2010 MCWC

The 2010 Mental Calculation World Cup was the fourth edition of the event. American competitor Gerald Newport won four of the six surprise tasks. In the overall rankings, Priyanshi Somani (India) won the event and remains the only female overall champion in MCWC history. Marc Jornet Sanz and Alberto Coto — both from Spain — finished second and third, respectively.

GMCA panelists Freddis Reyes and Hua Wei Chan finished 8th and 12th, respectively.

Surprise Task Categories (2010)

The 2010 MCWC surprise task category included the following six task types:

• A + (B × C)
  (where A and C were eight-digit numbers, and B ranged from 2 through 28)

• 3rd, 4th, or 5th root of a six-digit number

• (1/A) + (1/B) = 1/x

• Factorization of five-digit numbers

• (2π / 3) × 17.3³, computed to as many significant digits as possible
  (with the value of π provided)

• (a / b) ± (c / d) ≤ or ≥ e

Notes on Calculation League Modifications

In Calculation League, the factorization task is omitted, since the same task already exists in identical form in another category.

Additionally, the task involving π is omitted at the present time.

Example 1

((5724 / 44) + (−439 / 74)) × 8

(Easy Level)

This question format tends to be a relatively difficult version of the fraction addition category. For this type of problem, it is highly likely that at least three significant digits will be required.

In this case, the number 5724 contains a multiple of 44, which makes the problem noticeably easier.

Evaluating the First Term

We start by rewriting:

• 5724/44=5720/44+4/445724 / 44 = 5720 / 44 + 4 / 445724/44=5720/44+4/44

This gives:

• 5720/44=1305720 / 44 = 1305720/44=130

• 4/44=1/114 / 44 = 1 / 114/44=1/11

So:

• 5724/44=130+1/11=130.09090909095724 / 44 = 130 + 1/11 = 130.09090909095724/44=130+1/11=130.0909090909

Multiplying by 8 gives:

• 130×8=1040130 × 8 = 1040130×8=1040

• 8/118 / 118/11

So at this point we have 1040 and 8/11.

Estimating the Second Term

Now we estimate:

• 439/74439 / 74439/74

We only need this value to the nearest 0.1, which is approximately 5.9.

Multiplying by 8:

• 5.9×8≈47.25.9 × 8 ≈ 47.25.9×8≈47.2 (or 47 for quick subtraction)

Now subtract:

• 1040−47=9931040 − 47 = 9931040−47=993

Alternative Approaches

Multiplying First (for Strong Left-to-Right Multipliers)

For those very comfortable with left-to-right multiplication, it may be preferable to begin by multiplying by 8:

• 5724×8=457925724 × 8 = 457925724×8=45792
  → 45792/44=104045792 / 44 = 104045792/44=1040 and 32/4432 / 4432/44

• 439×8=3512439 × 8 = 3512439×8=3512
  → 3512/74=473512 / 74 = 473512/74=47 and 34/7434 / 7434/74

Subtracting:

• 1040−47=9931040 − 47 = 9931040−47=993

Dividing First (for Strong Dividers)

For those very comfortable with division, we can instead divide by:

• 44/8=5.544 / 8 = 5.544/8=5.5

• 74/8=9.2574 / 8 = 9.2574/8=9.25

This works well here because of the ease of working with 8.

Compute:

• 5724/5.5≈1040.75724 / 5.5 ≈ 1040.75724/5.5≈1040.7

• 439/9.25≈47.5439 / 9.25 ≈ 47.5439/9.25≈47.5

Subtracting:

• 1040.7−47.5=993.21040.7 − 47.5 = 993.21040.7−47.5=993.2

Example 2

1 / ((1 / 57) + (1 / 77))

(Advanced Level)

One approach to this question is to compute:

• (77×57)/(77+57)(77 × 57) / (77 + 57)(77×57)/(77+57)

This gives:

• 4389/134≈32.74389 / 134 ≈ 32.74389/134≈32.7

Algebraic Justification

Expanding the algebra:

(1/a)+(1/b)(1/a) + (1/b)(1/a)+(1/b)=(b/ab)+(a/ab)= (b / ab) + (a / ab)=(b/ab)+(a/ab)=(a+b)/(ab)= (a + b) / (ab)=(a+b)/(ab)

Therefore:

1/((1/a)+(1/b))=ab/(a+b)1 / ((1/a) + (1/b)) = ab / (a + b)1/((1/a)+(1/b))=ab/(a+b)

Decimal Estimation Alternative

Another approach is to convert each fraction to decimal form.

Estimating:

• 1/57≈0.01751 / 57 ≈ 0.01751/57≈0.0175

• 1/77≈0.0131 / 77 ≈ 0.0131/77≈0.013

Adding:

• 0.0175+0.013=0.03050.0175 + 0.013 = 0.03050.0175+0.013=0.0305

Then:

• 1/0.0305≈32.81 / 0.0305 ≈ 32.81/0.0305≈32.8

This is close enough for Calculation League purposes.

Example 3

442,883,186 + (676 × 219,792,902)

(Expert Level)

The approach to this question depends heavily on the competitor’s skill with addition, and — more importantly — multiplication.

While one could attempt cross-multiplication using the digits of 442,883,186, for many competitors it is more effective to break 676 into 67 and 6, and then perform two coordinated addition processes.

For most competitors, approaching this question right to left will be preferred.

General Procedure

We start with the right-most digit of 219,792,902, then proceed leftward digit-by-digit, applying the following steps:

1. Multiply by the last digit of 676 (which is 6).

2. Add the product to the corresponding remaining digit of 442,883,186. Carry any remainder and type the result.

3. Multiply by 67. Add back in the carried remainder.

4. Add the result from step (3) to the remaining digits of 442,883,186.

The final step involves discretion. You may:

• add one digit per step, or

• add multiple digits in a single step and skip additions in other steps.

However, you must not add digits from position x when you are already x places from the right in step (1).

My preference is to add blocks of digits, because if the carry is three digits, it is often easier to do one 3-digit + 3-digit addition than three separate single-digit additions.

Worked Execution

We compute:

442,883,186+(676×219,792,902)442,883,186 + (676 × 219,792,902)442,883,186+(676×219,792,902)

• 6×2=126 × 2 = 126×2=12 → +6=18+6 = 18+6=18
  Type 8, carry 1.

• 67×2=13467 × 2 = 13467×2=134 → +1=135+1 = 135+1=135 → +318=453+318 = 453+318=453
  Since the next digit is zero, we can type 3 as well.
  Running total: 45.

• 6×9=546 × 9 = 546×9=54 → +45=99+45 = 99+45=99
  Type 9, carry 9.

• 67×9=60367 × 9 = 60367×9=603 → +9=612+9 = 612+9=612

• 6×2=126 × 2 = 126×2=12 → +612=624+612 = 624+612=624
  Type 4, carry 62.

• 67×2=13467 × 2 = 13467×2=134 → +62=196+62 = 196+62=196

At this point, four digits from 442,883,186 have been used (the final digits 3186). This means that in the next step we must add more digits from 219,792,902.

• 6×9=546 × 9 = 546×9=54 → +196=250+196 = 250+196=250
  → +8+8+8 (next digit of 442,883,186) = 258
  Type 8, carry 25.

• 67×9=60367 × 9 = 60367×9=603 → +25=628+25 = 628+25=628
  Add the next three digits (428) to get 1056.

• 6×7=426 × 7 = 426×7=42 → +1056=1098+1056 = 1098+1056=1098
  Type 8, carry 109.

• 67×7=46967 × 7 = 46967×7=469 → +109=578+109 = 578+109=578

• 6×9=546 × 9 = 546×9=54 → +578=632+578 = 632+578=632
  Type 2, carry 63.

• 67×9=60367 × 9 = 60367×9=603 → +63=666+63 = 666+63=666

• 6×1=66 × 1 = 66×1=6 → +666=672+666 = 672+666=672
  Type 2, carry 67.

• 67×1=6767 × 1 = 6767×1=67 → +67=134+67 = 134+67=134

• 6×2=126 × 2 = 126×2=12 → +134=146+134 = 146+134=146
  → +4+4+4 (final remaining digit of 442,883,186) = 150
  Type 0, carry 15.

• 67×2=13467 × 2 = 13467×2=134 → +15=149+15 = 149+15=149
  Type 149.