{"title": "Ratio and Proportion (Algebra) - Grade 10 ICSE", "total_questions": 60, "questions": [{"id": 1, "difficulty": "easy", "question": "The ratio of 30 min to 1.5 hours is? ⏰", "options": {"A": "1:3", "B": "2:3", "C": "1:2", "D": "3:1"}, "correct_answer": "A"}, {"id": 2, "difficulty": "easy", "question": "If a:b = 2:3 and b:c = 4:5, find a:c. 🔗", "options": {"A": "8:15", "B": "3:4", "C": "2:5", "D": "6:5"}, "correct_answer": "A"}, {"id": 3, "difficulty": "easy", "question": "What is the fourth proportional to 3, 7, 9? 🔢", "options": {"A": "21", "B": "27", "C": "12", "D": "15"}, "correct_answer": "A"}, {"id": 4, "difficulty": "easy", "question": "If x:y = 3:4, find (3x+2y):(5x+3y). 🧮", "options": {"A": "17:27", "B": "27:17", "C": "12:23", "D": "23:12"}, "correct_answer": "A"}, {"id": 5, "difficulty": "easy", "question": "Divide ₹60 in ratio 3:2. 💰", "options": {"A": "₹36, ₹24", "B": "₹30, ₹30", "C": "₹40, ₹20", "D": "₹24, ₹36"}, "correct_answer": "A"}, {"id": 6, "difficulty": "easy", "question": "If 2:3 = x:12, find x. 🔍", "options": {"A": "8", "B": "6", "C": "18", "D": "10"}, "correct_answer": "A"}, {"id": 7, "difficulty": "easy", "question": "The mean proportional between 4 and 9 is? ⚖️", "options": {"A": "6", "B": "13", "C": "36", "D": "6.5"}, "correct_answer": "A"}, {"id": 8, "difficulty": "easy", "question": "If a/b = 3/5, then (a+b)/(a-b) = ? 🔄", "options": {"A": "4", "B": "2", "C": "1/4", "D": "8"}, "correct_answer": "A"}, {"id": 9, "difficulty": "easy", "question": "Which is greater: 3:4 or 5:8? ⚔️", "options": {"A": "3:4", "B": "5:8", "C": "Equal", "D": "Cannot say"}, "correct_answer": "A"}, {"id": 10, "difficulty": "easy", "question": "If ₹x divided in ratio 5:7 and smaller share is ₹25, find x. 🎯", "options": {"A": "₹60", "B": "₹50", "C": "₹70", "D": "₹35"}, "correct_answer": "A"}, {"id": 11, "difficulty": "easy", "question": "If p:q = 2:5, then (2p+3q):(7p-2q) = ? 🎪", "options": {"A": "19:4", "B": "4:19", "C": "16:5", "D": "5:16"}, "correct_answer": "A"}, {"id": 12, "difficulty": "easy", "question": "The third proportional to 9 and 12 is? 📐", "options": {"A": "16", "B": "15", "C": "18", "D": "12"}, "correct_answer": "A"}, {"id": 13, "difficulty": "easy", "question": "If (x+y):(x-y)=5:2, find x:y. 🧩", "options": {"A": "7:3", "B": "3:7", "C": "5:2", "D": "2:5"}, "correct_answer": "A"}, {"id": 14, "difficulty": "easy", "question": "The ratio 1 m : 60 cm in simplest form is? 📏", "options": {"A": "5:3", "B": "3:5", "C": "100:60", "D": "60:100"}, "correct_answer": "A"}, {"id": 15, "difficulty": "easy", "question": "If 15, x, 3, 5 are in proportion, find x. ⚓", "options": {"A": "25", "B": "9", "C": "1", "D": "75"}, "correct_answer": "A"}, {"id": 16, "difficulty": "easy", "question": "If a:b=3:4 and b:c=8:9, find a:b:c. 🎨", "options": {"A": "6:8:9", "B": "3:4:9", "C": "3:8:9", "D": "6:4:9"}, "correct_answer": "A"}, {"id": 17, "difficulty": "easy", "question": "Which are in proportion: 2,3,4,6? ✅", "options": {"A": "Yes", "B": "No", "C": "Only first three", "D": "Only last three"}, "correct_answer": "A"}, {"id": 18, "difficulty": "easy", "question": "If x/3 = y/4 = z/5, then (x+y+z)/z = ? 🌈", "options": {"A": "12/5", "B": "5", "C": "3", "D": "4"}, "correct_answer": "A"}, {"id": 19, "difficulty": "easy", "question": "The compound ratio of 2:3 and 4:5 is? 🌀", "options": {"A": "8:15", "B": "6:20", "C": "12:10", "D": "15:8"}, "correct_answer": "A"}, {"id": 20, "difficulty": "easy", "question": "If 30% of x = 45% of y, then x:y = ? 🍬", "options": {"A": "3:2", "B": "2:3", "C": "3:5", "D": "5:3"}, "correct_answer": "A"}, {"id": 21, "difficulty": "medium", "question": "If (2x+3y):(3x+5y)=18:29, find x:y. 🤔", "options": {"A": "3:4", "B": "4:3", "C": "2:5", "D": "5:2"}, "correct_answer": "A"}, {"id": 22, "difficulty": "medium", "question": "The ratio of ages of A and B is 5:7. After 6 years, ratio becomes 7:9. Find present age of A. 🎂", "options": {"A": "15 years", "B": "21 years", "C": "25 years", "D": "30 years"}, "correct_answer": "A"}, {"id": 23, "difficulty": "medium", "question": "If a/3 = b/4 = c/7, then (a+b+c)/b = ? 🔥", "options": {"A": "7/2", "B": "2", "C": "14/3", "D": "3"}, "correct_answer": "A"}, {"id": 24, "difficulty": "medium", "question": "The sides of a triangle are in ratio 3:5:7. If perimeter is 90 cm, find the longest side. 📐", "options": {"A": "42 cm", "B": "30 cm", "C": "18 cm", "D": "36 cm"}, "correct_answer": "A"}, {"id": 25, "difficulty": "medium", "question": "What number must be added to each term of 5:9 to make it 2:3? ➕", "options": {"A": "3", "B": "2", "C": "5", "D": "9"}, "correct_answer": "A"}, {"id": 26, "difficulty": "medium", "question": "If x:y=3:4, find the ratio (5x-2y):(x+3y). 🧨", "options": {"A": "7:15", "B": "15:7", "C": "3:13", "D": "13:3"}, "correct_answer": "A"}, {"id": 27, "difficulty": "medium", "question": "The ratio of copper and zinc in brass is 13:7. How much zinc in 100 kg brass? ⚖️", "options": {"A": "35 kg", "B": "65 kg", "C": "50 kg", "D": "70 kg"}, "correct_answer": "A"}, {"id": 28, "difficulty": "medium", "question": "If a:b=5:6 and b:c=8:11, find a:b:c. 🔗", "options": {"A": "20:24:33", "B": "5:8:11", "C": "5:6:11", "D": "40:48:66"}, "correct_answer": "A"}, {"id": 29, "difficulty": "medium", "question": "The mean proportional between (a-b) and (a³-a²b) is? 🎯", "options": {"A": "a(a-b)", "B": "b(a-b)", "C": "a²-b²", "D": "ab"}, "correct_answer": "A"}, {"id": 30, "difficulty": "medium", "question": "If (3x²+2y²):(3x²-2y²)=17:8, find x²:y². 🚀", "options": {"A": "25:4", "B": "4:25", "C": "5:2", "D": "2:5"}, "correct_answer": "A"}, {"id": 31, "difficulty": "medium", "question": "A bag has ₹1, 50p, 25p coins in ratio 5:6:8. If total is ₹210, find number of 50p coins. 💰", "options": {"A": "168", "B": "140", "C": "120", "D": "96"}, "correct_answer": "A"}, {"id": 32, "difficulty": "medium", "question": "If a:b=3:4, b:c=5:6, c:d=7:8, find a:d. 🎡", "options": {"A": "35:64", "B": "64:35", "C": "3:8", "D": "8:3"}, "correct_answer": "A"}, {"id": 33, "difficulty": "medium", "question": "What least number must be subtracted from each term of 15:19 to make it 3:4? ➖", "options": {"A": "3", "B": "2", "C": "1", "D": "4"}, "correct_answer": "A"}, {"id": 34, "difficulty": "medium", "question": "If (x³+3x²y+3xy²+y³):(x³+y³)=64:61, find x:y. 🧿", "options": {"A": "5:3", "B": "3:5", "C": "4:3", "D": "3:4"}, "correct_answer": "A"}, {"id": 35, "difficulty": "medium", "question": "Two numbers are in ratio 7:11. If 7 is added to each, ratio becomes 2:3. Find the numbers. 🔢", "options": {"A": "49, 77", "B": "35, 55", "C": "42, 66", "D": "28, 44"}, "correct_answer": "A"}, {"id": 36, "difficulty": "medium", "question": "If a:b=2:3, b:c=4:5, c:d=6:7, find a:b:c:d. 🌈", "options": {"A": "16:24:30:35", "B": "2:3:5:7", "C": "8:12:15:17.5", "D": "4:6:7.5:8.75"}, "correct_answer": "A"}, {"id": 37, "difficulty": "medium", "question": "The ratio of boys to girls in a class is 4:5. If 10 boys leave, ratio becomes 3:5. Find original boys. 👦👧", "options": {"A": "40", "B": "50", "C": "30", "D": "60"}, "correct_answer": "A"}, {"id": 38, "difficulty": "medium", "question": "If (a+b):(b+c):(c+a)=5:6:7, find a:b:c. 🎭", "options": {"A": "3:2:4", "B": "2:3:4", "C": "4:3:2", "D": "3:4:2"}, "correct_answer": "A"}, {"id": 39, "difficulty": "medium", "question": "What number must be subtracted from each term of 12:17 to make it equal to 2:3? 🎪", "options": {"A": "2", "B": "3", "C": "4", "D": "5"}, "correct_answer": "A"}, {"id": 40, "difficulty": "medium", "question": "If 2A=3B=4C, find A:B:C. ⚡", "options": {"A": "6:4:3", "B": "2:3:4", "C": "12:8:6", "D": "3:4:6"}, "correct_answer": "A"}, {"id": 41, "difficulty": "hard", "question": "If (x²+y²+z²):(xy+yz+zx)=11:6, and x+y+z=12, find x²+y²+z². 🏆", "options": {"A": "88", "B": "72", "C": "96", "D": "48"}, "correct_answer": "A"}, {"id": 42, "difficulty": "hard", "question": "If a,b,c are in continued proportion, prove that (a²+b²+c²)(b²+c²+d²)=(ab+bc+cd)². As MCQ: If a,b,c,d are in continued proportion, then (a²+b²+c²)(b²+c²+d²)/(ab+bc+cd)² = ? 🎲", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "A"}, {"id": 43, "difficulty": "hard", "question": "If (a²+b²):(a²-b²)=17:8, find a³+b³:a³-b³. 🚁", "options": {"A": "76:49", "B": "49:76", "C": "17:8", "D": "8:17"}, "correct_answer": "A"}, {"id": 44, "difficulty": "hard", "question": "If y is the mean proportional between x and z, prove that x²+y²+z²:(x+y+z)² = (x-z)²:(x+y+z)². As MCQ: If y²=xz, then (x²+y²+z²)/(x+y+z)² equals? 🪐", "options": {"A": "(x²+z²+2xz)/(x+z+2y)²", "B": "1/2", "C": "1/3", "D": "2/3"}, "correct_answer": "A"}, {"id": 45, "difficulty": "hard", "question": "If (4a+9b):(4c+9d) = (4a-9b):(4c-9d), show that a:b=c:d. As MCQ: Given proportion, then? 🎯", "options": {"A": "a/b = c/d", "B": "a/c = b/d", "C": "ad = bc", "D": "a+b=c+d"}, "correct_answer": "A"}, {"id": 46, "difficulty": "hard", "question": "If (a+b+c+d)(a-b-c+d)=(a-b+c-d)(a+b-c-d), prove that a:b=c:d. As MCQ: Given equality holds if? 🌠", "options": {"A": "a/b = c/d", "B": "a=c, b=d", "C": "a+b=c+d", "D": "a-b=c-d"}, "correct_answer": "A"}, {"id": 47, "difficulty": "hard", "question": "If x = (p+q)/(1+pq), y = (q+r)/(1+qr), z = (r+p)/(1+rp) and x,y,z are equal, show that p=q=r. As MCQ: If x=y=z given as above, then? 🧬", "options": {"A": "p=q=r", "B": "p+q+r=0", "C": "pq=qr=rp", "D": "p²+q²+r²=1"}, "correct_answer": "A"}, {"id": 48, "difficulty": "hard", "question": "If (a²+b²+c²)(x²+y²+z²)=(ax+by+cz)², then a:x = b:y = c:z. As MCQ: Given equality implies? 🌉", "options": {"A": "a,b,c proportional to x,y,z", "B": "a=x, b=y, c=z", "C": "a+b+c = x+y+z", "D": "ab+bc+ca=xy+yz+zx"}, "correct_answer": "A"}, {"id": 49, "difficulty": "hard", "question": "If (x³+ax²+bx+c) is exactly divisible by (x²+bx+ac), find relation between a,b,c. 🧠", "options": {"A": "c(a-1)=ab", "B": "c=ab", "C": "a=b=c", "D": "c=ab/(a-1)"}, "correct_answer": "A"}, {"id": 50, "difficulty": "hard", "question": "If (p²+qr)/(p(q+r)) = (q²+rp)/(q(r+p)) = (r²+pq)/(r(p+q)) = k, find k. 🔑", "options": {"A": "1", "B": "2", "C": "1/2", "D": "3"}, "correct_answer": "A"}, {"id": 51, "difficulty": "hard", "question": "If a/(b+c) = b/(c+a) = c/(a+b) = k, and a+b+c≠0, find k. 🎖️", "options": {"A": "1/2", "B": "1", "C": "2", "D": "0"}, "correct_answer": "A"}, {"id": 52, "difficulty": "hard", "question": "If x = (√(a+2b)+√(a-2b))/(√(a+2b)-√(a-2b)), show that bx² - ax + b = 0. As MCQ: If x is given as above, then which equation holds? 🏔️", "options": {"A": "bx² - ax + b = 0", "B": "ax² - bx + a = 0", "C": "x² - (a/b)x + 1 = 0", "D": "x² + (a/b)x + 1 = 0"}, "correct_answer": "A"}, {"id": 53, "difficulty": "hard", "question": "If (by+cz)/(b²+c²) = (cz+ax)/(c²+a²) = (ax+by)/(a²+b²), prove that x/a = y/b = z/c. As MCQ: Given equal ratios imply? 🛸", "options": {"A": "x/a = y/b = z/c", "B": "x=y=z", "C": "a=b=c", "D": "x/a + y/b + z/c = 1"}, "correct_answer": "A"}, {"id": 54, "difficulty": "hard", "question": "If a,b,c are in GP and x,y are arithmetic means between a,b and b,c respectively, prove that a/x + c/y = 2. As MCQ: If a,b,c in GP, x=(a+b)/2, y=(b+c)/2, then a/x + c/y = ? 🎇", "options": {"A": "2", "B": "1", "C": "3", "D": "4"}, "correct_answer": "A"}, {"id": 55, "difficulty": "hard", "question": "If (a²+b²+c²)/(x²+y²+z²) = (ax+by+cz)²/((a²+b²+c²)(x²+y²+z²)), prove that x/a = y/b = z/c. As MCQ: Given equality implies? 🌌", "options": {"A": "x/a = y/b = z/c", "B": "a/x = b/y = c/z", "C": "x,y,z proportional to a,b,c", "D": "All of these"}, "correct_answer": "A"}, {"id": 56, "difficulty": "hard", "question": "If (x²-yz)/a = (y²-zx)/b = (z²-xy)/c, then prove that (a+b+c)(x+y+z) = ax+by+cz. As MCQ: Given equal ratios, then (a+b+c)(x+y+z) = ? 🎲", "options": {"A": "ax+by+cz", "B": "a+b+c", "C": "x+y+z", "D": "abcxyz"}, "correct_answer": "A"}, {"id": 57, "difficulty": "hard", "question": "If (a+b)/c = (b+c)/a = (c+a)/b = k, and a+b+c≠0, find k². ⚡", "options": {"A": "4", "B": "1", "C": "9", "D": "1/4"}, "correct_answer": "A"}, {"id": 58, "difficulty": "hard", "question": "If (x²+1)/x = (y²+1)/y = (z²+1)/z and x+y+z=0, find x²+y²+z². 🤖", "options": {"A": "0", "B": "1", "C": "3", "D": "xyz"}, "correct_answer": "A"}, {"id": 59, "difficulty": "hard", "question": "If (a²+b²+c²)³ = (a³+b³+c³)², find a⁻¹+b⁻¹+c⁻¹. 🎡", "options": {"A": "0", "B": "1", "C": "2", "D": "abc"}, "correct_answer": "A"}, {"id": 60, "difficulty": "hard", "question": "If a,b,c are distinct and (a²+bc)/(a²-bc) = (b²+ca)/(b²-ca) = (c²+ab)/(c²-ab), find value of each ratio. 🔥", "options": {"A": "-1", "B": "1", "C": "0", "D": "2"}, "correct_answer": "A"}]}