Representing Numbers
Use each digit in 0, 2, 4, 6 and 8 exactly once to make a number with two decimal places. What is the greatest possible number less than 650?
What is 640.82?
Calculate 125 × 48 × 25 without a calculator. Find a strategy that avoids multiplying the numbers in the order given, and explain why it works.
What is 150,000?
Which is greater: ⅚ of ¾ metre or ¾ of ⅘ metre? By what fraction of a metre do they differ?
What is ⅚ of ¾ metre, which is greater by ¹⁄₄₀ metre?
A rectangular garden is 2.4 metres longer than it is wide. Its perimeter is 25.6 metres. What are its length and width?
What are a length of 7.6 metres and a width of 5.2 metres?
A prism has 24 vertices. What shape are its two matching end faces? How many faces and edges does it have? Explain the relationships you used.
What are dodecagonal end faces, 14 faces altogether and 36 edges?
Find the difference between the greatest and smallest numbers with exactly three decimal places that round to 7.46 when rounded to two decimal places.
What is 0.009?
A whole number divided by 12 leaves a remainder of 7. The same number divided by 8 also leaves a remainder of 7. What is the smallest possible number greater than 200?
What is 223?
Find three different fractions strictly between ⅝ and ¾. At least two must have different denominators. Prove that all three fit.
What are ¹³⁄₂₀, ⅔ and ⁷⁄₁₀? (Other correct examples accepted.)
Three consecutive vertices of a rectangle are A(−5, −3), B(4, −3) and C(4, 5). Find the fourth vertex and the perimeter. Then move every vertex 7 units left and 4 units down. What are the four new coordinates?
What are D(−5, 5), a perimeter of 34 units, and translated coordinates A′(−12, −7), B′(−3, −7), C′(−3, 1) and D′(−12, 1)?
A rectangular prism measures 12 cm × 8 cm × 6 cm. It is packed completely with cubes of the greatest possible whole-number side length, with no gaps or overlaps. What is each cube’s side length, and how many cubes are needed?
What are cubes with 2-centimetre sides, with 72 cubes needed?
A number between −10 and 0 is closer to −6 than to −2. Its distance from −2 is three times its distance from −6. What are the two possible numbers?
What are −5 and −8?
Insert one pair of brackets into 6 + 18 ÷ 3 × 2 − 1 to produce the greatest possible value. What is that value? Explain why your placement is best.
What is 23, using (6 + 18 ÷ 3) × 2 − 1?
Three ribbons have a combined length of 6 metres. The second ribbon is ½ metre longer than the first. The third is ¼ metre shorter than the second. How long is each ribbon?
What are 1¾ metres, 2¼ metres and 2 metres, respectively?
Four angles meet at a point. The second is twice the first, the third is 30° greater than the first, and the fourth is a right angle. Find all four angles.
What are 60°, 120°, 90° and 90°, respectively?
A solid is made from a 10 cm × 8 cm × 6 cm rectangular prism by removing a 4 cm × 3 cm × 6 cm rectangular prism from one corner. What volume remains? Describe the shape of the remaining solid when viewed from above.
What are 408 cubic centimetres and an L-shaped top view?
A shop offers two discounts on a $240 bicycle: 25% off, or $40 off followed by 10% off the reduced price. Which offer is better, or are they equal? Justify your answer.
What are two equal offers, both costing $180?
A school needs 350 notebooks. Supplier A sells packs of 24 for $42. Supplier B sells packs of 18 for $33, with one free pack for every five packs purchased. If the school buys from only one supplier, which supplier gives the lowest total cost?
What is Supplier B, costing $561 for 17 purchased packs and 3 free packs?
A tank is ⅘ full. After 18 litres are removed, it is ½ full. What is the tank’s capacity, and how much water must then be added to make it ¾ full?
What are a capacity of 60 litres and an additional 15 litres?
Two rectangular gardens share one complete side. Together they form a larger rectangle 18 metres long and 8 metres wide. The shared side is 8 metres long. Fencing is needed around the outside and along the shared boundary, with three separate 1.2-metre gate openings. Fencing is sold only in 5-metre rolls. How many rolls are needed, and how much fencing will be left over?
What are 12 rolls and 3.6 metres left over?
A rectangular tank has internal dimensions of 50 cm × 30 cm × 40 cm. It contains 36 litres of water. A solid rectangular block measuring 20 cm × 15 cm × 10 cm is placed completely underwater. What is the new water depth, and how many more litres can be added before the tank is full?
What are a water depth of 26 centimetres and room for another 21 litres?
A school needs 350 notebooks. Supplier A sells packs of 24 for $42. Supplier B sells packs of 18 for $33, with one free pack for every five packs purchased. If the school buys from only one supplier, which supplier gives the lowest total cost?
What are an original price of $240 and an equivalent discount of 32.5%?
Find every whole number between 100 and 200 that leaves a remainder of 3 when divided by 5, a remainder of 4 when divided by 6, and a remainder of 6 when divided by 8. Explain how you know your list is complete.
What is 118?
A school spends ¼ of its original fundraising money on books, then ⅖ of the remaining money on sports equipment. It has $540 left. How much did it raise, and what fraction of the original amount was spent on sports equipment?
What are $1,200 raised and ³⁄₁₀ spent on sports equipment?
A farmer has 48 metres of fencing to enclose three sides of a rectangle against a straight wall. Each side length must be a whole number of metres. Find all possible dimensions for which the distance around the complete rectangle, including the wall side, is at least 70 metres but less than 76 metres. Explain why there are no other possibilities.
What are 26 m × 11 m, 24 m × 12 m and 22 m × 13 m, with the longer side against the wall?
You have 120 identical unit cubes. Find every different rectangular prism you can build using all the cubes, where every dimension is at least 2 units. Rotations count as the same prism. Which prism has the smallest total length of its 12 edges? Prove that you have found every possibility.
What are 2 × 2 × 30, 2 × 3 × 20, 2 × 4 × 15, 2 × 5 × 12, 2 × 6 × 10, 3 × 4 × 10, 3 × 5 × 8 and 4 × 5 × 6, with 4 × 5 × 6 giving the shortest total edge length of 60 units?