Grade 5 Mixed Operations Practice

What this problem type covers

Twenty-four generated types across five levels: left-to-right calculations, full operation order, comparing expressions, identifying the calculation order, translating a short context into one expression, finding an order error, matching equivalent expressions, evaluating a conversion rule, solving nested missing-value expressions, balanced weights, empty-container weights, defined operations, searching one or two pairs of parentheses and operators, counting natural numbers in an inequality, correcting a mistaken calculation, comparing every four-operator arrangement, inferring a defined operation, score assumptions, two-price assumptions, unsold-item profit problems, three-price ratio problems, and linked age 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
   Choose the order that shows how to calculate the expression.
   37 + (②8 − 7)③ × 3④ ÷ 7①
   • ①③→②→④→①
   • ②②→③→④→①
   • ③①→②→③→④
   • ④②→④→③→①
   • ⑤②→③→①→④
   Show solution

   ②②→③→④→①

   1. Calculate inside the parentheses first.
   2. Then multiply and divide from left to right.
   3. Addition is last.
2. 2 · ★2
   Find the calculation that is incorrect.
   ㉠ 57+(17−13)÷2=59
   ㉡ 48−16+10=22
   ㉢ 23+8×5=63
   ㉣ (9+11)×4=80
   ㉤ 60÷6+3=13
   • ①㉤
   • ②㉠
   • ③㉢
   • ④㉣
   • ⑤㉡
   Show solution

   ⑤㉡

   1. Evaluate parentheses first and operations of equal priority from left to right.
   2. ㉡ uses the wrong order of operations.
   3. 48−16+10=42
3. 3 · ★3
   Find the number that makes the equation true.
   66 ÷ (14 − □) × 5 − 20 = 35
   • ①11
   • ②13
   • ③8
   • ④20
   • ⑤3
   Show solution

   ③8

   1. Undo the outer operations in reverse.
   2. 35+20=55
   3. 55÷5=11
   4. 14−□=66÷11
   5. □=8
4. 4 · ★4
   Find how many positive integers can replace □ and make the inequality true.
   11 + 7 × □ < 67
   • ①9
   • ②6
   • ③7
   • ④67
   • ⑤8
   Show solution

   ③7

   1. 11+7×□=67 gives
   2. □=8
   3. There are 7 positive integers below the boundary.
5. 5 · ★5
   There are three people: the youngest, the middle, and the oldest. The middle person's age is 2 years less than 2 times the youngest person's age. The oldest person's age is 8 years less than 2 times the middle person's age. The oldest person's age is 3 times the youngest person's age. Find the youngest person's age.
   • ①13
   • ②3
   • ③14
   • ④12
   • ⑤24
   Show solution

   ④12

   1. Let the youngest age be □ and combine both relations.
   2. ((□×2)−2)×2−8=□×3
   3. □=12

Where people usually go wrong

• Adding from left to right before doing multiplication or division.
• Ignoring parentheses, or treating every pair of parentheses as optional decoration.
• Undoing a missing-value expression from the inside instead of reversing the outside operations first.
• Dividing a total weight by the number of objects without removing the container mass.
• Counting the baseline kind instead of the replacements in an assume-all-one-kind problem.
• Using profit per item as though every purchased item was sold.
• Checking one operator placement and calling it the largest without testing all valid placements.