 # Mental arithmetic, how?

Hi,

Can someone help how to handle different kind of mental arithmetic tasks? I will do a test for a future education in some weeks.

Exemple of questions:

1:What is: (30612 ÷ 2132) x 13?
A) 14
B) 182
C) 156

2: What is: 1924 divided by 26?
A) 73
B) 74
C) 71

3: What is: 454.3 ÷ 18.9?
A) 24
B) 25
C) 23

4: What is: 13277 ÷ 22?
A) 686
B) 756
C) 603

5: What is: 4756 ÷ 58??
A) 82
B) 79
C) 75

6: What is: 1837 x 1368?
A) 2,562,264
B) 2,513,016
C) 2,511,179

Best regards,

Someone else is probably going to mention methods, so a short tip based on these question structures. You don’t need to evaluate much.

E.g

8x7=56

475/58 = 8…

Good luck for the test!

There is a mix of skill required here, and most of them can be done easily with approximation techniques, especially as the answers are multiple choice. [Edit: as demonstrated correctly by Nagime above]

I’m surprised to see that some of the answers are wrong (was Q1 supposed to be [B] 186 and [C] 152? The correct answer is between 186 and 187.

Do you get any information about whether the answers are exact or approximate? If you know that they are exact, then Q2 can be solved simply by seeing that the answer must end in 4 or 9, since we multiply it by something that ends in 6 (26) to get something that ends in 4 (1924). But Q3 and Q4 are not exact questions so this method wouldn’t work.

Finally for question 5, you should know the cross-multipliction technique. It’s the first method I learned when I started studying mental arithmetic, something that everyone I train in the JMCWC already knows, and something that I show most people I coach in mental maths. I’d definitely get some practise at that before your test.

Actually in this case if we assume that the multiple-choice answers are exact, then we can answer this in about 2 seconds by seeing that 7x8 ends in 6, so it must be 2,513,016.

Unless, you know that you’ll always get exact answers in which case you know that 2x8 ends in a 6. The ‘2’ from the 82 and the ‘8’ from the 58. Then you just make sure the other two aren’t possible… 9x8 ends in a 2 and 5x8 ends in a 0… no need to do any calculations past that then.

Oops - my error, I was looking at question 6, which is best solved using cross-multiplication (except that because of the multiple-choice you’ll get to the right answer immediately without further working.

I didn’t say anything that was intended for question 5, and I agree that if we assume all calculations are exact, the method you give is the best.

If you don’t know they’re exact, I’d round 58 up to 60, then round 4756 up to “more than 4800” to compensate, giving an answer of “more than 80” and so it must be 82. Or if I wasn’t confident with that, I’d just do the full division method, which is not too hard in this case.

These questions really should specify whether the answer is exact or not as it completely changes the strategy.

This is what I would do:

First, guesstimate 30612 ÷ 2132:
15 X 2100 = 31500, or 888 too much.
888 is less than 3% of 30612.
So the answer - guesstimated is 15 minus 3% ,or about 14.5.
14 X 13 = 182 and 14.5 X 13 = 188.5.

A closer guess is this:
2132 is roughly 2100 plus 1.5%.
So, instead of 15 minus 3%,we can take 15 minus (3% + 1.5%) or minus 4.5%.
4.5% of 15 is 0.675 (45 plus half of 45 is 67.5).
15 - 0.675 is 14.325.

13 X 14 is 182 and 13 X 0.325 is about 4, so 13 X 14.325 is about 186.

Another way is this:
15 X 2100 = 31500.

Then 15 X 2132 is 31500 plus 480 is 31980.
Then 14 X 2132 is 31980 minus 2132.
2132 - 1980 is 52, so 31980 minus 2132 is 30942 (1000’s complement of 52 is 942).
We are now 612 plus 52 off or 674.
2132 : 674 is about 0.3…

So take 14.3 X 13 is 182 plus 4 = 186.
Again, as Daniel also said, answer is around 186.

2000 : 25 is 80.
2000 - 75 is 1925 (75 is 3 times 25), so:
1925 : 25 is 77.
26 X 77 is then 1925 + 77 is 2002.
2002 - 1924 is 78 or 3 times 26.
77 - 3 is 74.

A quick way to answer (assuming you know the answers are exact) is to realise that 6 X 4 is 24.
So the answer that ends in a 4 is the right one.

25 X 18 is 450.
24 X 19 is 456 (450 - 18 + 24).
24 X 18.9 is 456 - 2.4 is 453.4.

Did you switch 2 numbers?
Because the answer is not exact.
However if the question was 453.4 ÷ 18.9, then the answer is 24 exact.

6 X 22 = 132, so the answer needs to be very close to 600.
600 X 22 is 13200 or 77 off.
77 : 22 is 3.5 exact.

So 603.5 is the right answer.

58 is close to 60 and 60 X 80 is 4800.
Then 58 X 80 is 4800 minus 2 X 80 or 160.
4800 - 160 is 4640.
So answer needs to be bigger than 80.
82 is the only answer possible.
2 times 58 is 116.
4640 + 116 is 4756, so 82 is correct.

8 X 7 is 56, so take the answer that ends in 6.

Alternatively, take the 9 or 11 proof:
1837 mod 9 is 1+8+3+7 is 19. 1+9 is 10. 1+0 is 1.
1368 mod 9 is 1+3+6+8 is 18. 1+8 is 9.

1 X 9 is 9, so the answer mod 9 needs to be 9 too.

A mod 9 is 9
B mod 9 is 9
C mod 9 is 8

So answer is A or B.

11 proof then:
1837 mod 11 is 0.
1368 mod 11 is 4
0X4 is 0, so answer mod 11 is 0

A is 1
B is 0
C is 0.

The only answer the satisfies both the 9 and 11 proof is answer B.

In general, what I do is round the numbers, get an easy estimate and refine from there.

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