the roots of the polynomial $x^4 - x^3 + 2x^2 + 3x + 1$, find the value of the expression:
\[ \frac{\sum x_i^3 + \sum (x_i x_j x_k)^3}{\sum (x_i x_j)^3 - (x_1 x_2 x_3 x_4)^3} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{20}{49}}
Let \(A \cup B \cup C = \{1, 2, 3, 4, 5, 6\}\), and \(A \cap B = \{1, 2\}\). Additionally, \(\{1, 2, 3, 4\} \subseteq B \cup C\). Determine the number of distinct groups \((A, B, C)\) that meet these conditions. (Note: The order of \(A\), \(B\), and \(C\) matters, so different orderings are considered different groups.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1600}
Find the greatest common divisor of all numbers of the form  $(2^{a^2}\cdot 19^{b^2} \cdot 53^{c^2} + 8)^{16} - 1$  where  $a,b,c$  are integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
The sequence of nonnegative integers $F_0, F_1, F_2, \dots$ is defined recursively as $F_0 = 0$, $F_1 = 1$, and $F_{n+2} = F_{n+1} + F_n$ for all integers $n \geq 0$. Let $d$ be the largest positive integer such that, for all integers $n \geq 0$, $d$ divides $F_{n+2020} - F_n$. Compute the remainder when $d$ is divided by $1001$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{638}
Given that \( f(x) \) is a function defined on \(\mathbf{R}\), and for any \(x, y \in \mathbf{R}\), \( f(x) + f(y) = 2 f\left(\frac{x+y}{2}\right) f\left(\frac{x-y}{2}\right) \) holds true. It is also known that \( f\left(\frac{\pi}{4}\right) = 0 \), but \( f(x) \) is not identically zero, and \( f(0) > 0 \). Find the period of the function \( f(x) \). Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Circles with centers $A$, $B$, and $C$ each have radius $r$, where $1 < r < 2$. The distance between each pair of centers is $2$. If $B'$ is the point of intersection of circle $A$ and circle $C$ which is outside circle $B$, and if $C'$ is the point of intersection of circle $A$ and circle $B$ which is outside circle $C$, then length $B'C'$ equals
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1+\sqrt{3(r^2-1)}}
In Ms. Johnson's class, each student averages two days absent out of thirty school days. What is the probability that out of any three students chosen at random, exactly two students will be absent and one will be present on a Monday, given that on Mondays the absence rate increases by 10%? Express your answer as a percent rounded to the nearest tenth.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.5\%}
What is the molecular weight of 7 moles of Aluminum iodide?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2853.76}
Let  $\phi(n)$  denote the number of positive integers less than or equal to  $n$  which are relatively prime to  $n$ . Over all integers  $1\le n \le 100$ , find the maximum value of  $\phi(n^2+2n)-\phi(n^2)$ .

*Proposed by Vincent Huang*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
A random triangle is produced as follows. A pair of standard dice is rolled independently three times to get three random numbers between 2 and 12, inclusive, by adding the numbers that come up on each pair rolled. Call these three random numbers $a$, $b$, and $t$. The random triangle has two sides of lengths $a$ and $b$ with the angle between them measuring $15(t - 1)$ degrees. What is the probability that the triangle is a right triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{113}{648}}
Alice and Bob are each given $2000 to invest. Alice puts all of her money in the stock market and doubles her money. Bob invests in real estate and makes five times more money than he invested. How much more money does Bob have now than Alice?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$6000}
On the Arctic Circle ($60^{\circ}$), there are two locations $A$ and $B$. The arc length between them on this latitude circle is equal to $\frac{\pi R}{2}$ ($R$ is the radius of the Earth). What is the spherical distance between locations $A$ and $B$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi R}{3}}
Exercise 12. Let $M N P Q$ be a rhombus such that $M P=180$. Suppose there exists a square $A B C D$ such that $A B=18, A \in[M N], B \in[N P], C \in[P Q]$ and $D \in[Q M]$. Determine the area of $M N P Q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1800}
A rhombus, which is divided by its diagonal into two equilateral triangles, has an inscribed circle with a radius of 2. Find the side of the rhombus.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{8\sqrt{3}}{3}}
Calculate the dimension of $H^0(T,\mathcal{L})$, where $T$ is a complex torus of dimension $3$ and $\mathcal{L}$ gives $T$ a $(p,q,r)$-polarization.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{pqr}
Determine the exact value of the series
\[\frac{1}{3 + 1} + \frac{2}{3^2 + 1} + \frac{4}{3^4 + 1} + \frac{8}{3^8 + 1} + \frac{16}{3^{16} + 1} + \dotsb.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Let $S$ be the set of integers of the form $2^x+2^y+2^z$, where $x,y,z$ are pairwise distinct non-negative integers.  Determine the $100$th smallest element of $S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{577}
In the diagram, right triangles PSQ and RSQ are shown. Given QR = 25, SR = 20, and PS = 8, find the perimeter of triangle PQR. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
Determine the values of $k$ for which the Fourier cosine transform of $e^{-x^k}$, defined as \[ g_k(z) = \int_0^\infty \cos(zx) e^{-x^k} \, dx, \] is non-negative and decreasing for $z \in \mathbb{R}^{+}$. Provide your answer as a set of values for $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 2]}
Tarek wants to shade two further squares on the given diagram so that the resulting pattern has a single axis of symmetry. Calculate the total number of different ways he can complete his pattern to achieve this symmetry. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Evaluate the expression $\frac{C}{D}$, where $C=\sum_{k=1}^{2000}\frac{1}{(2k-1)(2k)}$ and $D=\sum_{k=0}^{999}\frac{1}{(2000+k)(2999-k)}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.000}
Given 2022 points uniformly distributed on a circle, determine the maximum number of arcs that can be drawn between these points such that no arc is completely included in another, the lengths of arcs are pairwise different, and the smallest and biggest arcs do not overlap or share the same endpoint.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1011}
设黑板上写有128个1，每一步可以擦去黑板上的任意两个数a和b并写上ab+这样做了127次之后，只剩下1个数，将这样剩下的数的最大可能值记作A，试求A 的末位数字. （1992年圣彼得堡市代表队选拔试题）
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Consider $2003^{2009}$+$2009^{2017}$+$2015^{2025}$ in base 3, we get the very long string $a_{n} a_{n-1}... a_{1} a_{0}$. What are the values of the last 3 digits $a_2, a_1, a_0$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{110}
圆内接四边形 $A B C D$ 中, $A B=136, B C=80, C D=150, D A=102$, 求它的外接圆直径。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{170}
Determine the cardinality of the union of a collection $T = \{E_{\alpha}:\alpha \in I\}$, where $I$ is an uncountable directed set, each $E_{\alpha}$ is a countably infinite set, and $E_{\alpha} \subset E_{\beta}$ for $\alpha \le \beta$ with $T$ totally ordered by set inclusion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_1}
Determine the exponent that should accompany $i$ in the expression $i^n$ to achieve the most optimized change of direction without altering the modulus.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
There is a peculiar four-digit number (with the first digit not being 0). It is a perfect square, and the sum of its digits is also a perfect square. Dividing this four-digit number by the sum of its digits results in yet another perfect square. Additionally, the number of divisors of this number equals the sum of its digits, which is a perfect square. If all the digits of this four-digit number are distinct, what is this four-digit number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2601}
Tom, John, and Lily each shot six arrows at a target. Arrows hitting anywhere within the same ring scored the same number of points. Tom scored 46 points and John scored 34 points. Find Lily's score. Express your answer as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
In a three-day MLB game between Team A and Team B, the ratio of misses to hits and types of hits varies each day for both teams. On the first day, Team A has twice as many misses as hits (with 60 misses), while Team B has three times as many misses as hits (with 90 misses). For each team, half of their hits on the first day are home runs and the other half are singles.

On the second day, Team A has four times as many misses as hits (with 68 misses), while Team B has two times as many misses as hits (with 56 misses). On this day, one-third of each team's hits are doubles and the remainder are singles.

On the third day, the ratio of misses to hits for both teams is 5:1. Team A has 100 misses, and Team B has 120 misses. For third day hits of both teams, one-fourth are triples, one-fourth are home runs, and the remaining hits are singles.

Calculate the combined total of misses, singles, doubles, triples, and home runs for Team A and Team B made during the entire three-day game.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{643}
Determine all integers  $n$  such that both of the numbers:  $$ |n^3 - 4n^2 + 3n - 35| \text{ and }  |n^2 + 4n + 8| $$  are both prime numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \in \{-3, -1, 5\}}
on a sum of money , the simple interest for 2 years is rs . 326 , while the compound interest is rs . 340 , the rate of interest being the same in both the cases . the rate of interest is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8.59\%}