problem_key stringclasses 560
values | rollout_index int64 0 31 | level int64 1 5 | capability stringclasses 9
values | gt stringclasses 345
values | solution stringlengths 9 410k | correct bool 2
classes | committed bool 2
classes | finish_reason stringclasses 2
values | output_tokens int64 4 31.7k |
|---|---|---|---|---|---|---|---|---|---|
precalculus__8ee539 | 0 | 2 | precalculus | \left( 3, \frac{\pi}{2} \right) | .
Okay, so I need to convert the point (0, 3) from rectangular coordinates to polar coordinates. Hmm, polar coordinates are in the form (r, θ), right? Let me recall how to do this.
First, I remember that to convert from rectangular (x, y) to polar (r, θ), the formulas are r = sqrt(x² + y²) and θ = arctan(y/x). But wai... | true | true | stop | 682 |
precalculus__8ee539 | 1 | 2 | precalculus | \left( 3, \frac{\pi}{2} \right) | .
Okay, let's see. I need to convert the rectangular coordinates (0, 3) to polar coordinates. Hmm, polar coordinates are in the form (r, θ) where r is the distance from the origin and θ is the angle from the positive x-axis. Alright, so first, I should find r.
The formula for r is the square root of (x squared plus y ... | true | true | stop | 714 |
precalculus__8ee539 | 2 | 2 | precalculus | \left( 3, \frac{\pi}{2} \right) | .
Okay, so I need to convert the rectangular coordinates (0, 3) to polar coordinates. Let me remember how that works. Polar coordinates are (r, θ), where r is the distance from the origin, and θ is the angle from the positive x-axis.
First, let me recall the formulas. To convert from rectangular (x, y) to polar (r, ... | true | true | stop | 888 |
precalculus__8ee539 | 3 | 2 | precalculus | \left( 3, \frac{\pi}{2} \right) | .
Okay, let's see. I need to convert the rectangular coordinates (0, 3) to polar coordinates. Hmm, polar coordinates are (r, θ), right? So, first, I remember that to convert from rectangular (x, y) to polar, the formulas are r = sqrt(x² + y²) and θ = arctan(y/x). But wait, I have to be careful with the quadrant where ... | true | true | stop | 750 |
precalculus__8ee539 | 4 | 2 | precalculus | \left( 3, \frac{\pi}{2} \right) | .
Okay, let's see. I need to convert the point (0, 3) from rectangular coordinates to polar coordinates. Hmm, polar coordinates are usually written as (r, θ), right? So I need to find r and θ.
First, I remember that in polar coordinates, r is the distance from the origin to the point. So for the point (0, 3), that's 3... | true | true | stop | 573 |
precalculus__8ee539 | 5 | 2 | precalculus | \left( 3, \frac{\pi}{2} \right) | .
Okay, so I need to convert the point (0, 3) from rectangular coordinates to polar coordinates. Hmm, polar coordinates are in the form (r, θ), where r is the distance from the origin, and θ is the angle from the positive x-axis. Let me remember how to do this conversion.
First, for r. I think r is calculated using t... | true | true | stop | 500 |
precalculus__8ee539 | 6 | 2 | precalculus | \left( 3, \frac{\pi}{2} \right) | .
Okay, let's see. I need to convert the point (0, 3) from rectangular coordinates to polar coordinates. Hmm, polar coordinates are in the form (r, θ), right? So first, I remember that r is the distance from the origin to the point, calculated using the Pythagorean theorem. The formula for r is sqrt(x² + y²).
Given ... | true | true | stop | 551 |
precalculus__8ee539 | 7 | 2 | precalculus | \left( 3, \frac{\pi}{2} \right) | .
[asy]import graph; size(6cm); real r = 1.5; real s = 0.2; real t = 0.8; real R = 1; real x = 0; real y = 0; real y1 = 0; real y2 = 0; real y3 = 0; real x1 = 0; real x2 = 0; real x3 = 0; real y4 = 0; real y5 = 0; real x4 = 0; real x5 = 0; real y6 = 0; real x6 = 0; draw((0,0)--(0,-3),Arrows); draw((0,-3)--(0,-6),Arrows... | false | false | length | 31,744 |
precalculus__8ee539 | 8 | 2 | precalculus | \left( 3, \frac{\pi}{2} \right) | .
Okay, so I need to convert the rectangular coordinates (0, 3) to polar coordinates. Let me remember how that works. Polar coordinates are (r, θ), right? Where r is the distance from the origin, and θ is the angle made with the positive x-axis.
First, let's find r. The formula for r is the square root of (x squared ... | true | true | stop | 639 |
precalculus__8ee539 | 9 | 2 | precalculus | \left( 3, \frac{\pi}{2} \right) | .
Okay, so I need to convert the rectangular coordinates (0, 3) to polar coordinates. Hmm, polar coordinates are in the form (r, θ), where r is the distance from the origin and θ is the angle from the positive x-axis. Let me recall how to do this conversion.
First, I think the formula for r is the square root of (x s... | true | true | stop | 507 |
precalculus__8ee539 | 10 | 2 | precalculus | \left( 3, \frac{\pi}{2} \right) | .
Okay, so I need to convert the point (0,3) from rectangular coordinates to polar coordinates. Hmm, polar coordinates are (r, θ), right? Let me recall the formulas for converting.
I think the formula for r is the square root of (x squared plus y squared). So in this case, x is 0 and y is 3. Let me calculate that. r... | true | true | stop | 776 |
precalculus__8ee539 | 11 | 2 | precalculus | \left( 3, \frac{\pi}{2} \right) | .
Okay, so I need to convert the rectangular coordinates (0, 3) to polar coordinates (r, θ). Hmm, polar coordinates are based on the radius r and the angle θ from the positive x-axis. Let me recall the formulas for conversion.
I think the formula for r is the square root of (x squared plus y squared). So, r = sqrt(x... | true | true | stop | 686 |
precalculus__8ee539 | 12 | 2 | precalculus | \left( 3, \frac{\pi}{2} \right) | .
Okay, so I need to convert the rectangular coordinates (0, 3) to polar coordinates. Hmm, polar coordinates are (r, θ) where r is the distance from the origin and θ is the angle from the positive x-axis. Let me recall the formulas for conversion.
Right, the formulas are r = sqrt(x² + y²) and θ = arctan(y/x). But wa... | true | true | stop | 514 |
precalculus__8ee539 | 13 | 2 | precalculus | \left( 3, \frac{\pi}{2} \right) | .
Okay, let's see. I need to convert the rectangular coordinates (0, 3) to polar coordinates. Hmm, polar coordinates are in the form (r, θ), right? So, first, I remember that to convert from rectangular to polar, there are some formulas. Let me recall them.
The formulas are r squared equals x squared plus y squared, ... | true | true | stop | 833 |
precalculus__8ee539 | 14 | 2 | precalculus | \left( 3, \frac{\pi}{2} \right) | .
Okay, so I need to convert the rectangular coordinates (0, 3) to polar coordinates (r, θ). Hmm, polar coordinates are based on the distance from the origin and the angle from the positive x-axis, right? Let me recall the formulas for conversion.
I think the formulas are r = sqrt(x² + y²) and θ = arctan(y/x). Wait,... | true | true | stop | 653 |
precalculus__8ee539 | 15 | 2 | precalculus | \left( 3, \frac{\pi}{2} \right) | .
Okay, so I need to convert the point (0,3) from rectangular coordinates to polar coordinates. Hmm, polar coordinates are in the form (r, θ), right? Where r is the distance from the origin and θ is the angle from the positive x-axis. Let me recall the formulas for converting between rectangular and polar coordinates.... | true | true | stop | 637 |
algebra__e16674 | 0 | 3 | algebra | 3\sqrt{13} | .
Okay, so I need to find the distance between the points (2, -6) and (-4, 3). Hmm, I remember there's a formula for the distance between two points in a coordinate plane. Let me think. Oh yeah, the distance formula is derived from the Pythagorean theorem. Right, so if you have two points, (x1, y1) and (x2, y2), the di... | true | true | stop | 691 |
algebra__e16674 | 1 | 3 | algebra | 3\sqrt{13} | .
Okay, so I need to find the distance between the points (2, -6) and (-4, 3). Hmm, distance between two points in a coordinate plane... I think that's the distance formula. Let me recall. The distance formula is derived from the Pythagorean theorem, right? So if I have two points, (x1, y1) and (x2, y2), the distance b... | true | true | stop | 708 |
algebra__e16674 | 2 | 3 | algebra | 3\sqrt{13} | .
Okay, let's see. I need to find the distance between the points (2, -6) and (-4, 3). Hmm, I remember there's a formula for the distance between two points in a coordinate plane. Let me think. Oh right, the distance formula! It's derived from the Pythagorean theorem. So, the formula is the square root of [(x2 - x1) sq... | true | true | stop | 570 |
algebra__e16674 | 3 | 3 | algebra | 3\sqrt{13} | .
Okay, let's see. I need to find the distance between the points (2, -6) and (-4, 3). Hmm, how do I do that again? Oh right, the distance formula! It's like the Pythagorean theorem in two dimensions. So the formula is sqrt[(x2 - x1)^2 + (y2 - y1)^2]. Let me write that down.
First, I should identify the coordinates. T... | true | true | stop | 511 |
algebra__e16674 | 4 | 3 | algebra | 3\sqrt{13} | .
Okay, so I need to find the distance between the points (2, -6) and (-4, 3). Hmm, I remember there's a formula for the distance between two points in a coordinate plane. Let me think... Oh right, the distance formula! It's derived from the Pythagorean theorem. The formula is the square root of [(x2 - x1) squared plus... | true | true | stop | 853 |
algebra__e16674 | 5 | 3 | algebra | 3\sqrt{13} | .
Okay, let's see. I need to find the distance between the points (2, -6) and (-4, 3). Hmm, I remember there's a formula for the distance between two points in a coordinate plane. Let me recall... Oh right, the distance formula. It's derived from the Pythagorean theorem, right? So if you have two points, (x1, y1) and ... | true | true | stop | 671 |
algebra__e16674 | 6 | 3 | algebra | 3\sqrt{13} | .
Okay, let's see. I need to find the distance between the two points (2, -6) and (-4, 3). Hmm, distance formula, right? The formula is something like the square root of [(x2 - x1)^2 + (y2 - y1)^2]. Let me recall. Yeah, that's right. So first, I should identify the coordinates. The first point is (2, -6), so x1 is 2 an... | true | true | stop | 647 |
algebra__e16674 | 7 | 3 | algebra | 3\sqrt{13} | .
Okay, so I need to find the distance between the points (2, -6) and (-4, 3). Hmm, I remember there's a formula for the distance between two points in a coordinate plane. Let me think... Oh right, the distance formula! It's derived from the Pythagorean theorem, right?
The formula is something like the square root of... | true | true | stop | 625 |
algebra__e16674 | 8 | 3 | algebra | 3\sqrt{13} | .
Okay, so I need to find the distance between the points (2, -6) and (-4, 3). Hmm, I remember there's a formula for the distance between two points in a coordinate plane. Let me think... Oh right, the distance formula! It's derived from the Pythagorean theorem. The formula is \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)... | true | true | stop | 661 |
algebra__e16674 | 9 | 3 | algebra | 3\sqrt{13} | .
Okay, so I need to find the distance between the two points (2, -6) and (-4, 3). Hmm, I remember there's a formula for the distance between two points in a coordinate plane. Let me think... Oh right, the distance formula! It's derived from the Pythagorean theorem. The formula is sqrt[(x2 - x1)^2 + (y2 - y1)^2].
Al... | true | true | stop | 589 |
algebra__e16674 | 10 | 3 | algebra | 3\sqrt{13} | .
Okay, so I need to find the distance between the points (2, -6) and (-4, 3). Hmm, right, the distance formula. Let me recall how that works. I think it's based on the Pythagorean theorem. The formula is the square root of [(x2 - x1)^2 + (y2 - y1)^2]. Let me make sure I remember that correctly.
So, first, I should l... | true | true | stop | 721 |
algebra__e16674 | 11 | 3 | algebra | 3\sqrt{13} | .
Okay, let's see. I need to find the distance between the points (2, -6) and (-4, 3). Hmm, I remember there's a formula for the distance between two points in a coordinate plane. Let me recall. Oh right, the distance formula is derived from the Pythagorean theorem. So, if you have two points (x1, y1) and (x2, y2), th... | true | true | stop | 637 |
algebra__e16674 | 12 | 3 | algebra | 3\sqrt{13} | and to only include the answer, not any other text.
Okay, so I need to find the distance between the points (2, -6) and (-4, 3). Hmm, I remember there's a formula for the distance between two points in a coordinate plane. Let me think... Oh right, the distance formula! It's derived from the Pythagorean theorem. The f... | true | true | stop | 653 |
algebra__e16674 | 13 | 3 | algebra | 3\sqrt{13} | .
Okay, so I need to find the distance between the points (2, -6) and (-4, 3). Hmm, distance formula, right? Let me recall. The distance between two points (x1, y1) and (x2, y2) is sqrt[(x2 - x1)^2 + (y2 - y1)^2]. Yeah, that's the formula. Alright, let's plug in the values.
First, let's label the points. Let me call t... | true | true | stop | 581 |
algebra__e16674 | 14 | 3 | algebra | 3\sqrt{13} | .
Okay, so I need to find the distance between the points (2, -6) and (-4, 3). Hmm, I remember there's a formula for the distance between two points in a coordinate plane. Let me think... Oh right, the distance formula is derived from the Pythagorean theorem. The formula is sqrt[(x2 - x1)^2 + (y2 - y1)^2]. Let me make... | true | true | stop | 572 |
algebra__e16674 | 15 | 3 | algebra | 3\sqrt{13} | .
Okay, so I need to find the distance between the points (2, -6) and (-4, 3). Hmm, right, distance formula. Let me recall. The distance between two points (x1, y1) and (x2, y2) is the square root of [(x2 - x1)^2 + (y2 - y1)^2]. Yeah, that's right. So I should apply that here.
First, let's identify the coordinates. Th... | true | true | stop | 596 |
intermediate_algebra__687b74 | 0 | 1 | intermediate_algebra | -50 | .
Okay, let me see. I need to compute the sum 1 - 2 + 3 - 4 + 5 - ... + 99 - 100. Hmm, alternating signs, starting with positive. So the pattern is +, -, +, -, and so on. Let me think about how to approach this.
First, maybe I can group the terms in pairs. Like (1 - 2) + (3 - 4) + (5 - 6) + ... and so on. Each pair w... | true | true | stop | 1,290 |
intermediate_algebra__687b74 | 1 | 1 | intermediate_algebra | -50 | .
Okay, let me try to figure out this problem. So, the question is to compute the sum: 1 - 2 + 3 - 4 + 5 - ... + 99 - 100. Hmm, that's an alternating series where the signs switch between positive and negative. Let me see how to approach this.
First, let me write out the series to see the pattern. Starting from 1, su... | true | true | stop | 966 |
intermediate_algebra__687b74 | 2 | 1 | intermediate_algebra | -50 | .
Okay, so I need to compute the sum 1 - 2 + 3 - 4 + 5 - ... + 99 - 100. Hmm, let's see. This looks like an alternating series where the signs alternate between positive and negative. Let me try to figure out a pattern here.
First, let me write out the first few terms to see if I can spot a pattern. The series starts... | true | true | stop | 906 |
intermediate_algebra__687b74 | 3 | 1 | intermediate_algebra | -50 | .
Okay, so I need to compute the sum of the series 1 - 2 + 3 - 4 + 5 - ... + 99 - 100. Hmm, let's see. First, I should figure out if there's a pattern here. The signs are alternating, starting with positive for the first term. So the first term is positive, the second negative, third positive, and so on. The last term... | true | true | stop | 1,150 |
intermediate_algebra__687b74 | 4 | 1 | intermediate_algebra | -50 | .
Okay, let's see. I need to compute the sum of this series: 1 - 2 + 3 - 4 + 5 - ... + 99 - 100. Hmm, so it's alternating signs, starting with positive, then negative, positive, and so on. Let me try to figure out a pattern here.
First, maybe I can group the terms in pairs. Let's see, starting from the beginning:
1 -... | true | true | stop | 750 |
intermediate_algebra__687b74 | 5 | 1 | intermediate_algebra | -50 | .
Okay, so I need to compute the sum of the series 1 - 2 + 3 - 4 + 5 - ... + 99 - 100. Hmm, let's see. The pattern here is alternating between adding and subtracting consecutive integers. The first term is positive 1, then subtract 2, add 3, subtract 4, and so on until we reach 99 - 100.
First, I should figure out h... | true | true | stop | 1,387 |
intermediate_algebra__687b74 | 6 | 1 | intermediate_algebra | -50 | .
Okay, let's see. I need to compute the sum $1 - 2 + 3 - 4 + 5 - \dots + 99 - 100$. Hmm. So it's alternating addition and subtraction of consecutive numbers starting from 1. Let me try to figure out a pattern here.
First, maybe I should write out the first few terms to see if there's a pattern. Let's see:
1 - 2 = -1... | true | true | stop | 1,928 |
intermediate_algebra__687b74 | 7 | 1 | intermediate_algebra | -50 | .
Okay, so I need to compute the sum 1 - 2 + 3 - 4 + 5 - ... + 99 - 100. Hmm, let's see. This looks like an alternating series where each odd number is added and each even number is subtracted. Let me try to break it down step by step.
First, maybe I can group the terms in pairs. Like, (1 - 2) + (3 - 4) + (5 - 6) + .... | true | true | stop | 959 |
intermediate_algebra__687b74 | 8 | 1 | intermediate_algebra | -50 | .
Okay, so I need to compute the sum 1 - 2 + 3 - 4 + 5 - ... + 99 - 100. Hmm, let's see. The pattern here is alternating between adding an odd number and subtracting an even number. Let me write out the first few terms to see if there's a pattern.
Starting with 1, then subtract 2, add 3, subtract 4, add 5, subtract 6... | true | true | stop | 1,029 |
intermediate_algebra__687b74 | 9 | 1 | intermediate_algebra | -50 | .
Okay, so I need to compute the sum 1 - 2 + 3 - 4 + 5 - ... + 99 - 100. Hmm, let's see. This looks like an alternating series where each pair of numbers is subtracted. Let me break it down.
First, maybe I can group the terms into pairs. Each pair would be (1 - 2), then (3 - 4), and so on, up until the last pair whic... | true | true | stop | 2,069 |
intermediate_algebra__687b74 | 10 | 1 | intermediate_algebra | -50 | .
Okay, let's see. I need to compute this alternating sum: 1 - 2 + 3 - 4 + 5 - ... + 99 - 100. Hmm, so it's adding and subtracting consecutive integers starting from 1, alternating each time. Let me think about how to approach this.
First, maybe I can group the terms in pairs. Like, (1 - 2) + (3 - 4) + (5 - 6) + ... a... | true | true | stop | 1,024 |
intermediate_algebra__687b74 | 11 | 1 | intermediate_algebra | -50 | .
Okay, let's see. I need to compute the sum 1 - 2 + 3 - 4 + 5 - ... + 99 - 100. Hmm, this looks like an alternating series where each pair of numbers is subtracted and added. Let me try to break it down.
First, maybe I can group the terms in pairs. Like (1 - 2) + (3 - 4) + (5 - 6) + ... and so on. Each of these pair... | true | true | stop | 1,199 |
intermediate_algebra__687b74 | 12 | 1 | intermediate_algebra | -50 | .
Okay, let's see. I need to compute the sum of the series: 1 - 2 + 3 - 4 + 5 - ... + 99 - 100. Hmm, alternating addition and subtraction. So each pair is like (1 - 2), then (3 - 4), and so on. Let me check how many terms there are. The series goes from 1 to 100, so that's 100 terms. Since each pair is two terms, that... | true | true | stop | 504 |
intermediate_algebra__687b74 | 13 | 1 | intermediate_algebra | -50 | .
Okay, let's see. I need to compute the sum: 1 - 2 + 3 - 4 + 5 - ... + 99 - 100. Hmm, alternating signs, right? So it starts with positive 1, then negative 2, positive 3, and so on, ending with negative 100. Let me think about how to approach this.
First, maybe I should group the terms in pairs. Like, (1 - 2) + (3 - ... | true | true | stop | 859 |
intermediate_algebra__687b74 | 14 | 1 | intermediate_algebra | -50 | .
Okay, so I need to compute the sum: 1 - 2 + 3 - 4 + 5 - ... + 99 - 100. Hmm, let's see. This looks like an alternating series where the signs alternate between positive and negative. The pattern starts with 1, then subtracts 2, adds 3, subtracts 4, and so on up to 100.
First, I should figure out how many terms the... | true | true | stop | 1,020 |
intermediate_algebra__687b74 | 15 | 1 | intermediate_algebra | -50 | .
Okay, so I need to compute the sum of the series 1 - 2 + 3 - 4 + 5 - ... + 99 - 100. Hmm, let me think. First, I should figure out how the series is structured. It starts with 1, then subtracts 2, adds 3, subtracts 4, and so on, alternating between positive and negative terms. The last term is -100, which is even, s... | true | true | stop | 1,196 |
algebra__8dd85b | 0 | 5 | algebra | 10 | .
Okay, so the problem is about a worker who deposits $20,000 at the end of each year into a savings account. After three years, he wants to have at least $66,200. We need to find the minimal compound interest rate required. Hmm, let's see.
First, I remember that when dealing with compound interest, especially with r... | true | true | stop | 2,166 |
algebra__8dd85b | 1 | 5 | algebra | 10 | .
Okay, let's try to figure out this compound interest problem. So, the worker deposits $20,000 each year into a savings account, and by the end of the third year, he wants at least $66,200. We need to find the minimal compound interest rate required for this. Hmm, compound interest, so each deposit grows with interest... | true | true | stop | 1,694 |
algebra__8dd85b | 2 | 5 | algebra | 10 | .
Okay, let's see. The problem is about a worker who deposits $20,000 at the end of each year into a savings account. After three years, he wants to have at least $66,200. We need to find the minimal compound interest rate required for this to happen. Hmm, compound interest... so each deposit earns interest over the r... | true | true | stop | 1,767 |
algebra__8dd85b | 3 | 5 | algebra | 10 | .
Okay, let's see. The problem is about a worker who deposits $20,000 each year into a savings account, and after three years, he wants at least $66,200. We need to find the minimal compound interest rate required. Hmm, compound interest, right? So each deposit earns interest over the years they are in the account. Sin... | true | true | stop | 1,537 |
algebra__8dd85b | 4 | 5 | algebra | 10 | .
Okay, let's see. The problem is about compound interest. A worker deposits $20,000 at the end of each year into a savings account, and after three years, the total should be at least $66,200. We need to find the minimal compound interest rate required. Hmm.
First, I need to recall the formula for compound interest.... | true | true | stop | 2,192 |
algebra__8dd85b | 5 | 5 | algebra | 10 | .
Okay, so I need to figure out the minimal compound interest rate that the savings account must provide for the worker to have at least $66,200 by the end of the third year. Let's start by understanding the problem step by step.
First, the worker deposits $20,000 at the end of each year for three years. So, the depo... | true | true | stop | 1,968 |
algebra__8dd85b | 6 | 5 | algebra | 10 | .
Okay, so I need to find the minimal compound interest rate that the savings account must provide so that after three years, the total amount is at least $66,200. The worker deposits $20,000 each year at the end of each year, and it's a compound interest account. Hmm, let's break this down step by step.
First, compo... | true | true | stop | 2,175 |
algebra__8dd85b | 7 | 5 | algebra | 10 | .
Okay, let's see. The problem is about calculating the minimal compound interest rate needed so that after three years, the total amount in the savings account is at least $66,200. The worker deposits $20,000 each year at the end of each year. Hmm, so each deposit will earn interest for a different number of years.
... | true | true | stop | 2,015 |
algebra__8dd85b | 8 | 5 | algebra | 10 | .
Okay, let's see. The problem is about finding the minimal compound interest rate a worker needs to have at least $66,200 in his savings account after three years. He deposits $20,000 each year at the end of each year. So, it's a compound interest problem with annual deposits.
First, I need to recall the formula for ... | true | true | stop | 1,208 |
algebra__8dd85b | 9 | 5 | algebra | 10 | .
Okay, let's see. So the problem is about a worker who deposits $20,000 at the end of each year into a savings account, and after three years, the total should be at least $66,200. We need to find the minimal compound interest rate required. Hmm, compound interest... right, so each deposit earns interest for the remai... | true | true | stop | 1,775 |
algebra__8dd85b | 10 | 5 | algebra | 10 | .
Okay, so I need to figure out the minimal compound interest rate that the savings account must provide for the worker to have at least $66,200 by the end of the third year. Let me start by understanding the problem.
The worker deposits $20,000 each year at the end of the year. So, the first deposit is at the end of... | true | true | stop | 2,284 |
algebra__8dd85b | 11 | 5 | algebra | 10 | .
Okay, let's see. So the problem is about a worker who deposits $20,000 each year into a savings account, and by the end of the third year, he wants at least $66,200. We need to find the minimal compound interest rate required. Hmm, compound interest... so each deposit is made at the end of the year, and it compounds ... | true | true | stop | 1,960 |
algebra__8dd85b | 12 | 5 | algebra | 10 | .
Okay, let's see. The problem is about compound interest. A worker deposits $20,000 at the end of each year into a savings account, and after three years, he wants at least $66,200. We need to find the minimal compound interest rate required. Hmm.
First, I need to recall the formula for compound interest. Since the ... | true | true | stop | 1,793 |
algebra__8dd85b | 13 | 5 | algebra | 10 | .
Okay, let's try to solve this problem step by step. So, the worker gets $20,000 each year and deposits it at the end of each year into a savings account. After three years, he wants at least $66,200. We need to find the minimal compound interest rate required for this to happen. Alright, let's break this down.
Firs... | false | false | length | 31,744 |
algebra__8dd85b | 14 | 5 | algebra | 10 | .
Okay, so I need to figure out the minimum compound interest rate a worker needs so that after three years, with annual deposits of $20,000 each, the total in the account is at least $66,200. Hmm, let's break this down step by step.
First, the worker makes three deposits: one at the end of year 1, another at the end... | true | true | stop | 2,070 |
algebra__8dd85b | 15 | 5 | algebra | 10 | .
Okay, let's see. The problem is about a worker who deposits $20,000 at the end of each year into a savings account with compound interest. After three years, he wants to have at least $66,200. We need to find the minimal compound interest rate required. Hmm, okay.
First, I need to recall how compound interest works... | true | true | stop | 1,853 |
counting_probability__a9b462 | 0 | 4 | counting_probability | 720 | .
Okay, let's see. The problem is about arranging 8 people around a round table, but with the condition that Pierre, Rosa, and Thomas all want to sit together. And we need to find the number of distinct ways they can sit, considering that rotations are considered the same.
Hmm, so circular permutations can be a bit tr... | false | false | length | 31,744 |
counting_probability__a9b462 | 1 | 4 | counting_probability | 720 | .
Okay, let's try to figure out how many ways 8 people can sit around a round table with Pierre, Rosa, and Thomas all sitting together. Hmm, circular permutations can be tricky because rotations are considered the same. Let me recall some concepts.
First, in circular permutations, the number of ways to arrange n peopl... | true | true | stop | 1,747 |
counting_probability__a9b462 | 2 | 4 | counting_probability | 720 | and without any leading or trailing spaces.
Okay, so I need to figure out how many ways 8 people can sit around a round table with Pierre, Rosa, and Thomas all sitting together. Hmm, circular permutations can be a bit tricky because rotations are considered the same. Let me start by recalling the basics.
First, for ... | true | true | stop | 3,830 |
counting_probability__a9b462 | 3 | 4 | counting_probability | 720 | .
Okay, so I need to figure out how many ways 8 people can sit around a round table if Pierre, Rosa, and Thomas all want to sit together. Hmm, circular permutations can be a bit tricky because rotations are considered the same. Let me start by recalling some basics.
First, if there are n people sitting around a round... | true | true | stop | 3,910 |
counting_probability__a9b462 | 4 | 4 | counting_probability | 720 | .
Okay, let's see. The problem is asking how many ways 8 people can sit around a round table if Pierre, Rosa, and Thomas all want to sit together. Hmm, circular permutations can be a bit tricky because rotations are considered the same. Let me think step by step.
First, when dealing with circular arrangements, the num... | false | false | length | 31,744 |
counting_probability__a9b462 | 5 | 4 | counting_probability | 720 | .
Okay, so I need to figure out how many ways 8 people can sit around a round table with Pierre, Rosa, and Thomas all sitting together. Hmm, circular permutations can be tricky because rotations are considered the same. Let me start by recalling some basics.
First, for a circular arrangement of n people, the number of... | false | false | length | 31,744 |
counting_probability__a9b462 | 6 | 4 | counting_probability | 720 | .
Okay, let's see. The problem is about arranging 8 people around a round table, but three of them—Pierre, Rosa, and Thomas—want to sit together. And we need to find the number of distinct seating arrangements considering that rotations are considered the same. Hmm, circular permutations can be tricky because rotation... | true | true | stop | 1,256 |
counting_probability__a9b462 | 7 | 4 | counting_probability | 720 | .
Okay, so I need to figure out how many ways 8 people can sit around a round table if Pierre, Rosa, and Thomas all want to sit together. Hmm, circular permutations can be a bit tricky because rotations are considered the same. Let me recall how circular permutations work. For n people around a round table, the number... | true | true | stop | 901 |
counting_probability__a9b462 | 8 | 4 | counting_probability | 720 | .
Okay, let's see. The problem is about finding the number of ways 8 people can sit around a round table with Pierre, Rosa, and Thomas all sitting together. Hmm, circular permutations can be tricky because rotations are considered the same. Let me break it down step by step.
First, I remember that when arranging peop... | false | true | stop | 2,236 |
counting_probability__a9b462 | 9 | 4 | counting_probability | 720 | .
Okay, so I need to figure out how many ways 8 people can sit around a round table with Pierre, Rosa, and Thomas all sitting together. Hmm, circular permutations can be a bit tricky because rotations are considered the same. Let me think step by step.
First, I remember that for circular arrangements, the number of wa... | true | true | stop | 1,276 |
counting_probability__a9b462 | 10 | 4 | counting_probability | 720 | .
Okay, so I need to figure out how many ways 8 people can sit around a round table if Pierre, Rosa, and Thomas all want to sit together. Hmm, circular permutations can be a bit tricky because rotations are considered the same. Let me start by recalling some basics.
First, for circular arrangements, the number of way... | true | true | stop | 1,958 |
counting_probability__a9b462 | 11 | 4 | counting_probability | 720 | .
Okay, let's see. The problem is about arranging 8 people around a round table with the condition that Pierre, Rosa, and Thomas all want to sit together. Hmm. So, round table problems usually involve circular permutations where rotations are considered the same. But when there's a condition like certain people needin... | false | false | length | 31,744 |
counting_probability__a9b462 | 12 | 4 | counting_probability | 720 | .
Okay, so I need to figure out how many ways 8 people can sit around a round table with Pierre, Rosa, and Thomas all sitting together. Hmm, circular permutations can be a bit tricky because rotations are considered the same. Let me start by recalling the basic formula for circular permutations. If there are n people ... | true | true | stop | 537 |
counting_probability__a9b462 | 13 | 4 | counting_probability | 720 | .
Okay, so I need to figure out how many ways 8 people can sit around a round table if Pierre, Rosa, and Thomas all want to sit together. Hmm, circular permutations can be tricky because rotations are considered the same. Let me think.
First, if all the people were sitting around a round table without any restriction... | true | true | stop | 639 |
counting_probability__a9b462 | 14 | 4 | counting_probability | 720 | .
Okay, let's see. The problem is asking how many ways 8 people can sit around a round table if Pierre, Rosa, and Thomas all want to sit together. Hmm. So, rotations are considered the same, right? So, for circular permutations, the formula is usually (n-1)! because rotations are considered identical. But here, there's... | false | false | length | 31,744 |
counting_probability__a9b462 | 15 | 4 | counting_probability | 720 | .
Okay, so I need to figure out how many ways 8 people can sit around a round table where Pierre, Rosa, and Thomas all want to sit together. Hmm, circular permutations can be tricky because rotations are considered the same. Let me start by recalling some basics.
First, for a circular table with n people, the number ... | true | true | stop | 3,238 |
prealgebra__7e0761 | 0 | 2 | prealgebra | 12 | .
Okay, so I need to find the length of each side of a regular octagon that has the same perimeter as a regular hexagon with a side length of 16 cm. Let me think about how to approach this.
First, I remember that the perimeter of a regular polygon is the number of sides multiplied by the length of each side. So for t... | true | true | stop | 423 |
prealgebra__7e0761 | 1 | 2 | prealgebra | 12 | .
Okay, let's try to figure out this problem. So, we have a regular hexagon with a side length of 16 cm, and a regular octagon that has the same perimeter as the hexagon. We need to find the length of each side of the octagon. Hmm, let me start by recalling some basic formulas.
First, the perimeter of a regular polygo... | true | true | stop | 445 |
prealgebra__7e0761 | 2 | 2 | prealgebra | 12 | .
Okay, let's see. The problem is about a regular octagon and a regular hexagon having the same perimeter. The hexagon has a side length of 16 cm, and we need to find the side length of the octagon. Hmm, regular polygons, so all sides are equal.
First, I remember that the perimeter of a regular polygon is the number... | true | true | stop | 384 |
prealgebra__7e0761 | 3 | 2 | prealgebra | 12 | .
Okay, let's see. The problem says a regular octagon has the same perimeter as a regular hexagon with side length 16 cm. I need to find the length of each side of the octagon.
First, I remember that the perimeter of a regular polygon is the number of sides multiplied by the length of each side. So for the hexagon, si... | true | true | stop | 381 |
prealgebra__7e0761 | 4 | 2 | prealgebra | 12 | .
Okay, let's see. The problem is about a regular octagon and a regular hexagon that have the same perimeter. The hexagon has a side length of 16 cm, and we need to find the side length of the octagon. Hmm, regular polygons mean all sides are equal, right?
First, I should recall the formula for the perimeter of a reg... | true | true | stop | 386 |
prealgebra__7e0761 | 5 | 2 | prealgebra | 12 | .
Okay, so I have this problem where a regular octagon has the same perimeter as a regular hexagon with each side being 16 cm. I need to find the length of each side of the octagon. Let me think about how to approach this.
First, I know that the perimeter of a regular polygon is the number of sides multiplied by the ... | true | true | stop | 463 |
prealgebra__7e0761 | 6 | 2 | prealgebra | 12 | .
Okay, so I need to find the length of each side of a regular octagon that has the same perimeter as a regular hexagon with side length 16 cm. Let me start by recalling what a regular polygon is. A regular polygon has all sides equal and all angles equal. So, for both the hexagon and the octagon, all their sides are ... | true | true | stop | 480 |
prealgebra__7e0761 | 7 | 2 | prealgebra | 12 | .
Okay, let's see. The problem says that a regular octagon has the same perimeter as a regular hexagon with a side length of 16 cm. I need to find the length of each side of the octagon. Hmm, right.
First, I remember that the perimeter of a regular polygon is just the number of sides multiplied by the length of one s... | true | true | stop | 365 |
prealgebra__7e0761 | 8 | 2 | prealgebra | 12 | .
Okay, let's see. The problem says that a regular octagon has the same perimeter as a regular hexagon with side length 16 cm. I need to find the length of each side of the octagon. Hmm, right. So first, I should remember how the perimeter of regular polygons works. For any regular polygon, the perimeter is the number ... | true | true | stop | 463 |
prealgebra__7e0761 | 9 | 2 | prealgebra | 12 | .
Okay, so I have this problem where a regular octagon has the same perimeter as a regular hexagon with side length 16 cm. I need to find the length of each side of the octagon. Hmm, let's think about this step by step.
First, I remember that the perimeter of a regular polygon is the number of sides multiplied by the ... | true | true | stop | 400 |
prealgebra__7e0761 | 10 | 2 | prealgebra | 12 | .
Okay, so I need to find the length of each side of a regular octagon that has the same perimeter as a regular hexagon with side length 16 cm. Hmm, let me think.
First, let's recall what a regular octagon is. It's an eight-sided polygon with all sides equal and all internal angles equal. Similarly, a regular hexago... | true | true | stop | 473 |
prealgebra__7e0761 | 11 | 2 | prealgebra | 12 | .
Okay, let's see. The problem is about a regular octagon and a regular hexagon with the same perimeter. The hexagon has a side length of 16 cm, and I need to find the side length of the octagon. Hmm, regular polygons mean all sides are equal, right? So first, maybe I should figure out the perimeter of the hexagon.
Al... | true | true | stop | 406 |
prealgebra__7e0761 | 12 | 2 | prealgebra | 12 | .
Okay, let's see. The problem is about a regular octagon and a regular hexagon that have the same perimeter. The hexagon has side length 16 cm, and we need to find the side length of the octagon. Hmm, regular polygons have all sides equal, so their perimeters are just the number of sides multiplied by the side length... | true | true | stop | 409 |
prealgebra__7e0761 | 13 | 2 | prealgebra | 12 | .
Okay, let's try to solve this problem. So, we have a regular hexagon with side length 16 cm, and a regular octagon that has the same perimeter as this hexagon. We need to find the length of each side of the octagon. Hmm, okay. Let me start by recalling some basics about regular polygons.
First, a regular polygon ha... | true | true | stop | 524 |
prealgebra__7e0761 | 14 | 2 | prealgebra | 12 | .
Okay, so I have this problem where a regular octagon has the same perimeter as a regular hexagon that has side length 16 cm. I need to find the length of each side of the octagon. Hmm, let's see. Let me start by recalling some basic formulas.
First, the perimeter of a regular polygon is the number of sides multipli... | true | true | stop | 354 |
prealgebra__7e0761 | 15 | 2 | prealgebra | 12 | .
Okay, let's see. The problem is about a regular octagon and a regular hexagon having the same perimeter. The hexagon has a side length of 16 cm, and we need to find the side length of the octagon. Hmm, okay.
First, I need to recall the formula for the perimeter of a regular polygon. Since both are regular, their pe... | true | true | stop | 385 |
algebra__160786 | 0 | 2 | algebra | \frac{3}{2} | .
Okay, let's see. The problem is asking, "What power of 4 is equal to 8?" And I need to express the answer as a common fraction. Hmm, power of 4... So that means I need to find an exponent n such that 4 raised to the nth power equals 8. Right?
First, let's write down the equation: 4^n = 8. I know that 4 is 2 squared,... | true | true | stop | 490 |
algebra__160786 | 1 | 2 | algebra | \frac{3}{2} | .
The answer to this problem is 2. | false | false | stop | 11 |
algebra__160786 | 2 | 2 | algebra | \frac{3}{2} | .
Okay, let's see. The problem is asking what power of 4 equals 8, and I need to express the answer as a common fraction. Hmm, so I need to find an exponent n such that 4 raised to the power of n equals 8.
First, I know that 4 is 2 squared, right? So maybe I can rewrite 4 as 2². Let me try that. So 4^n becomes (2²)^... | true | true | stop | 707 |
algebra__160786 | 3 | 2 | algebra | \frac{3}{2} | .
Okay, so I need to figure out what power of 4 equals 8. Hmm, let's see. Powers of 4 are like 4 multiplied by itself a certain number of times. Let me start by writing down some powers of 4 and see where 8 fits in.
4^1 is 4, right? That's just 4. Then 4^2 would be 4 times 4, which is 16. Oh, wait, 16 is already bigg... | true | true | stop | 402 |
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