{
  "Id": "19902b4b-d5cb-40db-b9ad-ad91c538184b",
  "Title": "Class 6 Challenging Mathematics Worksheet",
  "Subject": "Mathematics",
  "Topic": "Multi-step Problem Solving from Class 6 Practice",
  "Instructions": "Read each question carefully and answer to the best of your ability. Show all working for word problems.",
  "EstimatedMinutes": 35,
  "StudentProfileId": "dd186faa-f489-46ee-84e4-26e2d01e2691",
  "SourceFileName": "combined_math_practice_sheets.pdf",
  "SessionMaterialId": "86139c68-3464-48b5-a7ec-2d15d3889839",
  "GeneratedAt": "2026-05-31T16:23:13.775802Z",
  "Questions": [
    {
      "Id": 1,
      "Type": "word_problem",
      "Text": "In the number 623541, determine the place values of the digits 6 and 5. Compute the difference between these place values and then multiply the result by 2. Show your working.",
      "Options": [],
      "Answer": "Step 1: Identify place values: 6 is in the hundred-thousands place (600000) and 5 is in the thousands place (5000).\nStep 2: Compute the difference: 600000 - 5000 = 595000.\nStep 3: Multiply by 2: 595000 \u00D7 2 = 1190000\nAnswer: 1190000",
      "Points": 3
    },
    {
      "Id": 2,
      "Type": "word_problem",
      "Text": "Using the digits 6, 0, 8, 4, and 1 exactly once, form the greatest possible number. Next, form the smallest possible number (ensuring the number does not begin with 0). Finally, subtract the smaller number from the greater one. Show your working.",
      "Options": [],
      "Answer": "Step 1: Arrange digits in descending order for the greatest number: 8, 6, 4, 1, 0 \u2192 86410.\nStep 2: Arrange digits in ascending order for the smallest number, ensuring the first digit isn\u2019t 0: 1, 0, 4, 6, 8 \u2192 10468.\nStep 3: Subtract: 86410 - 10468 = 75942\nAnswer: 75942",
      "Points": 3
    },
    {
      "Id": 3,
      "Type": "word_problem",
      "Text": "From the digits 3, 4, 9, and 0, determine how many unique 4-digit numbers can be formed without repetition. Then identify how many of these numbers are even. Finally, add these two results together. Show your working.",
      "Options": [],
      "Answer": "Step 1: Total 4-digit numbers: First digit choices (cannot be 0) = 3 options; remaining digits = 3! = 6; Total = 3\u00D76 = 18.\nStep 2: To count even numbers, the units digit must be 0 or 4. \n    - If unit digit is 0: First digit from {3,4,9} = 3 options, remaining 2 digits = 2! = 2; Count = 3\u00D72 = 6.\n    - If unit digit is 4: First digit from {3,9} (since 0 is allowed but not as first digit) = 2 options, remaining 2 digits = 2; Count = 2\u00D72 = 4.\n    Total even numbers = 6 \u002B 4 = 10.\nStep 3: Sum = 18 \u002B 10 = 28\nAnswer: 28",
      "Points": 3
    },
    {
      "Id": 4,
      "Type": "word_problem",
      "Text": "Two petrol tanks contain 6325 \u2113 and 4890 \u2113 respectively. First, compute the difference between these two volumes and round this difference to the nearest hundred. Then, calculate the percentage this rounded difference represents of the smaller tank\u2019s volume. Show your working.",
      "Options": [],
      "Answer": "Step 1: Compute the exact difference: 6325 - 4890 = 1435.\nStep 2: Round 1435 to the nearest hundred: 1435 \u2248 1400.\nStep 3: Percentage = (1400 / 4890) \u00D7 100 \u2248 28.62%\nAnswer: 28.62%",
      "Points": 3
    },
    {
      "Id": 5,
      "Type": "word_problem",
      "Text": "Estimate the product 958 \u00D7 387 by rounding each factor to the nearest hundred. Then, subtract 100 from the estimated product. Show your working.",
      "Options": [],
      "Answer": "Step 1: Round the numbers: 958 \u2192 1000 and 387 \u2192 400.\nStep 2: Multiply the rounded numbers: 1000 \u00D7 400 = 400000.\nStep 3: Subtract 100: 400000 - 100 = 399900\nAnswer: 399900",
      "Points": 3
    },
    {
      "Id": 6,
      "Type": "word_problem",
      "Text": "Estimate the quotient 2838 \u00F7 125 by rounding both the dividend and the divisor to the nearest hundred. Then, add 5 to the estimated quotient. Show your working.",
      "Options": [],
      "Answer": "Step 1: Round the numbers: 2838 \u2192 2800 and 125 \u2192 100.\nStep 2: Divide the rounded numbers: 2800 \u00F7 100 = 28.\nStep 3: Add 5: 28 \u002B 5 = 33\nAnswer: 33",
      "Points": 3
    },
    {
      "Id": 7,
      "Type": "word_problem",
      "Text": "A school management spends $26,756 on teachers\u0027 pay, $325,378 on building construction, $5,780 on games, and $8,235 on other work. Calculate the total expenditure and then find by how much this total exceeds $360,000. Show your working.",
      "Options": [],
      "Answer": "Step 1: Compute the total: 26756 \u002B 325378 = 352134; 352134 \u002B 5780 = 357914; 357914 \u002B 8235 = 366149.\nStep 2: Difference from 360000: 366149 - 360000 = 6149\nAnswer: 6149",
      "Points": 3
    },
    {
      "Id": 8,
      "Type": "word_problem",
      "Text": "Madhu had $9031 and after buying a TV she was left with $1356. Determine the price of the TV and then subtract 100 from this price. Show your working.",
      "Options": [],
      "Answer": "Step 1: Price of the TV = 9031 - 1356 = 7675.\nStep 2: Subtract 100: 7675 - 100 = 7575\nAnswer: 7575",
      "Points": 3
    },
    {
      "Id": 9,
      "Type": "word_problem",
      "Text": "The population of a country was 906450600 in 1990 and increased by 9889700 in 1991. Compute the population in 1991 and then determine 1% of that population. Show your working.",
      "Options": [],
      "Answer": "Step 1: New population = 906450600 \u002B 9889700 = 916340300.\nStep 2: Calculate 1%: (1/100) \u00D7 916340300 = 9163403\nAnswer: 9163403",
      "Points": 3
    },
    {
      "Id": 10,
      "Type": "word_problem",
      "Text": "From a set of numbers, compute the sum for two groups: (a) 1086, 2117, 3543, and 1284; (b) 2637, 1283, 2549, and 98713. Then, subtract the sum of group (a) from group (b). Show your working.",
      "Options": [],
      "Answer": "Step 1: Sum of group (a): 1086 \u002B 2117 = 3203; 3203 \u002B 3543 = 6746; 6746 \u002B 1284 = 8030.\nStep 2: Sum of group (b): 2637 \u002B 1283 = 3920; 3920 \u002B 2549 = 6469; 6469 \u002B 98713 = 105182.\nStep 3: Subtract: 105182 - 8030 = 97152\nAnswer: 97152",
      "Points": 3
    },
    {
      "Id": 11,
      "Type": "word_problem",
      "Text": "Determine whether 48 and 28 are perfect numbers. Assign a value of 1 if a number is perfect and 0 if it is not, then add these two values. Show your working.",
      "Options": [],
      "Answer": "Step 1: 48 is not a perfect number, so assign 0.\nStep 2: 28 is a perfect number (sum of proper divisors equals 28), so assign 1.\nStep 3: Sum = 0 \u002B 1 = 1\nAnswer: 1",
      "Points": 3
    },
    {
      "Id": 12,
      "Type": "word_problem",
      "Text": "Identify all prime numbers between 30 and 50. Then multiply these primes together and determine the last two digits of the product. Show your working.",
      "Options": [],
      "Answer": "Step 1: The primes between 30 and 50 are 31, 37, 41, 43, and 47.\nStep 2: Calculate the product step-wise modulo 100:\n        31 \u00D7 37 = 1147, last two digits = 47;\n        47 \u00D7 41 = 1927, last two digits = 27;\n        27 \u00D7 43 = 1161, last two digits = 61;\n        61 \u00D7 47 = 2867, last two digits = 67.\nAnswer: 67",
      "Points": 3
    },
    {
      "Id": 13,
      "Type": "word_problem",
      "Text": "Using Roman numerals, solve the equation: LXX \u002B x = LXXVI. First convert the numerals to decimals, solve for x, and then subtract 5 from the value of x. Show your working.",
      "Options": [],
      "Answer": "Step 1: Convert: LXX = 70 and LXXVI = 76.\nStep 2: Solve for x: 70 \u002B x = 76, so x = 6.\nStep 3: Subtract 5: 6 - 5 = 1\nAnswer: 1",
      "Points": 3
    },
    {
      "Id": 14,
      "Type": "word_problem",
      "Text": "Given the equation LIX = y \u002B IX in Roman numerals, convert to decimals and solve for y. Then multiply the resulting value of y by 3. Show your working.",
      "Options": [],
      "Answer": "Step 1: Convert: LIX = 59 and IX = 9.\nStep 2: Solve for y: 59 = y \u002B 9, so y = 50.\nStep 3: Multiply by 3: 50 \u00D7 3 = 150\nAnswer: 150",
      "Points": 3
    },
    {
      "Id": 15,
      "Type": "word_problem",
      "Text": "Determine the sum of the predecessor of 97801 and 90526. Then calculate how much this sum falls short of 200000. Show your working.",
      "Options": [],
      "Answer": "Step 1: Predecessor of 97801 is 97800.\nStep 2: Sum = 97800 \u002B 90526 = 188326.\nStep 3: Difference from 200000: 200000 - 188326 = 11674\nAnswer: 11674",
      "Points": 3
    },
    {
      "Id": 16,
      "Type": "word_problem",
      "Text": "Find the product of the successor of 147 and 389. Then divide the product by 7 and round down to the nearest whole number. Show your working.",
      "Options": [],
      "Answer": "Step 1: Successor of 147 is 148.\nStep 2: Compute the product: 148 \u00D7 389 = 57572.\nStep 3: Divide by 7: 57572 \u00F7 7 \u2248 8224.57, rounding down gives 8224\nAnswer: 8224",
      "Points": 3
    },
    {
      "Id": 17,
      "Type": "word_problem",
      "Text": "Using the distributive property, compute 632 \u00D7 1001. Then, calculate 99 \u00D7 82 and subtract this product from the first result. Show your working.",
      "Options": [],
      "Answer": "Step 1: Calculate 632 \u00D7 1001 = 632632.\nStep 2: Calculate 99 \u00D7 82 = 8118.\nStep 3: Subtract: 632632 - 8118 = 624514\nAnswer: 624514",
      "Points": 3
    },
    {
      "Id": 18,
      "Type": "word_problem",
      "Text": "Compute 1002 \u00D7 8716 using the distributive property. Then add 500 to the result. Show your working.",
      "Options": [],
      "Answer": "Step 1: Calculate 1002 \u00D7 8716 = (1000\u00D78716) \u002B (2\u00D78716) = 8716000 \u002B 17432 = 8733432.\nStep 2: Add 500: 8733432 \u002B 500 = 8733932\nAnswer: 8733932",
      "Points": 3
    },
    {
      "Id": 19,
      "Type": "word_problem",
      "Text": "Simplify the expression: 34 - [32 - {31 - (18 - 15 - 1)}]. Then add 10 to your simplified result. Show your working.",
      "Options": [],
      "Answer": "Step 1: Evaluate inner parentheses: (18 - 15 - 1) = 2.\nStep 2: Evaluate the curly braces: {31 - 2} = 29.\nStep 3: Evaluate the square brackets: [32 - 29] = 3.\nStep 4: Compute: 34 - 3 = 31; then add 10: 31 \u002B 10 = 41\nAnswer: 41",
      "Points": 3
    },
    {
      "Id": 20,
      "Type": "word_problem",
      "Text": "Simplify the expression: [4 \u00D7 {6 \u002B (9 - 4 \u002B 2)} \u002B 5] and then subtract 3 from the result. Show your working.",
      "Options": [],
      "Answer": "Step 1: Inside the parentheses: (9 - 4 \u002B 2) = 7.\nStep 2: Inside the curly braces: 6 \u002B 7 = 13.\nStep 3: Multiply: 4 \u00D7 13 = 52.\nStep 4: Add 5: 52 \u002B 5 = 57; then subtract 3: 57 - 3 = 54\nAnswer: 54",
      "Points": 3
    },
    {
      "Id": 21,
      "Type": "fill_blank",
      "Text": "The digit in the hundred-thousands place in 623541 is _____ .",
      "Options": [],
      "Answer": "6",
      "Points": 3
    },
    {
      "Id": 22,
      "Type": "fill_blank",
      "Text": "When forming the greatest number from the digits 6, 0, 8, 4, and 1, the digit placed in the ten-thousands position is _____ .",
      "Options": [],
      "Answer": "8",
      "Points": 3
    },
    {
      "Id": 23,
      "Type": "fill_blank",
      "Text": "In the set {3, 4, 9, 0}, when forming a 4-digit even number, the units digit must be _____ .",
      "Options": [],
      "Answer": "0",
      "Points": 3
    },
    {
      "Id": 24,
      "Type": "fill_blank",
      "Text": "Rounding 6325 to the nearest hundred gives _____ .",
      "Options": [],
      "Answer": "6300",
      "Points": 3
    },
    {
      "Id": 25,
      "Type": "fill_blank",
      "Text": "Rounding 4890 to the nearest hundred gives _____ .",
      "Options": [],
      "Answer": "4900",
      "Points": 3
    },
    {
      "Id": 26,
      "Type": "fill_blank",
      "Text": "When estimating the product 958 \u00D7 387 by rounding, 958 rounds to _____ .",
      "Options": [],
      "Answer": "1000",
      "Points": 3
    },
    {
      "Id": 27,
      "Type": "fill_blank",
      "Text": "When estimating the product 958 \u00D7 387 by rounding, 387 rounds to _____ .",
      "Options": [],
      "Answer": "400",
      "Points": 3
    },
    {
      "Id": 28,
      "Type": "fill_blank",
      "Text": "To estimate the quotient 2838 \u00F7 125, if 2838 is rounded to the nearest hundred it becomes _____ .",
      "Options": [],
      "Answer": "2800",
      "Points": 3
    },
    {
      "Id": 29,
      "Type": "fill_blank",
      "Text": "To estimate the quotient 2838 \u00F7 125, if 125 is rounded to the nearest hundred it becomes _____ .",
      "Options": [],
      "Answer": "100",
      "Points": 3
    },
    {
      "Id": 30,
      "Type": "fill_blank",
      "Text": "The additive identity for whole numbers is _____ .",
      "Options": [],
      "Answer": "0",
      "Points": 3
    },
    {
      "Id": 31,
      "Type": "fill_blank",
      "Text": "The sum of an odd number and an even number is always _____ .",
      "Options": [],
      "Answer": "odd",
      "Points": 3
    },
    {
      "Id": 32,
      "Type": "fill_blank",
      "Text": "The total expense for building construction and teachers\u0027 pay, amounting to $325378 and $26756 respectively, is _____ .",
      "Options": [],
      "Answer": "352134",
      "Points": 3
    },
    {
      "Id": 33,
      "Type": "fill_blank",
      "Text": "After spending on various school items, if the total expenditure is $366149, then it exceeds $360000 by _____ .",
      "Options": [],
      "Answer": "6149",
      "Points": 3
    },
    {
      "Id": 34,
      "Type": "fill_blank",
      "Text": "If Madhu had $9031 and is left with $1356 after buying a TV, then the price of the TV is _____ .",
      "Options": [],
      "Answer": "7675",
      "Points": 3
    },
    {
      "Id": 35,
      "Type": "fill_blank",
      "Text": "The population increased from 906450600 by _____ to become 916340300.",
      "Options": [],
      "Answer": "9889700",
      "Points": 3
    },
    {
      "Id": 36,
      "Type": "fill_blank",
      "Text": "The sum of the numbers 1086, 2117, 3543, and 1284 is _____ .",
      "Options": [],
      "Answer": "8030",
      "Points": 3
    },
    {
      "Id": 37,
      "Type": "fill_blank",
      "Text": "The sum of the numbers 2637, 1283, 2549, and 98713 is _____ .",
      "Options": [],
      "Answer": "105182",
      "Points": 3
    },
    {
      "Id": 38,
      "Type": "fill_blank",
      "Text": "Among the prime numbers between 30 and 50, the smallest prime is _____ .",
      "Options": [],
      "Answer": "31",
      "Points": 3
    },
    {
      "Id": 39,
      "Type": "fill_blank",
      "Text": "In Roman numerals, LXX represents _____ in decimal form.",
      "Options": [],
      "Answer": "70",
      "Points": 3
    },
    {
      "Id": 40,
      "Type": "fill_blank",
      "Text": "Converting the Roman numeral LIX yields the decimal number _____ .",
      "Options": [],
      "Answer": "59",
      "Points": 3
    }
  ],
  "TotalPoints": 120
}