Grade 5 Number Reasoning Problems

What this problem type covers

Each difficulty band is a different way of reasoning rather than the same calculation with new numbers: undoing a chain of operations, deducing a divisor from two remainders, counting common multiples in a range, recovering a fraction from its lowest terms, and finding the smallest number meeting two remainder conditions.

Example problems

One example from each difficulty band, with worked solutions.

These problems are generated automatically. The answers are derived as each problem is built and are checked by tests, but mistakes are still possible — please tell us if you find one.

1. 1 · ★1
   A workshop made 5 times its usual daily output, set aside 35 faulty pieces, and packed the rest evenly into 4 boxes. Each box holds 5 pieces. What is the usual daily output?
   • ①11
   • ②10
   • ③160
   • ④12
   • ⑤275
   Show solution

   ①11

   1. Work backwards. The last step divided by 4, so multiply 5 by 4.
   2. Before that 35 was subtracted, so add 35 back.
   3. The first step multiplied by 5, so divide by 5.
   4. The original number is 11.
2. 2 · ★2
   Packing 33 sweets equally into bags leaves 5 over, and packing 25 chocolates the same number per bag leaves 4 over. What is the largest number that could go in one bag?
   • ①84
   • ②14
   • ③5
   • ④1
   • ⑤7
   Show solution

   ⑤7

   1. A remainder of 5 means 33 − 5 divides exactly.
   2. In the same way 25 − 4 divides exactly.
   3. Take the greatest common divisor of those two results.
   4. It must also exceed the remainders, so the answer is 7.
3. 3 · ★3
   Someone goes to the library every 4 days and to the pool every 7 days. They went to both today. Over the next 120 days, how many days will they go to both?
   • ①4
   • ②30
   • ③5
   • ④28
   • ⑤17
   Show solution

   ①4

   1. A multiple of both 4 and 7 is a common multiple.
   2. Their least common multiple is 28.
   3. Divide 120 by 28 and take the whole-number part.
   4. So there are 4.
4. 4 · ★4
   A class has 136 students in total. Writing (boys)/(girls) and reducing gives 8/9. How many boys and how many girls are there?
   • ①64 and 72
   • ②72 and 64
   • ③72 and 72
   • ④8 and 9
   • ⑤68 and 68
   Show solution

   ①64 and 72

   1. Since it reduces to 8/9, the original is 8×□ over 9×□.
   2. The numerator and denominator add to (8 + 9) × □.
   3. 136 ÷ (8 + 9) = 8, so □ = 8.
   4. The original fraction is 64 and 72.
5. 5 · ★5
   Bundling sweets in 5s leaves 0 over, and in 8s leaves 6 over. If the number of sweets is a two-digit number, what is the smallest it can be?
   • ①30
   • ②70
   • ③40
   • ④15
   • ⑤31
   Show solution

   ①30

   1. List the numbers leaving remainder 0 when divided by 5.
   2. Among those, find the ones leaving 6 when divided by 8.
   3. Numbers satisfying both conditions repeat every 40.
   4. The smallest two-digit one is 30.

Where people usually go wrong

• Working forwards when the problem runs backwards. If the last step divided, the first step of your reverse solution must multiply.
• Undoing the operations in the original order. Reversing means going from the last step to the first, not the other way round.
• Forgetting to subtract the remainder. "Leaves 2" means the number minus 2 divides exactly — the divisor does not divide the original number.
• Using a × b instead of the least common multiple. For 4 and 6 that gives 24, but common multiples actually step by 12.
• Scaling only the numerator or only the denominator when rebuilding a fraction. Both parts get multiplied by the same number.