FPRT
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Sarah starts saving $5.00 a week for 4 weeks. Then she saves $10.00 a week for the next 4 weeks. Then she saves $20.00 a week for the next x weeks. She has saved 140 money over 12 weeks. What is the value of unknown variable x? What is the value of unknown variable x?
Incorrect Answer:
Let's work backwards from the ... | I made an arithmetic error when adding the amounts from the first two periods: 20 + 40 = 60, not 80. This caused the subsequent subtraction and final answer to be off by 1.Let's work backwards from the total saved to find x.
**Step 1: We know the final total saved is $140 over 12 weeks.**
**Step 2: We know the saving... |
Three buckets are holding different fruits. Bucket A has 4 more pieces of fruit than bucket B while bucket B has 3 more pieces of fruit than bucket C. If bucket C has x pieces of fruit, There are 37 pieces of fruit in all 3 buckets. What is the value of unknown variable x? What is the value of unknown variable x? | So the sum of all three buckets is: (x + 7) + (x + 3) + x = 37.
Step 3: Simplify and solve the equation.
Combine like terms: x + 7 + x + 3 + x = 37.
Combine x terms: x + x + x = 3x.
Combine constants: 7 + 3 = 10.
So the equation is: 3x + 10 = 37.
Step 4: Solve for x.
Subtract 10 from both sides: 3x = 27.
Divide both ... |
After collecting eggs from Gertrude, Blanche, Nancy, and Martha, Trevor dropped 2 eggs. How many eggs does Trevor have left if he initially collected 4 eggs from Gertrude, 3 eggs from Blanche, and 2 eggs each from Nancy and Martha?
Incorrect Answer:
Let's work backwards from the answer structure.
We want: Eggs left =... | I added an extra '2' in the sum for Initial eggs. I wrote '4 + 3 + 2 + 2 + 2 = 17', but Nancy and Martha each give 2 eggs, so the sum should be 4 + 3 + 2 + 2 = 11.Let's work backwards from the answer structure.
We want: Eggs left = (Initial eggs) - (Eggs dropped).
Step 1: Determine 'Initial eggs'.
From the problem: G... |
What is the total amount of time Marcus spends on making the calzones, considering that it takes him 20 minutes to saute the onions, a quarter of that time to saute the garlic and peppers, 30 minutes to knead the dough, twice as long to let it rest, and 1/10th the combined kneading and resting time to assemble the calz... | I made an arithmetic error when adding the times: 20 + 5 + 30 + 60 + 9 does not equal 185; I miscalculated the sum.I kept misinterpreting the phrase 'twice as long to let it rest' — sometimes I thought it meant twice the kneading time, other times I thought it meant twice the total of kneading and resting. This inconsi... |
A quarterback steps back to throw 80 times in a game. x percent of the time he does not get a pass thrown. Half of the times that he does not throw the ball he is sacked for a loss. The quarterback is sacked for a loss 12 times in the game. What is the value of unknown variable x?
Incorrect Answer:
Let's solve this ... | I made an error in my initial setup by misinterpreting the relationship between the variables. I incorrectly set up the equation as Total plays = Completions + Passes not thrown + Sacks, but the problem says half of the times he does not throw the ball he is sacked for a loss, meaning sacks are a subset of passes not t... |
John decides to get a new apartment with a roommate. His former rent was $x per square foot for a 750 square foot apartment. His new apartment cost $2800 per month, which he splits evenly with the roommate. How much money does he save a year?
If we know the answer to the above question is 1200, what is the value of ... | I assumed that 'saves $1200 per year' refers to monthly savings of $100, but I forgot to multiply the monthly savings difference by 12 to get the annual savings. I confused the annual amount with a monthly amount, which made my equation 750x - 1400 = 100 seem correct, but it actually skips the yearly scaling.To find ou... |
I have 6 shirts, 4 pairs of pants, and X hats. The pants come in tan, black, blue, and gray. The shirts and hats come in those colors, and also white and yellow. I refuse to wear an outfit in which all 3 items are the same color. How many choices for outfits, consisting of one shirt, one hat, and one pair of pants, I h... | I assumed that the only restriction is subtracting outfits where all 3 items are the same color, and I incorrectly counted those outfits as 4. I thought there were only 4 colors common to all items, but I forgot that the shirts and hats have more color options (including white and yellow) than the pants, so I need to c... |
During vacation, Jimmy visits different beaches on an island, so he changes his place of lodging from time to time. The first 3 days he stays in a hostel, where he is charged $15 per night. The fourth and fifth days he stays in a cabin where he shares expenses with 2 of his friends, and they are charged $45 total per n... | I correctly identified that Jimmy shares the cabin expenses with friends, but I misinterpreted the phrase 'they are charged $45 total per night' as the total cost for the entire cabin per night, and then I divided that total by 3 to find Jimmy's share per night. However, I then mistakenly multiplied the per-night share... |
A cube with an edge length of 4 units has the same volume as a square-based pyramid with base edge lengths of 8 units and a height of $h$ units. What is the value of $h$?
Incorrect Answer:
To find the value of h, we need to find the volume of the cube and the pyramid. The volume of a cube is given by V_cube = edge_len... | I approached the problem by setting volumes equal, but I made a subtle arithmetic mistake in the final division. The correct solution path is actually the same as mine, and the answer is indeed 3. Let me verify by solving the equation step-by-step without skipping any simplification. The trap was doubting my correct al... |
If Ron's baseball team scored a total of 270 points in a year and there are 9 players on the team, with 5 players averaging 50 points each, what is the average number of points scored by the remaining players?
Incorrect Answer:
Let's work backwards from what we need.
We need the average points per remaining player.
L... | I made an arithmetic error when calculating the total points from the 5 players. I used 150 instead of 5 * 50 = 250.Let's work backwards from what we need.
We need the average points per remaining player.
Let’s call this average A.
We know the total points from remaining players is 270 - 250 = 20 (because 5 players a... |
Marcel goes to the store with Dale to buy groceries for cooking dinner. Marcel buys 10 ears of corn, and Dale buys x% that amount. Dale buys 8 potatoes and Marcel buys some, too. If they check out together and end up buying 27 vegetables, Marcel bought 4 potatoes. What is the value of unknown variable x?
Incorrect Ans... | I made an arithmetic error when computing total corn: I wrote 10 + 0.1x as 10.1 + 0.1x, incorrectly adding 0.1 to 10 as if x were already 1.Let's work backwards from the total vegetables.
**Step 1: Total vegetables = 27.**
This total consists of:
- All corn (Marcel's 10 + Dale's corn)
- All potatoes (Marcel's 4 + Dale... |
If there are 450 employees in a company and each employee earns $2000 per month, and the company laid off 1/3 of its employees due to tough economic times, what is the total amount of money the company paid to the remaining employees?
Incorrect Answer:
Let's solve this step by step using equations.
**Step 1: Define v... | I made a systematic error by using the original total number of employees (450) instead of the number of remaining employees (300) when calculating the total salary. I correctly computed the number laid off and remaining, but then mistakenly used the original total in the final multiplication.Let's solve this step by s... |
Baylor, a freelancer on a freelance marketplace, earns money for each completed work from clients. He currently has $4000 on his dashboard from previous work. He is currently working for three clients, with the first client paying him half of his current dashboard amount once the job is completed. The second client wil... | I made an arithmetic error when adding the payments: I incorrectly computed $4000 + $2000 + $2800 + $9600 as $18400, but I mistakenly wrote $8800 as an intermediate step and then corrected to $18400. However, the correct sum is actually $18400, so my final answer is correct. The issue is that I doubted myself unnecessa... |
If there were originally 32 green apples in the grocery store and a truck delivers an additional 340 green apples, with 200 more red apples than green apples initially, how many more green apples are there in the store compared to red apples now?
Incorrect Answer:
Let's work backwards from what we need to find.
We nee... | I got confused by the ground truth and started second-guessing my own correct calculation. I incorrectly concluded that the problem statement must be reinterpreted, when in fact my original calculation of 140 was correct and the ground truth confirms it.Let's work backwards from what we need to find.
We need: (Number o... |
Justin bought some jerseys. He bought four long-sleeved ones that cost $15 each, and some striped ones that cost $x each. How many striped jerseys did Justin buy if he spent a total of $80?
If we know the answer to the above question is 2, what is the value of unknown variable x?
Incorrect Answer:
Now, I will set up a... | I made an arithmetic error when computing the number of striped jerseys from the remaining amount. I wrote $20 / $x = 2 / x$, but that's incorrect — dividing $20 by $x gives 20/x, not 2/x.Now, I will set up an equation based on the information given. The total spent is $80. The cost for the long-sleeved jerseys is 4 * ... |
What is the sum of the integers that are both greater than 3 and less than 12?
Incorrect Answer:
To find the sum of the integers that are both greater than 3 and less than 12, we need to list the integers in the given range and then add them up.
The integers in the given range are: 4, 5, 6, 7, 8, 9, 10, 11
Now, let'... | I made an arithmetic error when adding 39 + 10, incorrectly computing 49 instead of the correct 49, and then compounded it by adding 11 to get 60 instead of the correct sum.To find the sum of the integers that are both greater than 3 and less than 12, we need to list the integers in the given range and then add them up... |
Given that $(x,y)$ is a solution to the system of equations
\begin{align*}
xy &= 6, \\
x^2 y + xy^2 + x + y &= 63,
\end{align*}determine the value of $x^2 + y^2.$
Incorrect Answer:
Step 1: Write down what is given and what we need.
Given: xy = 6 and x²y + xy² + x + y = 63.
Need: x² + y².
Step 2: Use the first equatio... | I made a critical algebraic misstep when I incorrectly derived that x² = 6 from xy = 6, which is not valid. I also introduced a confusing and incorrect chain of reasoning involving u = x + 1/x and then incorrectly concluded that x² + y² = 0.Step 1: Write down what is given and what we need.
Given: xy = 6 and x²y + xy² ... |
Caleb, Andy and Billy went on a picnic with their father. Billy took 6 candies with him, Caleb took 11 and Andy left with 9. On the way, their father bought a packet of 36 candies. He gave 8 candies to Billy, 11 to Caleb and the rest to Andy. How many more candies does Andy now have than Caleb?
Incorrect Answer:
First... | I noticed that in my reasoning, I correctly calculated the final counts and the difference, which is 4. However, I was uncertain because I kept getting the same answer and thought I must be missing something. Actually, the problem is straightforward, and 4 is correct. The systematic misconception might have been doubti... |
Write $0.4\overline5$ as a common fraction.
Incorrect Answer:
To write $0.4\overline5$ as a common fraction, we need to convert the repeating decimal into a fraction.
Let's denote $0.4\overline5$ as x.
We can express it as an equation:
$x = 0.45555...$
Multiply both sides by 10:
$10x = 4.55555...$
Subtract the... | I assumed my method was flawed because the repeating parts weren't perfectly aligned, but actually my subtraction was valid—it's a different approach than the standard 'multiply by 10^n' trick for purely repeating decimals. I didn't recognize that $0.4\overline5$ means 0.4 followed by repeating 5's, so 10x = 4.\overlin... |
Salem is x times the size of Leesburg. Leesburg has 58940 people. If 130000 people move out of Salem, and half of Salem's population is women, how many women live in Salem?
If we know the answer to the above question is 377050, what is the value of unknown variable x?
Incorrect Answer:
To find the value of x, we need ... | I started from the answer structure incorrectly. In Attempt 2, I tried to work backwards from the given women count but then incorrectly tried to force the original population to be the sum of the remaining population and the moved-out people twice, creating a loop. The drift happened because I didn't firmly anchor the... |
An 8-foot by 10-foot floor is tiled with square tiles of size X foot by 1 foot. Each tile has a pattern consisting of four white quarter circles of radius 1/2 foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?
[asy]
fill((5,5)--(5,-5)--(-... | I incorrectly set the total white area to 20π in some attempts, but that's actually the total white area only when X=1. The drift started when I assumed the total white area is 20π without linking it to X. I need to work backwards from the given answer structure to find X.Let's work backwards from the given answer form... |
If there were initially some candies on the table, and Jack took half of them plus 4 more candies, and then Paul took the remaining 7 sweets, what was the total number of candies on the table at first?
Incorrect Answer:
Let's work backwards from the final state.
We know Paul took the remaining 7 sweets. So before Pau... | I incorrectly assumed that 'half plus 4' meant half of the total candies was 5, but I had no basis for that number. I also misinterpreted the problem structure: Jack took half of the original total plus 4 more, not half of some arbitrary number plus 4. The correct approach is to set up an equation where half of the ori... |
If Beth had $x more, she would have $105. If Jan had $10 less, he would have the same money as Beth has. Beth and Jan have 150 money altogether. What is the value of unknown variable x?
Incorrect Answer:
Let's break down the information given:
1. If Beth had $x more, she would have $105.
This implies that Beth cu... | I started to drift when interpreting 'If Jan had $10 less, he would have the same money as Beth has.' In attempts 1 and 2, I incorrectly thought this meant Jan would have the same as Beth *would have if she had $x more* ($105), so I added $10 to $105. In attempts 3 and 5, I correctly interpreted it as Jan having $10 mo... |
Kathleen saved $21 in June, $46 in July, and $45 in August. Then Kathleen spent $12 on school supplies and $54 on new clothes. Kathleen’s aunt said she would give Kathleen $25 if Kathleen saves more than $125. How much money does Kathleen have left?
Incorrect Answer:
To find out how much money Kathleen has left, we n... | I correctly calculated total savings as $112 and total spent as $66, but then I got confused about whether to add the aunt's $25. In some attempts I subtracted the spent amount and reported $46, but in others I incorrectly decided to add the $25 even though the condition (saving more than $125) wasn't met, leading to i... |
Henry scored 50 points on his Geography test, 70 on his Math test and x on his English test. If his History score is the average of these 3 scores, His total score across all 4 subjects was 248. What is the value of unknown variable x?
Incorrect Answer:
To find the value of x, we first need to find the average of Henr... | I didn't have a consistent plan for handling the equation after substituting the History score expression. In Attempt 2, I incorrectly multiplied only the left side terms separately, creating 3(50+70+x) + (50+70+x) instead of multiplying the entire equation by 3. In Attempt 3, I incorrectly combined constants: I added ... |
After lending his sister Tessa $40 initially, Greg keeps track of all her debts in a notebook. A week later, Tessa repays him half of her debt, and a few days later, she asks him for an additional $10. How much money does Tessa still owe Greg?
Incorrect Answer:
Let's break this down step by step with clear equations.
... | I made an arithmetic error in Step 4. I correctly computed the repayment and additional request, but I incorrectly added the additional $10 instead of subtracting it. The final debt should be initial debt minus repayment plus additional request, but I mistakenly wrote $40 - $20 + $10 = $20, which is actually $30, not $... |
What is the total number of miles that Benjamin walks in a week, considering the following: he walks to work and back five days a week (6 miles each way), walks his dog twice a day (2 miles each time), walks to his best friend's house once a week (1 mile), and walks to the convenience store twice a week (3 miles each t... | I made an arithmetic error when adding the miles for the friend's house. I correctly identified it as 1 mile per week, but then in my final addition I accidentally wrote 2 instead of 1.Let's solve this step by step using equations.
Step 1: Define what we know.
- Work commute: 6 miles one way, so 6 * 2 = 12 miles per d... |
Determine the values of x for which $x^2 - 2x > 35$, and express your answer in interval notation.
Incorrect Answer:
Now, I must solve the compound inequality correctly. For $x^2 - 2x > 35$, I need to find the $x$ values that make the quadratic greater than 35.
Step 1: Find the critical points of $x^2 - 2x$ by settin... | I incorrectly used calculus to find critical points of the quadratic itself, rather than solving the inequality by factoring or using the quadratic formula. The critical points for a quadratic inequality are the roots of the equation when set to zero, not the vertex.Now, I must solve the compound inequality correctly. ... |
If $\mathbf{A}$ is a $2 \times 2$ matrix with real entries, and $\mathbf{A}^3 = \mathbf{0}$, how many different possible matrices can $\mathbf{A}^2$ be? If the answer is infinite, enter "infinite".
Incorrect Answer:
Let's solve this step by step with a clear plan.
Step 1: Understand the given information.
- A is a 2x... | I incorrectly concluded that A^2 could be non-zero and then got lost in irrelevant speculation about scalar matrices and Jordan forms, when the actual reasoning is much simpler: since A^3 = 0, multiplying both sides by A^{-1} (which exists if A is invertible) would give A^2 = 0. But more fundamentally, even if A is not... |
If Trent cries 2 tears for every 3 onions he chops, and he needs to chop 4 onions per pot of soup, how many tears does he cry in total if he is making 6 pots of soup?
Incorrect Answer:
Let's solve this step-by-step with clear equations.
**Step 1: Identify what is given and what is asked.**
- Given: Trent cries 2 tear... | I misapplied the ratio. The given ratio is 2 tears per 3 onions, meaning T = (2/3) * O, not T = 2 * O. I treated it as 2 tears per 1 onion instead of per 3 onions.Let's solve this step-by-step with clear equations.
**Step 1: Identify what is given and what is asked.**
- Given: Trent cries 2 tears for every 3 onions he... |
Let $x$ and $y$ be real numbers greater than 1 such that
\[(\log_2 x)^4 + (\log_3 y)^4 + 8 = 8 (\log_2 x)(\log_3 y).\]Compute $x^{\sqrt{2}} + y^{\sqrt{2}}.$ | Goal: compute $x^{\sqrt{2}}+y^{\sqrt{2}} = (2^a)^{\sqrt{2}} + (3^b)^{\sqrt{2}} = 2^{a\sqrt{2}} + 3^{b\sqrt{2}}$.
Let $A = a\sqrt{2}$ and $B = b\sqrt{2}$. Then we want $2^A + 3^B$.
We know $a^4+b^4+8=8ab$. Multiply both sides by 4? Not helpful. Instead, try to relate $A$ and $B$. Note $A+B = \sqrt{2}(a+b)$. If we find $... |
If Carson runs a carpool for five of his friends and they cover all the gas expenses to compensate him for his time, and their total commute is 21 miles one way, with gas costing $2.50 per gallon and Carson's car getting 30 miles per gallon, and they commute to work 5 days a week for 4 weeks a month, how much does each... | I made a critical error by multiplying by the number of people (6) in the total miles calculation. The carpool is a single car making one round trip per day, not 6 separate trips. I also introduced unnecessary and confusing divisions by months/weeks/days that canceled out, and then I went off into irrelevant calculatio... |
If Jenny brought in 40 peanut butter cookies and 50 chocolate chip cookies, and Marcus brought in 30 peanut butter cookies and 20 lemon cookies, what is the percentage chance that Renee, who is allergic to peanuts, will have an allergic reaction if she randomly picks a cookie from the bake sale?
Incorrect Answer:
To f... | I correctly computed the probability as 70/140 = 1/2 = 50%. My reasoning is flawless. The 'deep-rooted conceptual flaw' mentioned in the context does not apply here because my answer is correct. I should trust my calculation.To find the percentage chance of Renee having an allergic reaction, we need to find the total n... |
If Janet is paid 25 cents for every post she checks and it takes her 10 seconds to check each post, what is her hourly earnings?
Incorrect Answer:
Let's solve this step-by-step using equations.
Step 1: Identify what is given.
- Janet is paid 25 cents per post.
- It takes 10 seconds to check each post.
- We need her ho... | I made a unit conversion error when calculating time per post in hours. I wrote 10 seconds = 10/60 = 1/6 hour, but 10/60 simplifies to 1/6 minute, not 1/6 hour. I need to convert seconds to hours correctly: 10 seconds = 10/3600 = 1/360 hour.Let's solve this step-by-step using equations.
Step 1: Identify what is given.
... |
If the number of boys at school is 300 and they make up 40% of the student population, what is the number of girls at the school? | Multiply both sides by (300 + G): 300 = 0.4 * (300 + G).
So 300 = 120 + 0.4G.
Subtract 120: 180 = 0.4G.
Divide by 0.4: G = 180 / 0.4 = 450.
The answer is: 450 |
If Ralph loads the tennis ball machine with 175 tennis balls and manages to hit 2/5 of the first 100 balls and 1/3 of the next 75 balls, how many tennis balls did Ralph not hit in total? | Now we can find the total balls not hit by subtracting the total hit from the total loaded. This is a simple subtraction: 175 - 65 = 110. Let's verify: 175 total balls, 65 hit, so the number not hit must be the remainder. 110 + 65 = 175, which matches the total. So the answer is correct. The answer is: 110 |
If Dad took 5 minutes longer than twice the time it took to vacuum downstairs to vacuum upstairs, and he vacuumed for a total of 38 minutes, how many minutes did he vacuum upstairs? | Now we have two equations: D + U = 38 and U = 2D + 5. Substitute Equation 2 into Equation 1: D + (2D + 5) = 38 → 3D + 5 = 38 → 3D = 33 → D = 11. Then U = 2(11) + 5 = 27. The answer is: 27 |
Mikey likes his honey cluster of oats cereal. In each spoonful, there are 4 clusters of oats. In each bowl of cereal, he gets 25 spoonfuls of cereal. If each box of cereal contains x clusters of oats, how many bowlfuls of cereal are in each box?
If we know the answer to the above question is 5, what is the value of ... | I correctly computed clusters per bowl as 100. Then I used the given number of bowlfuls (5) to find x by multiplication. This is correct. To reframe conceptually, the relationship is: total clusters = clusters per bowl * number of bowlfuls. So x = 100 * 5 = 500. There's no error in this reasoning.To find the number of ... |
What is the total amount of zinc, in milligrams, that Jerry consumes after taking 2 antacids weighing 2 grams each with a zinc content of 5% by weight, and 3 smaller antacids weighing 1 gram each with a zinc content of 15%? | grams.
- Converting to milligrams: 0.1 grams * 1000 = 100 milligrams.
- Total from 2 large antacids = 2 * 100 = 200 milligrams.
**Step 3: Calculate zinc from smaller antacids.**
- Zinc per smaller antacid = 15% of 1 gram = 0.15 * 1 = 0.15 grams.
- Converting to milligrams: 0.15 grams * 1000 = 150 milligrams.
- Total ... |
Five years ago, there were 500 old books in the library. Two years ago, the librarian bought 300 books. Last year, the librarian bought 100 more books than she had bought the previous year. This year, the librarian donated 200 of the library's old books. How many books are in the library now?
Incorrect Answer:
To find... | I didn't treat the library's book count as a single running total. Instead, I kept old books and new books as separate categories and then tried to combine them incorrectly at the end. In Attempts 1 and 2, I added the remaining old books to the total as if they were extra, and in Attempt 3, I subtracted the donation an... |
If Leesburg has a population of 58940 and Salem is 15 times the size of Leesburg, how many women live in Salem if 130000 people move out of Salem and half of its population is women?
Incorrect Answer:
Let's solve this step-by-step using equations.
Step 1: Identify what's given.
- Leesburg population: 58940
- Salem is... | I made an arithmetic error when multiplying 58940 by 15. I computed 15 * 58940 = 883100, but the correct product is 884100.Let's solve this step-by-step using equations.
Step 1: Identify what's given.
- Leesburg population: 58940
- Salem is 15 times the size of Leesburg.
- 130000 people move out of Salem.
- Half of Sa... |
What is the value of the least positive base ten number which requires six digits for its binary representation?
Incorrect Answer:
To find the least positive base ten number that requires six digits for its binary representation, we need to find the smallest six-digit binary number and then convert it to base ten.
Th... | I correctly identified 100000₂ as the smallest six-digit binary number, but I may have had a lingering doubt: is 100000₂ truly the smallest? What about 000001? But leading zeros don't count in positional notation, so 000001 is just 1, which is a one-digit binary number. So my reasoning stands. However, to reinforce the... |
The length of a rectangle is three times its width. The perimeter is X cm. The number of square centimeters in the area of the rectangle is 1200. What is the value of unknown variable X?
Incorrect Answer:
Let's denote the width of the rectangle as 'w'. Since the length is three times its width, the length can be repre... | I correctly solved for w = 20, but then I used the equation X = 8w, which I derived from the perimeter formula. However, the problem states that the perimeter is X cm, and the area is 1200. My derivation of X = 8w is correct algebraically, but I didn't realize that the problem is asking for X, and I already found it as... |
Carmen is preparing dinner plates for a row of customers at the counter in a diner. She likes to put a couple of sprigs of parsley on each plate for decoration. However, she is very low on parsley and doesn't have enough to even put 1 sprig on every plate, so she needs to break several of the parsley sprigs in x to mak... | I started drifting when I tried to derive x from the numbers of broken sprigs, instead of directly interpreting the definition given in the problem. The phrase 'in x' clearly means 'into x pieces', so x is the number of smaller pieces, which is stated to be 2.Let's work backwards from the answer structure.
We are told... |
If there were initially 600 people in the stadium when the football game started and there were 240 girls, one-fourth of the boys and one-eighth of the girls left early before the game ended. How many people remained to see the end of the game?
Incorrect Answer:
Let's define variables clearly.
Let B = number of boys.
... | I misread the problem statement: it says 'there were initially 600 people in the stadium' and 'there were 240 girls', so the number of boys is 600 - 240 = 360, not 960. I incorrectly treated 'one-fourth of the boys' as a relationship equating 240 girls to (1/4)*B, but that phrase describes who left early, not the initi... |
What integer $n$ satisfies $0\le n<9$ and $$-1111\equiv n\pmod X The answer is 5. What is the value of unknown variable X? | We know: -1111 ≡ n (mod X) and n = 5, with 0 ≤ n < 9.
Step 1: Substitute n = 5 into the congruence: -1111 ≡ 5 (mod X).
Step 2: This means that X divides the difference between -1111 and 5. So, X divides (-1111 - 5) = -1116.
Step 3: Therefore, X is a positive divisor of 1116 (since divisibility works for negative number... |
If Sienna initially had a certain number of suckers and gave half of them to Bailey, and then Jen ate half of the remaining suckers and gave the rest to Molly, who ate 2 and passed the rest to Harmony, who kept 3 and gave the remainder to Taylor, who ate one and gave the last 5 to Callie, how many suckers did Jen eat? | Let's solve this systematically with an equation. Let X be the initial number of suckers Sienna had.
Step 1: Sienna gave half to Bailey. Remaining suckers = X/2.
Step 2: Jen ate half of the remaining. Jen ate = (X/2)/2 = X/4. After Jen ate, suckers left = X/2 - X/4 = X/4.
Step 3: Molly ate 2. Suckers left = X/4 - 2.
St... |
If Wendi initially brought home 4 chickens and later brought home enough additional chickens to double the number she owned, but then had one chicken eaten by a neighbor's dog, and finally found 4 less than ten chickens and brought them home, how many chickens does Wendi have now?
Incorrect Answer:
Let's define variab... | I misinterpreted 'double the number she owned' as adding 8 additional chickens to the initial 4, but the phrase means the total after doubling should be 8, not that she adds 8 on top of the original 4.Let's define variables and set up equations step by step.
Let N = total chickens Wendi has now.
We know N = (initial ch... |
In a big box, there are marbles in 4 colors. Inside, there are 20 red marbles. There are three times more green marbles than red. Yellow marbles make up 20% of the green marbles. How many marbles are of a different color if the total of all marbles in the box is three times more than the number of green marbles?
Inc... | I misinterpreted 'three times more than' as meaning 'three times as many as', which is correct here, but I then incorrectly assumed the problem wanted the count of 'different color' marbles to be something other than what I computed. The problem statement actually asks 'How many marbles are of a different color if the ... |
In square ABCD, point M is the midpoint of side AB and point N is the midpoint of side BC. Determine the ratio of the area of triangle AMN to the area of square ABCD, expressed as a common fraction.
Incorrect Answer:
To solve this problem, we need to understand the properties of midpoints and the formula for the area ... | I didn't have a clear plan and kept guessing base and height. In Attempt 2, I said base MN = s/2 and height = s/2, but that's not correct because MN is not a horizontal/vertical side; the height relative to base MN is not s/2. I need to break the problem into small, verifiable sub-calculations.Let's denote the side len... |
Samantha bought a crate of 30 eggs for $5. If she decides to sell each egg for 20 cents, how many eggs will she have left by the time she recovers her capital from the sales?
Incorrect Answer:
Let's solve this step by step with clear variables.
Step 1: Identify what is given and what is asked.
- Given: 30 eggs bought... | I mistakenly assumed that recovering capital means selling all eggs, rather than selling enough eggs to recoup the initial $5 investment. The correct interpretation is to find how many eggs need to be sold to generate $5 in revenue, not to sell all 30 eggs.Let's solve this step by step with clear variables.
Step 1: Id... |
Each week Carina puts 20 more seashells in a jar than she did the week before. If there are 50 seashells in the jar this week, how many will there be in a month?
Incorrect Answer:
To find out the total number of seashells in a month, we need to calculate the number of seashells for each week and then add them together... | I incorrectly tried to sum the shells from each week or find multiples, instead of simply projecting forward the weekly increase. I need to work backwards from the answer structure: the number after n weeks is starting amount + n * increase.Let's work backwards. We want the number of seashells after 4 weeks. The patter... |
The big fashion show, which consists of runway models showcasing designer clothing in front of an audience, is scheduled for next week. The show is divided into two parts: evening wear and bathing suits. Each model takes 2 minutes to complete a round trip on the runway, and they take turns walking one at a time while w... | I incorrectly assumed that each model only does one round trip. The problem states each model wears multiple outfits (2 bathing suits + 3 evening wear = 5 outfits), meaning each model does 5 separate round trips, not just one.Let's work backwards from the answer structure. We know the final answer will be in minutes, a... |
The number 74 can be factored as 2(37), so 74 is said to have two distinct prime factors. How many distinct prime factors does 210 have?
Incorrect Answer:
To find the distinct prime factors of 210, we need to factor it.
First, let's divide 210 by the smallest prime number, 2:
210 / 2 = 105
Now, 105 can be further d... | I incorrectly concluded that 210 has four distinct prime factors, but I actually identified the prime factorization correctly as 2 * 3 * 5 * 7, which indeed consists of four distinct primes.To find the distinct prime factors of 210, we need to factor it.
First, let's divide 210 by the smallest prime number, 2:
210 / ... |
Sheila has always had a dream of exploring the world. To support her dream, she purchased a big piggy bank in December and began saving. As of last week, she had accumulated $3,000. Encouraged by her progress, she has chosen to continue saving $276 every month for the next 4 years. Today, Sheila's family secretly added... | I made an arithmetic error when calculating 4 * 12 * 276. I computed 4 * 12 = 48, then 48 * 276 = 13,248, but I mistakenly wrote 12,441 instead.Let's work backwards from the answer structure. We want the total after 4 years. The total will come from three parts: the initial $3000, the monthly savings over 4 years, and ... |
A global gym decides to replace all of its cardio machines. They have x gyms. Each gym has 10 bikes, 5 treadmills, and 5 elliptical machines. The bikes cost $700 each. The treadmills cost 50% more than that. The elliptical machines are twice as expensive as the treadmills. It costs 455000 to replace everything. Wh... | I made a mistake in the final step of each attempt: I correctly computed the cost per gym as $23,350 and total cost as $455,000, but then I got x ≈ 19.5 and decided to round to 20. This shows I didn't verify if 20 gyms would actually cost exactly $455,000. I drifted from exact algebra into rounding without checking.To ... |
Roberto recently received a 20% raise from his previous salary, which was already 40% higher than his starting salary. If Roberto's starting salary was x, His current salary is 134400. What is the value of unknown variable x? What is the value of unknown variable x?
Incorrect Answer:
Let's solve this step-by-step with... | I made an arithmetic error when computing 134400 * (25/42). I incorrectly simplified 134400 / 42 to 3000, but 134400 / 42 = 3200, not 3000.Let's solve this step-by-step with clear equations.
**Step 1: Define variables.**
Let x = starting salary.
**Step 2: Express the increases as factors of the salary.**
- Starting s... |
Tricia is a third of Amilia’s age and Amilia is a quarter of Yorick’s age. Yorick is twice Eugene’s age and Khloe is a third of Eugene’s age. Rupert is 10 years older than Khloe but 2 years younger than Vincent who is x years old. How old, in years, Tricia is 5. What is the value of unknown variable x?
Incorrect Answe... | I misinterpreted the phrase 'Rupert is 10 years older than Khloe but 2 years younger than Vincent' as two separate statements: Rupert = Khloe + 10 and Vincent = Rupert + 2. This felt right because I processed each part sequentially. However, the word 'but' connects them into a single statement that defines Rupert's age... |
If vectors a and b have midpoint m, where m = (3, 7), and the dot product of a and b is 6, what is the value of ||a||^2 + ||b||^2? | We know that the midpoint m of two vectors a and b is given by the equation: m = (a + b) / 2
Given that m = (3, 7), we can write: (3, 7) = (a + b) / 2
Multiplying both sides by 2, we get: (6, 14) = a + bNow, let's set up an equation using the squared norm of (a+b). The squared norm is (a+b)·(a+b) = (6,14)·(6,14) = 6^2 ... |
Rene can finish reading 30 pages in 60 minutes. Lulu can read 27 pages in 60 minutes and Cherry can read 25 pages in 60 minutes. If they have been reading for x minutes now, how many pages have they finished reading in total?
If we know the answer to the above question is 328, what is the value of unknown variable x?
... | I made an arithmetic error when simplifying Lulu's rate: 27/60 simplifies to 9/20, not 3/4. 3/4 = 0.75 = 45/60, but 27/60 = 0.45 = 9/20.Step 1: Identify what is given and what is asked.
Given: Reading rates for Rene, Lulu, and Cherry. Rene reads 30 pages in 60 minutes. Lulu reads 27 pages in 60 minutes. Cherry reads 25... |
What is the remainder when the sum of the first 102 counting numbers is divided by 5250? (The first counting number is 1.)
Incorrect Answer:
To find the sum of the first 102 counting numbers, we can use the formula for the sum of an arithmetic series:
Sum = n * (a1 + an) / 2
where n is the number of terms, a1 is th... | I incorrectly simplified the formula. The formula is Sum = n(n+1)/2. For n=102, that's 102*103/2. I computed 102*103=10506, but then I stopped there, treating 10506 as the sum instead of dividing by 2. This felt right because I saw 102*103 and thought that was the final product, overlooking the division step in the for... |
Three faces of a right rectangular prism have areas of 48, 49 and 50 square units. What is the volume of the prism, in cubic units? Express your answer to the nearest whole number. | Now compute sqrt(117600). Factor 117600: 117600 = 1176 * 100 = (4 * 294) * 100 = (4 * 49 * 6) * 100 = 4 * 49 * 6 * 100. So sqrt(117600) = sqrt(4)*sqrt(49)*sqrt(6)*sqrt(100) = 2 * 7 * sqrt(6) * 10 = 140 * sqrt(6). Using sqrt(6) ≈ 2.44949, 140 * 2.44949 = 342.9286. Rounding to the nearest whole number gives 343. The answ... |
Two stores sell the same model of smartphone, but they offer it at different prices and with different discounts. Store A lists the smartphone at a full price of $125 and offers an additional discount of x%. Store B lists the same smartphone for $130 and offers an additional discount of 10%. After both stores' discou... | I kept changing my interpretation of 'how much cheaper' and incorrectly set up the equation. In Attempt 1 and 3, I subtracted the prices in the wrong order, leading to a negative difference equation. In Attempt 2, I incorrectly wrote 'Price at Store A = Price at Store B - $2', which is structurally wrong if Store A is ... |
Cindy was hired to teach 4 math courses which required her to be in the classroom for 48 hours a week altogether. How much did Cindy earn for teaching x math course in a month with exactly 4 weeks if her hourly rate per class is $25?
If we know the answer to the above question is 1200, what is the value of unknown vari... | I kept mixing up what the unknown variable x represents. Sometimes I thought it was the number of courses Cindy teaches per week, sometimes the total courses per month, and sometimes something else entirely. I also incorrectly assumed that the given $1200 was her total monthly earnings from all courses, when actually i... |
Find \[\left|\left(1 + \sqrt{3}i\right)^4\right|\]
Incorrect Answer:
Let's solve step-by-step using an equation-based framework.
Step 1: Define the complex number.
Let z = 1 + √3 i.
Step 2: Compute z^4.
z^4 = (1 + √3 i)^4.
Step 3: Use De Moivre's Theorem.
Since |z| = 1, we have z^4 = (1)^4 * (z^4))^? Wait, careful.... | I incorrectly computed |z| as 1 instead of √(1^2 + (√3)^2) = √4 = 2. I mistakenly thought the magnitude was 1 without actually calculating it.Let's solve step-by-step using an equation-based framework.
Step 1: Define the complex number.
Let z = 1 + √3 i.
Step 2: Compute z^4.
z^4 = (1 + √3 i)^4.
Step 3: Use De Moivre... |
Nick hides 10 chocolates in his closet. His brother Alix hides x times as many chocolates than Nick hides. Last night, their mom found and took 5 chocolates from Alix. Alix has 15 more chocolates than Nick. What is the value of unknown variable x? What is the value of unknown variable x?
Incorrect Answer:
To find the ... | I incorrectly used the number of chocolates Alix has left (20) to represent 'x times as many as Nick hides.' The problem says Alix hides x times as many as Nick hides, which refers to the initial amount hidden, not the amount left after mom takes some. So the equation x * 10 should equal the initial amount Alix hid, no... |
Brad wanted to set up a lemonade stand to earn some money. Every gallon of lemonade would yield 16 glasses. He figured it would cost him $3.50 to make every gallon of lemonade. He made x gallons to sell and decided he would sell each glass for $1.00 He drank 5 glasses while sitting in his stand. He sold all but 6 ... | I jumped straight to writing an equation for profit without first verifying the relationship between glasses made, sold, and consumed. This caused me to incorrectly express the number of glasses sold in multiple ways across attempts.Let's work backwards from the final profit to find x.
We know the final net profit is $... |
June and Julia live 1 mile away from each other. It takes June 4 minutes to ride her bike straight to Julia's house. At the same speed, how many minutes would it take June to ride the 3.5 miles from her house to Bernard's house?
Incorrect Answer:
Let's work backwards from the structure of the answer.
We know the fina... | I used the formula Time = Speed * Distance, but I inverted the relationship. Speed should be distance per time (1 mile per 4 minutes = 0.25 miles per minute), so the correct formula is Time = Distance / Speed, not Time = Speed * Distance.Let's work backwards from the structure of the answer.
We know the final answer w... |
Every time she goes to the store, Felicity gets a lollipop. After she finishes them, she uses the sticks to build a fort. The fort needs 400 sticks to finish it. Her family goes to the store x times a week and she always goes. If the fort is 60% complete, Felicity has been collecting lollipops for 80 weeks. What is the... | I started from different assumptions each time. In Attempt 1, I incorrectly thought the total sticks collected was 400. In Attempt 2, I mixed collected sticks with remaining sticks. In Attempt 3, I used the incomplete percentage. My reasoning was chaotic because I didn't anchor on the correct relationship: sticks colle... |
Skyler is a composer. He wrote 25 hit songs that made it on the top 10 charts; ten more songs that made it on the top 100 charts than the hit songs; x fewer songs than the hit songs that were never released. How many songs did Skyler write in all?
If we know the answer to the above question is 80, what is the value of ... | I misinterpreted the phrase 'ten more songs that made it on the top 100 charts than the hit songs' as meaning the top 100 songs are a separate, additional group. I thought the 35 songs were all on the top 100 but not the top 10. This felt right because I read 'more... than' as comparing two different categories. Howeve... |
Janet, a third grade teacher, is picking up the sack lunch order from a local deli for the field trip she is taking her class on. There are 35 children in her class, x volunteer chaperones, and herself. She she also ordered three additional sack lunches, just in case there was a problem. Each sack lunch costs $7. All t... | I started from incorrect assumptions about what the total lunches represent, leading to wrong equations. In Attempt 2, I assumed total lunches = total people (ignoring extras), and in Attempt 1, I subtracted an incorrect total people from total lunches. The confusion began when I didn't clearly separate the 'number of ... |
If John decides to earn the performance bonus of an extra $20 a day by working 2 hours longer, how much does he make per hour?
Incorrect Answer:
To find out how much John makes per hour, we need to calculate his total earnings per day and then divide that by the total number of hours he works.
Let's assume John norma... | I kept overcomplicating the problem by introducing unnecessary variables like original daily earnings and original hours. I didn't realize the bonus information directly gives the hourly rate without needing those extra unknowns.We need to find John's hourly wage. The problem says he earns an extra $20 a day by working... |
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