Level 3: Whole-Number Operations in Practical Contexts
Level 3 problems ask you to apply one or two basic operations - addition, subtraction, multiplication, division - to whole numbers in a plain everyday situation. Typical settings: counting inventory, totaling a bill, splitting items into equal groups, or finding a remainder. The math itself is simple; the challenge is reading carefully and choosing the right operation.
Concepts to know
- Place value and regrouping when adding or subtracting multi-digit numbers.
- Multiplication as repeated groups and division as fair sharing or grouping.
- Remainders: when a division does not come out even, decide whether the question wants the remainder, the rounded-down quotient, or the rounded-up quotient.
- Order of operations at a basic level: work inside parentheses first, then multiply/divide, then add/subtract.
Traps to watch
- Keyword mismatch. "How many are left" signals subtraction, but "how many more are needed" often signals a subtraction too - read what is being compared.
- Rounding direction. If you are packing 47 items into boxes of 6, you need 8 boxes, not 7 remainder 5. Questions about "how many containers are needed" always round up.
- Extra numbers. Some questions include a number you do not need. Identify the goal before computing.
- Units. Answers may be requested in dozens, cases, or pairs - convert before answering.
Rule of thumb
Restate the question in your own words as a single sentence with a blank: "I need to find ___." Then ask whether the answer should be bigger or smaller than the starting number; that tells you whether to multiply/add or divide/subtract.
Worked example
A shelf holds 24 cans per row and there are 5 full rows, plus 9 loose cans. How many cans in all? 24 × 5 = 120, then 120 + 9 = 129 cans.
Level 5: Multi-Step Problems with Rates, Ratios, and Proportions
Level 5 problems chain two or more operations and often involve rates (per hour, per mile, per unit) or proportions (scaling a recipe, mixing a solution, comparing speeds). The arithmetic is familiar; the difficulty is sequencing.
Concepts to know
- Unit rate: divide to get "per one" (e.g., 180 miles in 3 hours = 60 mph).
- Proportion setup: a/b = c/d, cross-multiply to solve.
- Work and time: if a machine makes 40 parts per hour, in 2.5 hours it makes 100 parts.
- Averages: total ÷ count; to find a missing value from an average, multiply the average by the count and subtract the known values.
- Combining rates: two workers together add their rates.
Traps to watch
- Mismatched units. Minutes vs. hours, ounces vs. pounds - convert first, then compute.
- Averaging averages. You cannot average two averages unless the group sizes are equal; go back to totals.
- Direction of the proportion. Set up the ratio with matching labels on top and bottom so the units cancel.
- Stopping too early. Multi-step questions often require a second operation after the first answer; reread the question to confirm you answered what was asked.
Rule of thumb
Label every number with its unit as you write it down. If a computation leaves you with a unit that does not match the question (e.g., "hours" when they want "minutes"), you have one more conversion to do.
Worked example
A crew of 3 painters paints 12 rooms in 4 days. At the same rate, how many rooms will 5 painters paint in 6 days? Rate per painter per day = 12 ÷ (3 × 4) = 1 room. Then 5 × 6 × 1 = 30 rooms.