Sunday, April 13, 2014

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.
143
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.
144
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
145
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.
146
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.
147
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.
148
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.
149
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.

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