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Practice · Quant prep · Mental arithmetic

Mental arithmetic interview questions

66 questions with worked solutions. The fast, exact arithmetic trading interviews test under time pressure.

36 easy · 25 medium · 5 hard · Question type reported at:

Take the 80 in 8 and other timed tests Practise these: 30 questions · 15s each All Quant prep topics

Applying to trading firms? The Quant Careers Report: what 13 firms test and pay, and when they hire. £79.

1 Easy Type asked at
17×2317\times23
Show answer
Answer: 391391
(20−3)(20+3)=400−9(20-3)(20+3)=400-9.
2 Easy Type asked at
48×5248\times52
Show answer
Answer: 24962496
(50−2)(50+2)=2500−4(50-2)(50+2)=2500-4.
3 Easy Type asked at
99×8799\times87
Show answer
Answer: 86138613
100⋅87−87=8700−87100\cdot87-87=8700-87.
4 Easy Type asked at
125×64125\times64
Show answer
Answer: 80008000
125×8=1000125\times8=1000, and 64=8×864=8\times8.
5 Easy Type asked atSIG
38\tfrac38 as a decimal
Show answer
Answer: 0.3750.375
18=0.125\tfrac18=0.125, times 3.
6 Easy Type asked at
716\tfrac{7}{16} as a decimal
Show answer
Answer: 0.43750.4375
116=0.0625\tfrac1{16}=0.0625, times 7.
7 Easy Type asked at
15%15\% of 360360
Show answer
Answer: 5454
10%=3610\%=36, 5%=185\%=18.
8 Easy Type asked at
2025\sqrt{2025}
Show answer
Answer: 4545
Squares ending in 25 come from numbers ending in 5: 452=4⋅5 ∣ 25=202545^2=4\cdot5\,|\,25=2025.
9 Medium Type asked at
1.0521.05^2
Show answer
Answer: 1.10251.1025
1+2(0.05)+0.0521+2(0.05)+0.05^2.
10 Easy Type asked at
2162^{16}
Show answer
Answer: 65,53665{,}536
210=10242^{10}=1024, 216=1024⋅642^{16}=1024\cdot64.
11 Medium Type asked atSIG
37237^2
Show answer
Answer: 13691369
(40−3)2=1600−240+9(40-3)^2=1600-240+9.
12 Medium Type asked at
17\tfrac17 to four decimal places
Show answer
Answer: 0.14290.1429
17=0.142857‾\tfrac17=0.\overline{142857}.

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The Quant Careers Report

Speed with numbers is screened first at market makers. The report covers the arithmetic tests candidates describe, how 13 firms interview, and the pay they state.

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13 Easy Type asked at
0.125×880.125\times88
Show answer
Answer: 1111
0.125=180.125=\tfrac18.
14 Easy Type asked at
624÷8624\div8
Show answer
Answer: 7878
640÷8=80640\div8=80, minus 16÷8=216\div8=2.
15 Easy Type asked at
1001×71001\times7
Show answer
Answer: 70077007
7000+77000+7.
16 Easy Type asked atSIG
55255^2
Show answer
Answer: 30253025
5⋅6 ∣ 255\cdot6\,|\,25.
17 Easy Type asked at
12.5%12.5\% of 480480
Show answer
Answer: 6060
18\tfrac18 of 480.
18 Medium Type asked atSIG
999×999999\times999
Show answer
Answer: 998,001998{,}001
(1000−1)2=1,000,000−2000+1(1000-1)^2=1{,}000{,}000-2000+1.
19 Easy Type asked at
6.25×166.25\times16
Show answer
Answer: 100100
6.25=2546.25=\tfrac{25}4, and 16/4=416/4=4.
20 Easy Type asked at
3.75×123.75\times12
Show answer
Answer: 4545
3.75=1543.75=\tfrac{15}4.
21 Medium Type asked atSIG
81×1981\times19
Show answer
Answer: 15391539
81⋅20−8181\cdot20-81.
22 Easy Type asked at
23\tfrac23 of 0.90.9
Show answer
Answer: 0.60.6
0.9/3=0.30.9/3=0.3, doubled.
23 Medium Type asked at
144÷0.12144\div0.12
Show answer
Answer: 12001200
14400÷1214400\div12.
24 Easy Type asked at
7×8×97\times8\times9
Show answer
Answer: 504504
56×956\times9.
25 Medium Type asked at
1.131.1^3
Show answer
Answer: 1.3311.331
Binomial coefficients 1, 3, 3, 1.
26 Easy Type asked at
0.3×0.070.3\times0.07
Show answer
Answer: 0.0210.021
3×7=213\times7=21, three decimal places.
27 Medium Type asked at
56−34\tfrac56-\tfrac34 (as a decimal)
Show answer
Answer: 112≈0.0833\tfrac1{12}\approx0.0833
1012−912\tfrac{10}{12}-\tfrac{9}{12}.
28 Easy Type asked atSIG
250×36250\times36
Show answer
Answer: 90009000
1000×91000\times9.
29 Easy Type asked at
11×8711\times87
Show answer
Answer: 957957
Write 8, then 8+7=158+7=15 (carry): 957957.
30 Easy Type asked atSIG
1÷0.041\div0.04
Show answer
Answer: 2525
0.04=1250.04=\tfrac1{25}.
31 Easy Type asked atSIG
13×1713\times17
Show answer
Answer: 221221
(15−2)(15+2)=225−4(15-2)(15+2)=225-4.
32 Easy Type asked at
96×10496\times104
Show answer
Answer: 99849984
(100−4)(100+4)=10,000−16(100-4)(100+4)=10{,}000-16.
33 Easy Type asked at
45245^2
Show answer
Answer: 20252025
Numbers ending in 5: 4⋅5 ∣ 254\cdot5\,|\,25.
34 Easy Type asked at
18\tfrac18 as a percentage
Show answer
Answer: 12.5%12.5\%
18=0.125\tfrac18=0.125.
35 Easy Type asked at
2122^{12}
Show answer
Answer: 40964096
210=10242^{10}=1024, doubled twice.
36 Easy Type asked at
7×11×137\times11\times13
Show answer
Answer: 10011001
Worth memorising: 7⋅11⋅13=10017\cdot11\cdot13=1001, which is why abc abcabc\,abc is always divisible by 7, 11 and 13.
37 Easy Type asked at
0.75×640.75\times64
Show answer
Answer: 4848
34\tfrac34 of 64.
38 Easy Type asked atSIG
58\tfrac58 as a decimal
Show answer
Answer: 0.6250.625
18=0.125\tfrac18=0.125, times 5.
39 Medium Type asked atSIG
111×111111\times111
Show answer
Answer: 12,32112{,}321
Repunits square to a palindrome of digits 1,2,3,2,11,2,3,2,1.
40 Easy Type asked at
18×2518\times25
Show answer
Answer: 450450
25=100425=\tfrac{100}4, so 18⋅100/418\cdot100/4.
41 Medium Type asked atSIG
1.541.5^4
Show answer
Answer: 5.06255.0625
1.52=2.251.5^2=2.25, and 2.252=5.06252.25^2=5.0625.
42 Easy Type asked at
900÷0.6900\div0.6
Show answer
Answer: 15001500
9000÷69000\div6.
43 Medium Type asked at
67×6367\times63
Show answer
Answer: 42214221
Same tens digit, units adding to 10: multiply the tens digit by one more than itself, 6×7=426\times7=42, then the units, 7×3=217\times3=21. Write them side by side: 42214221.
44 Medium Type asked at
86×8486\times84
Show answer
Answer: 72247224
Same tens, units add to 10: 8×9=728\times9=72, then 6×4=246\times4=24, giving 72247224.
45 Medium Type asked at
73×7773\times77
Show answer
Answer: 56215621
7×8=567\times8=56 and 3×7=213\times7=21: 56215621. The rule is (10a+b)(10a+10−b)=100a(a+1)+b(10−b)(10a+b)(10a+10-b)=100a(a+1)+b(10-b).
46 Medium Type asked at
98×9798\times97
Show answer
Answer: 95069506
Both near 100: take the shortfalls, 2 and 3. Subtract one shortfall from the other number, 98−3=9598-3=95, and append their product, 0606: 95069506. It is (100−a)(100−b)=100(100−a−b)+ab(100-a)(100-b)=100(100-a-b)+ab.
47 Medium Type asked at
103×107103\times107
Show answer
Answer: 11,02111{,}021
Both just over 100: 103+7=110103+7=110, then append 3×7=213\times7=21 as two digits: 11,02111{,}021. The same identity as below 100, with the signs flipped.
48 Medium Type asked atSIG
58×6258\times62
Show answer
Answer: 35963596
Equidistant from 60: (60−2)(60+2)=3600−4=3596(60-2)(60+2)=3600-4=3596.
49 Hard Type asked at
1182118^2
Show answer
Answer: 13,92413{,}924
(120−2)2=14,400−480+4=13,924(120-2)^2=14{,}400-480+4=13{,}924. Square the round number, subtract twice the product, add the small square.
50 Medium Type asked atSIG
37\tfrac37 to four decimal places
Show answer
Answer: 0.42860.4286
Sevenths cycle through the digits of 17=0.142857‾\tfrac17=0.\overline{142857}, starting at a different place: 37=0.428571‾\tfrac37=0.\overline{428571}, so 0.42860.4286 to four places.
51 Hard Type asked at
29+311\tfrac29+\tfrac3{11}, as a decimal
Show answer
Answer: 0.49‾≈0.49490.\overline{49}\approx0.4949
Over 99: 29=2299\tfrac29=\tfrac{22}{99} and 311=2799\tfrac3{11}=\tfrac{27}{99}, so the sum is 4999=0.494949…\tfrac{49}{99}=0.494949\ldots. Any two-digit n99\tfrac{n}{99} repeats its digits: 0.nn‾0.\overline{nn}.
52 Medium Type asked at
17.5%17.5\% of 360360
Show answer
Answer: 6363
Build it: 10%=3610\%=36, 5%=185\%=18, 2.5%=92.5\%=9. Total 6363.
53 Medium Type asked at
37.5%37.5\% of 8888
Show answer
Answer: 3333
37.5%=3837.5\%=\tfrac38: 88÷8=1188\div8=11, times 3 is 3333.
54 Easy Type asked at
0.0625×1440.0625\times144
Show answer
Answer: 99
0.0625=1160.0625=\tfrac1{16}, and 144÷16=9144\div16=9.
55 Hard Type asked at
1.0831.08^3
Show answer
Answer: 1.2597121.259712
Binomial with x=0.08x=0.08: 1+3x+3x2+x3=1+0.24+0.0192+0.000512=1.2597121+3x+3x^2+x^3=1+0.24+0.0192+0.000512=1.259712. In a hurry, 1+3x+3x2≈1.25921+3x+3x^2\approx1.2592 is already within a thousandth.
56 Hard Type asked at
3000\sqrt{3000} to one decimal place
Show answer
Answer: ≈54.8\approx54.8
552=302555^2=3025, which is 25 too many. Correct with a2−d≈a−d2a=55−25110≈54.77\sqrt{a^2-d}\approx a-\tfrac d{2a}=55-\tfrac{25}{110}\approx54.77.
57 Medium Type asked at
999×1001999\times1001
Show answer
Answer: 999,999999{,}999
(1000−1)(1000+1)=1,000,000−1(1000-1)(1000+1)=1{,}000{,}000-1.
58 Easy Type asked at
4.8×2.54.8\times2.5
Show answer
Answer: 1212
×2.5\times2.5 is ×10÷4\times10\div4: 48÷4=1248\div4=12.
59 Medium Type asked atSIG
7.2÷0.087.2\div0.08
Show answer
Answer: 9090
Move both decimal points two places: 720÷8=90720\div8=90.
60 Medium Type asked at
10.125+10.0625\dfrac1{0.125}+\dfrac1{0.0625}
Show answer
Answer: 2424
0.125=180.125=\tfrac18 and 0.0625=1160.0625=\tfrac1{16}, so the reciprocals are 8 and 16: 2424.
61 Hard Type asked atSIG
24×25−23×2624\times25-23\times26
Show answer
Answer: 22
Both pairs sum to 49, so they sit around the same centre, 24.5. The products are 24.52−0.5224.5^2-0.5^2 and 24.52−1.5224.5^2-1.5^2; the difference is 1.52−0.52=21.5^2-0.5^2=2. No multiplication needed.
62 Medium Type asked at
1+3+5+⋯+991+3+5+\dots+99
Show answer
Answer: 25002500
The first nn odd numbers add to n2n^2. There are 50 odd numbers from 1 to 99, so 502=250050^2=2500.
63 Easy Type asked at
72×1.2572\times1.25
Show answer
Answer: 9090
1.25=541.25=\tfrac54: 72÷4=1872\div4=18, times 5 is 9090.
64 Medium Type asked at
0.3520.35^2
Show answer
Answer: 0.12250.1225
35235^2: 3×4=123\times4=12, then 25, so 12251225. Four decimal places: 0.12250.1225.
65 Easy Type asked atSIG
36 is what percent of 80?
Show answer
Answer: 45%45\%
3680=920=45100\tfrac{36}{80}=\tfrac9{20}=\tfrac{45}{100}.
66 Medium Type asked atSIG
A price falls by 25%. By what percent must it rise to get back to where it started?
Show answer
Answer: 3313%33\tfrac13\%
After the fall it is 34\tfrac34 of the start. To return you multiply by 43\tfrac43, a rise of 13\tfrac13, or 3313%33\tfrac13\%. Falls need bigger rises to undo them because the base has shrunk.

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