4.3 Quick Numerical Problem-Solving
Key Takeaways
- Multiplying by 5 is fastest done as multiply by 10 then divide by 2, and multiplying by 9 is fastest done as multiply by 10 then subtract the original number once
- Estimating and rounding numbers before calculating lets you eliminate two or three clearly wrong answer options in a few seconds, even before finding the exact value
- Squaring a number near 50 or 100 has a shortcut pattern that avoids long multiplication entirely
- Plugging in the answer options directly into a word problem is often faster than setting up and solving a full equation, especially for ratio, average, and simple work-rate items
- Quick numerical problem-solving items still deserve a full read of the question, since a fast wrong setup is worse than a slightly slower correct one
Quick Numerical Problem-Solving
Quick Answer: This item type tests basic arithmetic and simple word problems under the same roughly 22-second-per-item pace as the rest of Mental Ability. The winning approach is not doing the full calculation faster - it is avoiding the full calculation whenever a shortcut, an estimate, or a plug-in from the answer options gets you to the correct choice sooner.
Why Speed Math Belongs in Mental Ability
Quick numerical problem-solving sits inside Mental Ability rather than the separate Mathematics subtest because it measures raw calculation speed and number sense, not subject-specific math content like algebra or geometry. The numbers involved are usually simple - whole numbers, basic fractions, everyday percentages - but the time pressure is real, so the value of this section comes entirely from mental-math shortcuts rather than from knowing more advanced formulas.
Shortcut 1: Multiplying by 5
Multiplying any number by 5 is fastest as multiply by 10, then divide by 2, since both of those steps are close to instant.
Worked example: 48 times 5. Multiply by 10 first: 480. Divide by 2: 240. The answer is 240, found in two easy mental steps instead of one harder one.
Shortcut 2: Multiplying by 9
Multiplying by 9 is fastest as multiply by 10, then subtract the original number once.
Worked example: 63 times 9. Multiply by 10 first: 630. Subtract the original 63: 630 minus 63 equals 567. The answer is 567.
Shortcut 3: Squaring Numbers Near 50 or 100
For numbers close to 50 or 100, squaring has a pattern that skips long multiplication.
Worked example: 49 squared. Since 49 is one less than 50, use the pattern (50 minus 1) squared, which equals 50 squared minus (2 times 50) plus 1: 2500 minus 100 plus 1 equals 2401. With practice, numbers just above or below a round base like 50 or 100 can be squared almost as fast as reading the question.
Shortcut Reference Table
| Operation | Shortcut | Worked check |
|---|---|---|
| Multiply by 5 | x10, then divide by 2 | 48 x 5 = 480 / 2 = 240 |
| Multiply by 9 | x10, then subtract the original number | 63 x 9 = 630 - 63 = 567 |
| Multiply by 25 | x100, then divide by 4 | 16 x 25 = 1600 / 4 = 400 |
| Percent of a number | Find 10 percent first, then scale up or down | 15 percent of 80 = (10 percent = 8) + (5 percent = 4) = 12 |
| Square near a round base | Use (base plus or minus difference) squared expansion | 49 squared = 2500 - 100 + 1 = 2401 |
Estimate First, Calculate Only If Needed
Before doing exact arithmetic, round the numbers in the question to the nearest convenient value and estimate the answer. If your estimate clearly matches only one answer option, you may not need the exact calculation at all.
Worked example: What is approximately 31 percent of 199?
Round 31 percent to 30 percent and 199 to 200. Thirty percent of 200 is 60. If the four answer options are 40, 60, 95, and 120, only 60 is close to your estimate, so you can select it immediately without computing the exact value of 31 percent of 199.
Plug-In Strategy for Word Problems
For certain word problems - especially ratio, average, and simple work-rate items - testing the answer options directly in the word problem is often faster than building and solving a full equation from scratch.
Worked example: The average of three consecutive whole numbers is 15. What is the smallest of the three numbers?
Rather than writing an algebraic equation, test the options directly. If an option says 14, check: 14, 15, 16 averages to (14 plus 15 plus 16) divided by 3, which equals 45 divided by 3, which equals 15. This matches the given average exactly, so 14 is the correct smallest number - confirmed in one quick check rather than a full algebraic setup.
When Plugging In Is Not the Faster Choice
Plugging in works best when the options are simple whole numbers and the check itself is quick, such as the consecutive-numbers example above. It works poorly when checking a single option requires just as much work as solving the equation directly, such as complex multi-step ratio problems with several unknowns - in those cases, a direct, careful setup is usually faster overall.
Time-Pressure Strategy
- Read the question fully once before choosing a method - a fast wrong setup wastes more time than a slightly slower correct one, since you may need to start over.
- Default to estimation first for any percentage, average, or large-number question; only calculate exactly if the estimate does not clearly separate the answer options.
- Use the plug-in strategy specifically for ratio, average, and simple work-rate items with clean whole-number options.
- Reserve full written calculation for the small number of items where no shortcut applies, and accept that these will take closer to 30 seconds rather than the average 22.
Common Traps
- Reaching for a full calculation out of habit when a two-step shortcut (like the multiply-by-5 or multiply-by-9 tricks) would reach the same answer faster.
- Estimating so loosely that two answer options both seem plausible, which means the estimate needs to be tightened rather than trusted as final.
- Plugging in answer options for a problem type where checking each option is actually just as slow as solving the equation directly.
- Rushing the initial reading of the question and solving for the wrong quantity, such as finding the largest of three numbers when the question asked for the smallest.
What is 48 times 5, using the multiply-by-10-then-halve shortcut?
Approximately what is 31 percent of 199, using rounding to estimate?
The average of three consecutive whole numbers is 15. Using the plug-in strategy, what is the smallest of the three numbers?