Mental Math & Number Tricks

Correct answer: .

Adding up all of a number's digits and checking whether that sum is divisible by 3 is the standard test for divisibility by 3, and it works here because 4+7+3+1 equals 15, which is a multiple of 3, so 4,731 is too. The option that looks at the last digit and calls it even is wrong on two counts: 4,731 actually ends in an odd digit, and even/odd only ever tells you about divisibility by 2, never by 3. The option that reaches the same correct digit sum of 15 but then concludes divisibility by 9 confuses two different rules -- a digit sum divisible by 3 only guarantees divisibility by 3, not the stricter divisibility by 9, which needs the digit sum itself to be a multiple of 9. The option using an alternating sum of digits is applying the divisibility-by-11 test to the wrong number, and even that test's own result (1) would correctly say 4,731 fails it -- it just has nothing to do with divisibility by 3.

Source: National curriculum in England: mathematics programmes of study, Key Stage 2 (GOV.UK); Wolfram MathWorld: 'Divisibility Rules'

Correct answer: .

4,731 is not divisible by 9, even though it is divisible by 3, because the two rules require a different strength of digit sum: divisibility by 3 only needs the digit sum to be a multiple of 3, while divisibility by 9 needs that same digit sum to be a multiple of 9. Here the digit sum is 4+7+3+1=15, which is a multiple of 3 but not of 9 (9x1=9, 9x2=18, and 15 sits between them), so the stricter test fails. The option claiming that any multiple of 3 is automatically a multiple of 9 gets the logic backwards: because 9 is 3x3, every multiple of 9 is guaranteed to be a multiple of 3, but the reverse is not true, and 4,731 is exactly the kind of number that shows why. The option that accepts the same digit sum of 15 as sufficient for divisibility by 9 skips the actual requirement that the digit sum itself be a multiple of 9. The option pointing at the last digit, 1, is testing an irrelevant property; divisibility by 9 depends on the sum of all the digits, not on any single digit in isolation.

Source: National curriculum in England: mathematics programmes of study, Key Stage 3 (GOV.UK); Wolfram MathWorld: 'Divisibility Rules'

Correct answer: .

Alternately subtracting and adding the digits of 594 from the right (4 − 9 + 5 = 0) gives a result that is itself a multiple of 11 (zero counts), which is exactly the standard test for divisibility by 11, confirming that 594 divided by 11 equals 54. The option that instead sums all the digits (5+9+4=18) is applying the digit-sum test used for divisibility by 3 and 9, not 11, and following it here would wrongly suggest the number fails since 18 is not a multiple of 11. The option that doubles the last digit and subtracts it from the rest is the standard test for divisibility by 7, not 11, and applying it to 594 (59 − 8 = 51) again leads to the wrong conclusion since 51 is not a multiple of 11 either. The option relying on the number being a palindrome is a common but false pattern: while it's true that many small multiples of 11 happen to read the same forwards and backwards (11, 22, 121), 594 is not a palindrome yet is still perfectly divisible by 11, which disproves the idea that palindrome-ness has anything to do with the actual rule.

Source: Wolfram MathWorld: 'Divisibility Rules'; Brilliant.org Wiki: 'Divisibility Rules'

Correct answer: .

For any number ending in 5, squaring it always produces a result ending in 25, with the digits in front found by multiplying the tens digit by the next whole number up. For 65, the tens digit is 6, so 6x7=42, and attaching 25 gives 4,225, which matches exactly what long multiplication confirms (65x65=4,225). The option using 6x6=36 instead makes the common mistake of squaring the tens digit by itself rather than multiplying it by the number one greater. The option using 7x7=49 makes the opposite mistake, rounding the tens digit up before multiplying instead of pairing it with the next number. The option that correctly computes 6x7=42 but then attaches 35 instead of 25 breaks the one fixed part of this shortcut: the last two digits of the square of any number ending in 5 are always 25, never anything else.

Source: Math Fun Facts, Harvey Mudd College Department of Mathematics: 'Squares Ending in 5'

Correct answer: .

Multiplying by 9 is the same as multiplying by 10 and then subtracting the original number once, because 9 times any number equals 10 times that number minus one more copy of it. For 47, that means 47x10=470, and then 470−47=423, which is the correct answer. The option that stops after the x10 step and never subtracts anything is only calculating 47x10, not 47x9. The option that subtracts 37 instead of 47 has transposed the digits of the number being subtracted, taking away too little. The option that adds 47 to 470 instead of subtracting it moves in the wrong direction entirely, computing something closer to 47x11 rather than 47x9.

Source: cut-the-knot.org: 'Multiplication by 9, 99, 999, etc.'

Correct answer: .

To multiply a two-digit number by 11, add its two digits together and place that sum between them -- but when the sum is 10 or more, the extra ten has to be carried into the leading digit rather than just written down as a two-digit chunk. For 68, the digits sum to 6+8=14; since that's 10 or more, the 4 goes in the middle and the carried 1 is added to the leading 6, making it 7, for a final answer of 748. The option that simply inserts the full two-digit sum '14' between the original digits without carrying anything produces a number with too many digits and never actually carries the extra ten where it belongs. The option that keeps only the ones digit of the sum (4) but forgets to carry the ten into the leading digit misses exactly the step that matters once the digit sum reaches double figures. The option that does carry the 1 into the leading digit but reports the middle digit as 5 instead of 4 has simply added the two original digits incorrectly.

Source: Math Fun Facts, Harvey Mudd College Department of Mathematics: 'Multiplication by 11'

Correct answer: .

Breaking 35% into 30% plus 5% turns a single awkward percentage into two easy building blocks: 10% of 840 is found by moving the decimal point, giving 84, so 30% is three of those (84x3=252), and 5% is exactly half of that 10% block (42); adding 252+42 gives 294. The option landing on 336 comes from mistakenly treating 30% plus 10% as if it added up to 35%, when that combination is actually 40% of 840. The option landing on 273 comes from only adding half of the needed 5% block on top of 30%, effectively computing 32.5% instead of the full 35%. The option landing on 315 corresponds to 37.5% of 840, overshooting by adding too generous an extra sliver on top of the 30% block instead of the correct 5%.

Source: National curriculum in England: mathematics programmes of study, Key Stage 3 (GOV.UK) -- ratio, proportion and rates of change

Correct answer: .

Reducing each factor to its digital root -- repeatedly summing digits until one digit remains -- turns 47 into 2 (4+7=11, then 1+1=2) and 38 into 2 (3+8=11, then 1+1=2) as well, so the expected digital root of their true product is 2x2=4. Reducing the claimed answer, 1,786, the same way gives 1+7+8+6=22, then 2+2=4, which matches, meaning the multiplication is almost certainly correct; the check can't guarantee it's flawless, though, since casting out nines can occasionally miss specific errors, such as two digits in the final answer being swapped with each other, so a match is strong evidence of correctness rather than absolute proof. The option claiming a matching digital root proves the answer is exactly correct with no possible error overstates what the check can actually guarantee. The option that stops at the two-digit total of 22 without reducing it down to the single digit 4 never finishes the digital-root process, so comparing 22 directly against 4 produces a false 'mismatch' that isn't real. The option that suggests adding the two original numbers together (47+38) instead of finding their digital roots abandons the actual casting-out-nines procedure altogether, since the check works by comparing digital roots of the numbers being multiplied against the digital root of the product, not their sum.

Source: Wolfram MathWorld: 'Casting Out Nines'; Wikipedia: 'Casting out nines' (historical origin in Fibonacci's Liber Abaci, 1202)

Correct answer: .

The complements-from-100 shortcut multiplies two numbers close to 100 by first finding how far below 100 each one sits -- 3 for 97 and 4 for 96 -- then cross-subtracting one number's complement from the other factor (97−4, or equivalently 96−3, both give 93) to form the leading digits, and finally multiplying the two complements together (3x4=12) to form the trailing digits, combining them into 9,312. The option landing on 9,300 correctly finds the leading part but drops the complements' product entirely, effectively treating it as zero instead of 12. The option landing on 9,212 makes an error in the cross-subtraction step, computing 96−4=92 instead of pairing each number with the other's complement correctly. The option landing on 9,316 keeps the correct leading digits but miscalculates the complements' product as 16 (as if multiplying 4x4) instead of the correct 3x4=12.

Source: cut-the-knot.org: 'Product of numbers close to 100'

Correct answer: .

When two numbers are the same distance from a convenient middle value, their product equals the middle value squared minus that distance squared, because (m−d)(m+d) always works out to m squared minus d squared. Here 47 and 53 both sit 3 away from the midpoint 50, so the product is 50 squared minus 3 squared, or 2,500−9=2,491, which matches direct multiplication (47x53=2,491). The option landing on 2,500 uses only the midpoint squared and forgets to subtract anything for the distance at all. The option landing on 2,494 subtracts twice the distance (2x3=6) rather than the distance squared (3 squared=9), mixing up the two different roles distance plays in the formula. The option landing on 2,509 adds the squared distance instead of subtracting it, moving in the wrong direction entirely.

Source: Wikipedia: 'Difference of two squares'; cut-the-knot.org: 'Fast Arithmetic Tips'

Correct answer: .

The pairing trick young Carl Friedrich Gauss reportedly used pairs the first and last numbers, the second and second-to-last, and so on, so that every pair sums to the same total. For the numbers 1 through 50, pairing 1 with 50, 2 with 49, and so on all the way to 25 with 26 produces 25 pairs that each sum to 51, so the total is 25 times 51, or 1,275. The option that uses 50 pairs makes the pairing itself wrong: pairing every number with itself instead of with its partner from the opposite end doubles the pair count and breaks the method entirely. The option using a pair-sum of 50 gets the pair count right but miscalculates what each pair actually adds up to -- 1+50 is 51, not 50. The option landing on 51 correctly identifies each pair's sum but stops there, forgetting that this sum has to be multiplied by how many pairs there are.

Source: National Council of Teachers of Mathematics: 'The Story of Gauss'; NRICH, University of Cambridge: 'Clever Carl'

Correct answer: .

Since 5 is exactly half of 10, multiplying any number by 5 gives the same result as multiplying it by 10 and then taking half of that -- for 48, that means 48x10=480, then half of 480 is 240, which matches direct multiplication (48x5=240). The option landing on 480 performs the multiply-by-10 step correctly but never applies the halving step that the shortcut depends on, leaving the answer twice as large as it should be. The option landing on 24 does the opposite: it halves 48 first, which is a valid order for this shortcut, but then never multiplies by 10 afterward, leaving the answer only a tenth of the correct size. The option landing on 250 tries to halve first but makes an arithmetic slip, treating half of 48 as 25 instead of the correct 24, before multiplying by 10.

Source: Mathnasium: 'Mental Math Tricks for Multiplying by 5, 9, 11, and 25'

Correct answer: .

Since 25 is exactly a quarter of 100, multiplying any number by 25 gives the same result as dividing it by 4 and then multiplying by 100 -- for 36, that means 36÷4=9, then 9x100=900, matching direct multiplication (36x25=900). The option landing on 90 divides by 4 correctly but then multiplies by 10 instead of 100, losing a factor of ten along the way. The option landing on 800 keeps the right second step (multiplying by 100) but makes an arithmetic slip in the division, treating 36÷4 as 8 instead of the correct 9. The option landing on 3,600 skips the division step entirely and just multiplies 36 by 100, which would only be the correct shortcut for multiplying by 100 itself, not by 25.

Source: cut-the-knot.org: 'Division/Multiplication by 25'; Mathnasium: 'Mental Math Tricks for Multiplying by 5, 9, 11, and 25'

Correct answer: .

Because 100 is itself divisible by 4, any multiple of 100 -- and therefore every digit before the last two -- contributes nothing to whether the whole number is divisible by 4; only the two-digit number formed by the last two digits decides it. For 3,148, that's 48, and 48÷4=12 exactly, so 3,148 is divisible by 4 (3,148÷4=787). The option checking only the last digit happens to reach the right conclusion here since 8 is divisible by 4, but this single-digit test is not a real rule -- 18 also ends in 8, yet 18÷4 is not a whole number, so the test fails elsewhere. The option using the digit sum (16) borrows the test for divisibility by 3 or 9, which has nothing to do with divisibility by 4; it happens to land on a true conclusion here purely by coincidence. The option checking only for evenness tests divisibility by 2, a weaker condition -- 6 and 18 are both even but neither is divisible by 4, so evenness alone can never establish divisibility by 4.

Source: Wikipedia: 'Divisibility rule'

Correct answer: .

Because 6 equals 2 times 3, a number is divisible by 6 exactly when it passes both the divisibility test for 2 (it's even) and the divisibility test for 3 (its digit sum is divisible by 3); 522 is even and its digit sum, 9, is divisible by 3, so both conditions hold and 522 is divisible by 6 (522÷6=87). The option checking divisibility by 3 alone reaches the right conclusion for 522 but only because 522 happens to also be even -- a number like 9 passes the divisibility-by-3 test just as easily but isn't divisible by 6 at all. The option checking evenness alone has the same gap in reverse: 8 is even but fails the divisibility-by-3 test, so it isn't divisible by 6 either. The option that tests whether the digit sum itself is divisible by 6 applies a rule that doesn't exist -- there is no such direct digit-sum test for 6 -- and it leads to the wrong conclusion here, since 522 actually is divisible by 6 even though its digit sum (9) is not.

Source: Wikipedia: 'Divisibility rule'

Correct answer: .

The compensation shortcut rounds the number being subtracted up to a friendly value, subtracts that instead, and then adds back the extra amount that rounding introduced; rounding 197 up to 200 means subtracting 3 more than the problem actually asked for, so 503-200=303 has to be corrected by adding that 3 back, giving 306, which matches direct subtraction (503-197=306). The option landing on 303 performs the rounded subtraction correctly but never applies the compensation step at all, leaving the answer exactly 3 too small. The option landing on 300 does remember that a compensation step exists but applies it in the wrong direction, subtracting the extra 3 again instead of adding it back. The option landing on 293 has nothing to do with the compensation step itself -- it comes from an arithmetic slip in the rounded subtraction 503-200, which should be 303, not 293.

Source: Shelley Gray Teaching: 'Compensation: An Addition Strategy'; SplashLearn: 'What is Compensation in Math?'

Correct answer: .

Since 101 equals 100 plus 1, multiplying any number by 101 gives the same result as multiplying it by 100 and then adding the original number back once -- for 34, that's 34x100=3,400, plus 34, giving 3,434, which matches direct multiplication (34x101=3,434); for a two-digit number like this, that shortcut also has the neat side effect of just writing the number twice in a row. The option landing on 3,400 gets the multiply-by-100 step right but never adds the original number back at all, leaving out the extra one copy that 101 contributes beyond 100. The option landing on 3,468 adds the original number back twice instead of once, as if multiplying by 102 rather than 101. The option landing on 340 loses a zero by treating the first step as multiplying by 10 instead of 100, even though it does correctly add the extra 34 afterward.

Source: MathNinja.org: 'Multiplying by 101'

Correct answer: .

Halving one factor while doubling the other always leaves the product unchanged, because dividing one number by 2 and multiplying the other by 2 cancels out; starting from 16x25, halving 16 to 8 and doubling 25 to 50 gives 8x50=400, and repeating the move once more (halving 8 to 4, doubling 50 to 100) gives 4x100=400 as well, both matching direct multiplication (16x25=400). The option landing on 200 breaks the pairing rule by halving 16 twice but only doubling 25 once, so the two adjustments no longer offset each other and the product shrinks to half its true value. The option landing on 100 makes a different pairing error, halving both factors instead of halving one and doubling the other, which shrinks the product to a quarter of its true value. The option landing on 1,600 doubles both factors instead of halving one and doubling the other, which quadruples the product instead of leaving it unchanged.

Source: Komodo Math: 'Multiplication - the Halving and Doubling Strategy'; Magoosh: 'Mental Math: Doubling and Halving'

Correct answer: .

A 20% increase multiplies the original price by 1.20, and a later 20% decrease multiplies whatever the price is at that point by 0.80; applying both in sequence gives 1.20x0.80=0.96, so the final price is 96% of the original -- a net 4% loss, not a return to the starting price -- because the decrease is calculated on the larger, already-increased amount, so it removes more in absolute terms than the increase added. The option claiming the changes cancel out to 100% is the most common version of this mistake, treating equal percentage changes as if they were equal dollar amounts rather than percentages of two different bases. The option landing on 80% makes a different error, simply adding the two percentages together as a flat 40% swing instead of applying them one after another multiplicatively. The option landing on 104% gets the direction of the net effect backwards -- it's the decrease that's calculated on the larger base and therefore dominates, not the increase.

Source: The Math Doctors: 'Percent Change: Working Backward'

Correct answer: .

Recognizing that 12.5% is exactly the fraction 1/8 turns a percentage calculation into a simple division: 12.5% of 96 is 96÷8=12, which matches the decimal calculation (0.125x96=12). The option landing on 9.6 skips the fraction shortcut entirely and instead calculates 10% of 96, treating 12.5% as if it were the more familiar 10% benchmark rather than its own distinct fraction. The option landing on 24 correctly identifies 12.5% as 1/8 but then inverts the operation, multiplying by 8 instead of dividing by it, which is the operation that would undo an eighth rather than find it. The option landing on 6 uses the right kind of shortcut but the wrong fraction, mistaking 12.5% for 1/16 (which is actually 6.25%) instead of the correct 1/8.

Source: Math Is Fun: 'Decimals, Fractions and Percentages'

Correct answer: .

Dropping the last digit of 826 (the 6), doubling it to get 12, and subtracting that from the number left over (82) gives 82 minus 12 = 70, which is a multiple of 7 (7 x 10), so 826 is divisible by 7 as well -- and indeed 826 divided by 7 works out to exactly 118. The option checking whether the last digit is even is testing the wrong thing entirely: even-or-odd only ever reveals divisibility by 2, and 826 happens to be even anyway, but that tells you nothing about divisibility by 7. The option using an alternating sum of digits is applying the test for divisibility by 11, not 7; it even reaches a correct conclusion about 11 (12 is not a multiple of 11) while saying nothing valid about 7. The option adding up all the digits is applying the test for divisibility by 3 or 9, and although 16 genuinely is not divisible by 3 or 9 either, that rule simply has no bearing on whether a number is divisible by 7 -- each small prime has its own distinct shortcut, and they are not interchangeable with one another.

Source: Mathnasium: 'Divisibility Rule for Seven'; GeeksforGeeks: 'Divisibility Rules From 1 to 19'

Correct answer: .

Multiplying by 99 works like the multiply-by-101 shortcut but in reverse: 99 is 100 minus 1, so 68 x 99 equals 68 x 100 minus 68 one time, which is 6,800 minus 68 = 6,732. The option that adds 68 back instead of subtracting it is applying the multiply-by-101 rule, which fits 100 + 1 and needs an addition, to a problem built around 100 minus 1, which needs a subtraction instead -- a mix-up between two closely related but opposite shortcuts. The option subtracting 99 instead of 68 confuses the number being multiplied by (99, which only shows up as 100 minus 1 so the shortcut works, and is never itself the thing subtracted) with the original number the shortcut is meant to act on. The option subtracting 86 performs the right kind of step, a subtraction from 6,800, but with the digits of 68 accidentally swapped, exactly the sort of quick transposition slip that creeps in when working without pen and paper.

Source: cut-the-knot.org: 'Multiplication by 9, 99, 999, etc.'

Correct answer: .

97 is 3 below 100, so d = 3, and the shortcut's formula gives 10,000 minus 200 times 3 plus 3 squared, which is 10,000 minus 600 plus 9 = 9,409 -- and that checks out, since 97 x 97 really does equal 9,409. This formula comes from expanding (100 minus d) squared algebraically: squaring the 100 gives 10,000, the middle cross-term doubles to 200d and is subtracted, and the small d squared term is added back at the end, so all three pieces have to appear with the right size and the right sign for the shortcut to work. The option subtracting d squared instead of adding it drops that final correction in the wrong direction, undershooting the true answer by twice d squared. The option using the wrong distance (4 instead of 3) shows why getting d itself right matters before anything else -- the rest of the arithmetic can be flawless and still land on the wrong number if d is wrong going in. The option using 100d instead of 200d forgets that the middle term in (100 minus d) squared is doubled, since multiplying out (100 minus d)(100 minus d) produces two identical cross-terms of 100 times d, not just one.

Source: cut-the-knot.org: 'Product of numbers close to 100'

Correct answer: .

A 20% discount leaves 80% of the original price behind, so the $64 final price represents 80%, or 0.8, of whatever the backpack started at -- meaning the original price is found by dividing, not multiplying: $64 divided by 0.8 equals $80. The option that adds 20% of the discounted price back onto $64 makes a very natural but incorrect assumption: 20% of $64 is a smaller amount than 20% of the true original $80, because the discounted price is already the smaller, shrunken base, so adding a percentage of it back can never fully undo a percentage that was taken off the larger original. The option dividing by 1.2 reverses the wrong kind of change altogether -- dividing by 1.2 undoes a 20% increase, but this problem describes a 20% decrease, which is undone by dividing by 0.8 instead. The option adding a flat 20 misreads the 20% as though it were a dollar amount rather than a percentage of a price that was never stated directly in the question.

Source: Third Space Learning: 'Reverse Percentages'; Math Is Fun: 'Decimals, Fractions and Percentages'

Correct answer: .

This list runs from 4 to 100 going up by 4 each time, which is 25 terms in total; pairing the first with the last, the second with the second-to-last, and so on always produces the same sum, 104 (4+100, 8+96, and so on), and since 25 is an odd number of terms, one term in the exact middle (52) is left over unpaired. That gives 12 complete pairs worth 104 each, plus the lone middle term, for 12 x 104 + 52 = 1,300. The option landing on 1,248 undercounts the terms as 24 by forgetting that counting from 4 to 100 in steps of 4 requires adding 1 after dividing the span by the step size, the same kind of off-by-one slip that trips up fence-post counting problems generally. The option landing on 2,600 multiplies the paired total by every term instead of by the number of pairs, effectively double-counting since each pair already accounts for two terms at once. The option landing on 1,200 breaks the premise of the whole trick by subtracting the endpoints instead of adding them, which does not produce a constant, repeatable pair total at all.

Source: National Council of Teachers of Mathematics: 'The Story of Gauss'; NRICH, University of Cambridge: 'Clever Carl'

Correct answer: .

Because 1,000 is itself divisible by 8 (1,000÷8=125), every multiple of 1,000 -- and therefore every digit before the last three -- contributes nothing to whether the whole number is divisible by 8; only the three-digit number formed by the last three digits decides it. For 7,136, that's 136, and 136÷8=17 exactly, so 7,136 is divisible by 8 (7,136÷8=892). The option checking only the last digit tests a rule that only ever works for divisibility by 2 or 5, since a single final digit can never capture a full factor of 8; it also reaches the wrong conclusion here, since 6 alone isn't divisible by 8 even though the whole number is. The option checking only the last two digits borrows the real rule for divisibility by 4 (where 100's divisibility by 4 makes two digits enough), but 100 itself is not divisible by 8 (100÷8=12.5), so two digits can never reliably settle the question for 8; it likewise reaches the wrong conclusion, since 36 isn't divisible by 8 even though the full number is. The option summing all the digits is applying the test for divisibility by 3 or 9, a completely different rule based on remainders after dividing by 9, with no connection to divisibility by 8; it too lands on the wrong conclusion, since 17 isn't divisible by 8 despite the whole number being so.

Source: Wikipedia: 'Divisibility rule'

Correct answer: .

Multiplying by 15 is the same as multiplying by 10 and then by 1.5, so multiplying by 10 first and adding back half of that result finishes the job in two easy steps: 24x10=240, half of 240 is 120, and 240+120=360, which matches direct multiplication (24x15=360). The option adding the original number (24) instead of half the product confuses this shortcut with a different one -- the useful adjustment here is half of the already-scaled-up product (120), not the small original number, and adding only 24 falls 96 short of the true answer. The option halving the original number (to 12) instead of the x10 product makes a sizing mistake: it adjusts a number ten times too small, so the correction added back (12) is only a tenth of what's needed (120), landing 108 short. The option doubling the x10 product instead of halving it moves in the wrong direction entirely -- 15 is 1.5 times 10, which calls for adding back half of the product, not doubling it, and doubling instead overshoots the true answer by 360.

Source: StudyRaid: 'Spoken rule for multiplying by 15'; MTG Learning Media: 'Mental Math Shortcuts to Solve Complex Calculations'

Correct answer: .

Square roots grow more slowly as numbers get larger, so the gap between consecutive perfect squares stretches out even though it represents the same single step up in the root -- between 49 (7 squared) and 64 (8 squared) there's a 15-unit gap, and 50 sits just 1 unit into that gap, so the root should land only a small fraction of the way from 7 toward 8, close to 7+1/14=7.07 using the standard correction, which matches the true value of roughly 7.071. The option landing on exactly 7 captures the right starting bracket but skips the adjustment step entirely, treating 50 as if it were exactly 49. The option landing on 7.5 makes the opposite mistake, assuming the fraction of the gap is always one-half regardless of where the number actually sits -- that would only be correct if the number were close to 56 or 57, roughly the midpoint between 49 and 64, not 50. The option landing on roughly 8 skips over 8 squared (64) entirely and instead brackets 50 between 7 squared and 9 squared (49 and 81), a pair of perfect squares that aren't even consecutive, so splitting evenly between 7 and 9 has no connection to where 50 actually falls among the real, consecutive perfect squares surrounding it.

Source: Casual Inference: 'Speed Math 1: How to estimate square roots'; The Axiom Academy mental-math square-root estimation lesson

Correct answer: .

The Rule of 72 estimates doubling time by dividing 72 by the growth rate expressed as a whole number, not a decimal: at 8%, that's 72÷8=9 years, which closely matches the exact value from compound-growth math (about 9.01 years), since 72 was chosen specifically because it divides evenly by many common small rates. The option landing on roughly 900 years makes a very natural but costly mistake, treating the 8% rate as 0.08 and dividing 72 by that decimal instead of by the whole number 8 -- off by a factor of 100. The option landing on roughly 80 years adds the rate to 72 instead of dividing, treating the rule as if it meant something like 'add the rate to 72' rather than 'divide 72 by the rate.' The option landing on roughly 8.75 years swaps in 70 instead of 72 as the numerator -- 70÷8=8.75 -- which is close but uses the wrong constant for this particular shortcut; 70 (or more precisely about 69.3, based on the natural logarithm of 2) is the more accurate numerator for continuous compounding, while 72 is the conventional choice for the Rule of 72 because it has more small whole-number divisors and fits typical annual compounding better.

Source: Wikipedia: 'Rule of 72'

Correct answer: .

Picking 60 as a round reference near the middle of the list, the five values sit +2, −2, +5, 0, and +10 away from it; those deviations sum to 15, and dividing that sum by the count of values (5) gives an average deviation of 3, so the true average is 60+3=63 -- which matches adding the numbers directly (62+58+65+60+70=315, 315÷5=63). The option landing on 75 skips the division step, adding the raw deviation total (15) straight back onto the reference instead of first spreading it evenly across the five values, overshooting by 12. The option landing on 57 gets the direction backwards, subtracting the average deviation from the reference instead of adding it -- appropriate only if the values mostly sat below the reference, which they don't here, since the deviations are mostly positive. The option landing on 63.75 keeps the right idea but divides the deviation sum by the wrong count, 4 instead of the actual 5 values in the list, nudging the estimate slightly too high.

Source: SpeedMath: 'Average Calculation Shortcut: Find the Mean of Any Data Set in 3 Seconds'; Open Exam Prep: 'Averages, Estimation Shortcuts, and Quick Approximations'