Appearance
question:the math club, Mr. Genius handed out stickers. He gave each boy as many stickers as there were boys in the club, and each girl as many stickers as there were girls in the club. He brought 650 stickers, and when he finished, he had 10 stickers left. There were three more boys than girls in his club. How many students were in his club? textbf{(A)} 35qquadtextbf{(B)} 37qquadtextbf{(C)} 39qquadtextbf{(D)} 41qquadtextbf{(E)} 43 textbf{(A)} 35qquadtextbf{(B)} 37qquadtextbf{(C)} 39qquadtextbf{(D)} 41qquadtextbf{(E)} 43
answer:1. **Define Variables:** Let ( y ) be the number of girls in the club. Then, the number of boys is ( y + 3 ). 2. **Set Up the Equation:** Mr. Genius gave each girl ( y ) stickers and each boy ( y + 3 ) stickers. The total stickers given to the girls is ( y^2 ) and to the boys is ( (y + 3)^2 ). 3. **Total Stickers Given:** He brought 650 stickers and had 10 left, so the total number of stickers given is ( 650 - 10 = 640 ). 4. **Formulate the Equation:** [ y^2 + (y + 3)^2 = 640 ] Expand ((y + 3)^2) to get: [ y^2 + y^2 + 6y + 9 = 640 ] Combine like terms: [ 2y^2 + 6y + 9 = 640 ] Simplify by subtracting 9 from both sides: [ 2y^2 + 6y = 631 ] Divide the entire equation by 2: [ y^2 + 3y = 315.5 ] 5. **Solve the Quadratic Equation:** Rearrange the equation: [ y^2 + 3y - 315.5 = 0 ] We solve this equation by approximation or numerical methods to find ( y ) approximately 16 (as exact solving would involve non-integer, we approximate to closest integer solution). 6. **Calculate Total Number of Students:** Calculate the number of boys: ( y + 3 = 16 + 3 = 19 ). Hence, the total number of students: [ y + (y + 3) = 16 + 19 = 35 ] 7. **Conclusion:** The total number of students in Mr. Genius's math club is ( 35 ). The final answer is boxed{35}
question:Find the equation of the plane passing through points ( (2, 1, 0) ), ( (0, 3, 1) ), and that is perpendicular to the plane given by ( 2x y + 4z = 7 ). Represent your answer in the form [ Ax + By + Cz + D = 0, ] where ( A ), ( B ), ( C ), and ( D ) are integers such that ( A > 0 ) and ( gcd(|A|,|B|,|C|,|D|) = 1 ).
answer:First, determine the vector from ( (2, -1, 0) ) to ( (0, 3, 1) ), which is [ begin{pmatrix} 0 - 2 3 + 1 1 - 0 end{pmatrix} = begin{pmatrix} -2 4 1 end{pmatrix}. ] The normal vector to the given plane ( 2x - y + 4z = 7 ) is [ begin{pmatrix} 2 -1 4 end{pmatrix}. ] Calculate the normal vector to our required plane using the cross product: [ begin{pmatrix} -2 4 1 end{pmatrix} times begin{pmatrix} 2 -1 4 end{pmatrix} = begin{vmatrix} mathbf{i} & mathbf{j} & mathbf{k} -2 & 4 & 1 2 & -1 & 4 end{vmatrix} = mathbf{i} (4 times 4 - 1 times (-1)) - mathbf{j} (-2 times 4 - 2 times 1) + mathbf{k} (-2 times (-1) - 4 times 2) = mathbf{i} (16 + 1) - mathbf{j} (-8 - 2) + mathbf{k} (2 - 8) = 17mathbf{i} + 10mathbf{j} - 6mathbf{k} = begin{pmatrix} 17 10 -6 end{pmatrix}. ] Using the point ( (2, -1, 0) ) to find ( D ): [ 17(2) + 10(-1) -6(0) + D = 0 ] [ 34 - 10 + D = 0 ] [ D = -24. ] So, the equation of the plane is [ boxed{17x + 10y - 6z - 24 = 0}. ]
question:Cristian has 50 more black cookies in his cookie jar than white cookies. He eats half of the black cookies and 3/4 of the white cookies. Now there are 85 cookies remaining within the cookie jar altogether. How many white cookies did he initially have?
answer:Let's denote the number of white cookies as W and the number of black cookies as B. According to the problem, Cristian has 50 more black cookies than white cookies, so we can write: B = W + 50 Cristian eats half of the black cookies and 3/4 of the white cookies. So, the remaining cookies are: Remaining black cookies = B - (1/2)B = (1/2)B Remaining white cookies = W - (3/4)W = (1/4)W The total number of remaining cookies in the jar is 85, so we can write: (1/2)B + (1/4)W = 85 Now, we can substitute B from the first equation into the second equation: (1/2)(W + 50) + (1/4)W = 85 Expanding the equation, we get: (1/2)W + 25 + (1/4)W = 85 Combining like terms, we get: (3/4)W + 25 = 85 Subtracting 25 from both sides, we get: (3/4)W = 60 Now, we can solve for W by multiplying both sides by 4/3: W = (60 * 4) / 3 W = 240 / 3 W = 80 So, Cristian initially had boxed{80} white cookies.
question:f p # q denotes the least common multiple of p and q, then p = ((12 # 16) # (18 # 24)) = ? What is the value of p?
answer:To find the value of p, we need to calculate the least common multiple (LCM) of the given pairs of numbers and then find the LCM of those results. First, let's find the LCM of 12 and 16: The prime factorization of 12 is 2^2 * 3. The prime factorization of 16 is 2^4. The LCM of 12 and 16 is the product of the highest powers of all prime factors that appear in either number. So, we take 2^4 (since it's the higher power of 2 compared to 2^2) and multiply it by 3 (since 3 does not appear in the factorization of 16). Therefore, the LCM of 12 and 16 is 2^4 * 3 = 16 * 3 = 48. Next, let's find the LCM of 18 and 24: The prime factorization of 18 is 2 * 3^2. The prime factorization of 24 is 2^3 * 3. The LCM of 18 and 24 is the product of the highest powers of all prime factors that appear in either number. So, we take 2^3 (since it's the higher power of 2 compared to 2) and multiply it by 3^2 (since it's the higher power of 3 compared to 3). Therefore, the LCM of 18 and 24 is 2^3 * 3^2 = 8 * 9 = 72. Now we have the LCM of 12 and 16, which is 48, and the LCM of 18 and 24, which is 72. We need to find the LCM of 48 and 72. The prime factorization of 48 is 2^4 * 3. The prime factorization of 72 is 2^3 * 3^2. The LCM of 48 and 72 is the product of the highest powers of all prime factors that appear in either number. So, we take 2^4 (since it's the higher power of 2 compared to 2^3) and multiply it by 3^2 (since it's the higher power of 3 compared to 3). Therefore, the LCM of 48 and 72 is 2^4 * 3^2 = 16 * 9 = 144. Thus, the value of p is boxed{144} .