MENTAL MATH PART 1

MENTAL MATH PART 1

1. If an item costs $36.78, how much change would you get from $100?

Because the dollars sum to 99 and the cents sum to 100, the change

is $63.22.

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2. Do the mental subtraction problem: 1618 – 789.

1618 – 789 = 1618 – (800 – 11) – 818 + 11 = 829

Do the following multiplication problems.

3. 13 × 18 = (13 + 8) × 10 + (3 × 8) = 210 + 24 = 234

4. 65 × 65 = 60 × 70 + 52 = 4200 + 25 = 4225

5. 997 × 996 = (1000 × 993) + (􀀐3) × (􀀐4) = 993,012

6. Is the number 72,534 a multiple of 11?

Yes, because 7 􀀐 2 + 5 – 3 + 4 = 11.

7. What is the remainder when you divide 72,534 by a multiple of 9?

Because 7 + 2 + 5 + 3 + 4 = 21, which sums to 3, the remainder is 3.

8. Determine 23/7 to 6 decimal places.

23/7 = 3 2/7 = 3.285714 (repeated)

9. If you multiply a 5-digit number beginning with 5 by a 6-digit

number beginning with 6, then how many digits will be

in the answer?

Just from the number of digits in the problem, you know the answer

must be either 11 digits or 10 digits. Then, because the product of

the initial digits in this particular problem (5 × 6 = 30), is more than

10, the answer is de􀂿 nitely the longer of the two choices, in this

case 11 digits.

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Solutions

10. Estimate the square root of 70.

70 ÷ 8 = 8 3/4 = 8.75. Averaging 8 and 8.75 gives us an estimate

of 8.37.

(Exact answer begins 8.366… .)

Do the following problems on paper and just write down the answer.

11. 509 × 325 = 165,425 (by criss-cross method).

12. 21,401 ÷ 9: Using the Vedic method, we get 2 3 7 7 R 8.

13. 34,567 ÷ 89: Using the Vedic method, with divisor 9 and multiplier

1, we get:

7 6 2

3 8 8 R35

89 3 4 5 6 7

1

90

􀀎

14. Use the phonetic code to memorize the following chemical

elements: Aluminum is the 13th element, copper is the 29th element,

and lead is the 82nd element.

Aluminum = 13 = DIME or TOMB. An aluminum can 􀂿 lled with

DIMEs or maybe a TOMBstone that was “Aluminated”?

Copper = 29 = KNOB or NAP. A doorKNOB made of copper or a

COP taking a NAP.

Lead = 82 = VAN or FUN. A VAN 􀂿 lled with lead pipes or maybe

being “lead” to a FUN event.

15. What day of the week was March 17, 2000? Day = 2 + 17 + 0 – 14

= 5 = Friday.

16. Compute 2122 = 200 × 224 + 122 = 44,800 + 144 = 44,944

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17. Why must the cube root of a 4-, 5-, or 6-digit number be a

2-digit number?

The largest 1-digit cube is 93 = 729, which has 3 digits, and a 3-digit

cube must be at least 1003 = 1,000,000, which has 7 digits.

Find the cube roots of the following numbers.

18. 12,167 has cube root 23.

19. 357,911 has cube root 71.

20. 175,616 has cube root 56.

21. 205,379 has cube root 59.

The next few problems will allow us to 􀂿 nd the cube root when the original

number is the cube of a 3-digit number. We’ll 􀂿 rst build up some ideas to

􀂿 nd the cube root of 17,173,512, which is the cube of a 3-digit number.

22. Why must the 􀂿 rst digit of the answer be 2?

2003 = 8,000,000 and 3003 = 27,000,000, so the answer must be in

the 200s.

23. Why must the last digit of the answer be 8?

Because 8 is the only digit that, when cubed, ends in 2.

24. How can we quickly tell that 17,173,512 is a multiple of 9?

By adding its digits, which sum to 27, a multiple of 9.

25. It follows that the 3-digit number must be a multiple of 3 (because

if the 3-digit number was not a multiple of 3, then its cube could not

be a multiple of 9). What middle digits would result in the number

2_8 being a multiple of 3? There are three possibilities.

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Solutions

For 2_8 to be a multiple of 3, its digits must sum to a multiple of 3.

This works only when the middle number is 2, 5, or 8 because the

digit sums of 228, 258, and 288 are 12, 15, and 18, respectively.

26. Use estimation to choose which of the three possibilities is

most reasonable.

Since 17,000,000 is nearly halfway between 8,000,000 and

27,000,000, the middle choice, 258, seems most reasonable. Indeed,

if we approximate the cube of 26 as 30 × 30 × 22 = 19,800, we get

2603, which is about 20 million, consistent with our answer.

Using the steps above, we can do cube roots of any 3-digit cubes. The 􀂿 rst

digit can be determined by looking at the millions digits (the numbers before

the 􀂿 rst comma); the last digit can be determined by looking at the last digit

of the cube; the middle digit can be determined through digit sums and

estimation. There will always be three or four possibilities for the middle

digit; they can be determined using the following observations, which you

should verify.

27. Verify that if the digit sum of a number is 3, 6, or 9, then its cube

will have digit sum 9.

If the digit sum is 3, 6, or 9, then the number is a multiple of 3,

which when cubed will be a multiple of 9; thus, its digits will sum

to 9.

28. Verify that if the digit sum of a number is 1, 4, or 7, then its cube

will have digit sum 1.

A number with digit sum 1, when cubed, will have a digit sum

that can be reduced to 13 = 1. Likewise, 43 = 64 reduces to 1 and

73 = 343 reduces to 1.

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29. Verify that if the digit sum of a number is 2, 5, or 8, then its cube

will have digit sum 8.

Similarly, a number with digit sum 2, 5, or 8, when cubed, will have

the same digit sum as 23 = 8, 53 = 125, and 83 = 512, respectively, all

of which have digit sum 8.

Using these ideas, determine the 3-digit number that produces the

cubes below.

30. Find the cube root of 212,776,173.

Since 53 < 212 < 63, the 􀂿 rst digit is 5, and since 73 ends in 3, the

last digit is 7. Thus, the answer looks like 5_7. The digit sum of

212,776,173 is 36, which is a multiple of 9, so the number 5_7 must

be a multiple of 3. Hence, the middle digit must be 0, 3, 6, or 9

(because the digit sums of 507, 537, 567, and 597 are all multiples

of 3). Given that 212,000,000 is so close to 6003 (= 216,000,000),

we pick the largest choice: 597.

31. Find the cube root of 374,805,361.

Since 73 < 374 < 83, the 􀂿 rst digit is 7, and since only 13 ends in

1, the last digit is 1. Thus, the answer looks like 7_1. The digit

sum of 74,805,361 is 37, which has digit sum 1; by our previous

observation, 7_1 must have a digit sum that reduces to 1, 4, or 7.

Hence, the middle digit must be 2, 5, or 8 (because 721, 751, and

781 have digit sums 10, 13, and 16, which reduce to 1, 4, and 7,

respectively). Given that 374 is much closer to 343 than it is to 512,

we choose the smallest possibility, 721. To be on the safe side, we

estimate 723 as 70 × 70 × 76 = 372,400, which means that 7203 is

about 372,000,000; thus, the answer 721 must be correct.

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Solutions

32. Find the cube root of 4,410,944.

Here, 13 < 4 < 23, so the 􀂿 rst digit is 1, and (by examining the last

digit) the last digit must be 4. Hence, the answer looks like 1_4.

The digit sum of 4,410,944 is 26, which reduces to 8, so 1_4 must

reduce to 2, 5, or 8. Thus, the middle digit must be 0, 3, 6, or 9.

Given that 4 is comfortably between 13 and 23, it must be 134 or

164. Since 163 = 16 × 16 × 16 = 256 × 8 × 2 = 2048 × 2 = 4096, we

choose the answer 164.

Compute the following 5-digit squares in your head! (Note that the necessary

2-by-3 and 3-digit square calculations were given in the solutions to

Lecture 11.)

33. 11,2352

11 × 235 × 2 = 2585 × 2 = 5,170. So 11,000 × 235 × 2 = 5,170,000.

We can hold the 5 on our 􀂿 ngers and turn 170 into DUCKS. 11,0002

= 121,000,000, which when added to 5 million gives us 126 million,

which we can say. Next, we have 2352 = 55,225, which when added

to 170,000, gives us the rest of the answer: 225,225. Final answer

= 126,225,225.

34. 56,7532

56,000 × 753 × 2 = 56 × 753 × 2 × 1000 = 42,168 × 2 × 1000

= 84,336,000 = FIRE, MY MATCH.

56,0002 = 3,136,000,000, so we can say “3 billion.” After adding

136 to 84 (FIRE), we can say “220 million.” Then, 7532 = 567,009,

which when added to 336,000 (MY MATCH) gives the rest of the

answer, 903,009. Final answer = 3,220,903,009.

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35. 82,6822

82,000 × 682 × 2 = 82 × 682 × 2 × 1000 = 55,924 × 2 × 1000

= 111,848 = DOTTED, VERIFY.

82,0002 = 6,724,000,000, so we can say “6 billion,” then add

724 to 111 (DOTTED) to get 835 million, but because we see a

carry coming (from 848,000 + 6822), we say “836 million.” Next,

6822 = 465,124 (turning 124 into TENOR, if helpful). Now,

465,000 + 848,000 (VERIFY) = 1,313,000, but we have already

taken care of the leading 1, so we can say “313 thousand,” followed

by (TENOR) 124. Final answer = 6,836,313,124.