Difference of Sequences

For the sequence 1,-2,3,-4,5,-6,7,\cdots , what is the difference between the mean of the sequence’s first 400 terms and the mean of its first 200 terms?
Source: NCTM Mathematics Teacher February 2008

Solution
If we add the first 400 terms by pairs 1+(-2),3+(-4),5+(-6),\cdots we end up with 200(-1)=-200. The mean of the first 400 terms is -200/400=-1/2.
Likewise, if we add the first 200 terms by pairs, we get 100(-1)=-100. The mean of the first 200 terms is -100/200=-1/2.
The difference between the two means is -1/2-(-1/2)=0.

Answer: 0

 

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Sums of Reciprocals

The sum of the positive divisors of 480 is 1512. Find the sum of the reciprocals of the positive divisors of 480.
Source: NCTM Mathematics Teacher, February 2008

Solution
Given that 480=2^5\cdot 3^1\cdot 5^1, the number of divisors is (5+1)(1+1)(1+1)=24. The first twelve divisors are small and easy to guess: 1,2,3,4,5,6,8,10,12,15,16, and 20 and as a bonus we get the last twelve by dividing 480 by the first twelve divisors. For example, 480/1=480,480/2=240,480/3=160, etc. : 480,240,160,120,96,80,60,48,40,32,30,24.
When we add the reciprocals of the divisors, the divisors appear as denominators in a set of 24 fractions
\dfrac{1}{1}+\dfrac{1}{2}+\dfrac{1}{3}+\cdots+\dfrac{1}{240}+\dfrac{1}{480}
When we reduce the fractions to 480 (the least common denominator), the 24 divisors appear as numerators
\dfrac{480}{480}+\dfrac{240}{480}+\cdots+\dfrac{2}{480}+\dfrac{1}{480}=\dfrac{480+240+\cdots+2+1}{480}=\dfrac{1512}{480}

Answer: 1512/480

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Divisible by Units Digit

The number 64 is divisible by its units digit (4). How many whole numbers less than 64 are divisible by their respective units digit?
Source: NCTM Mathematics Teacher, February 2008

Solution
Nine numbers: 1,2,3,4,5,6,7,8,9 divisible by themselves
Six numbers: 11,21,31,41,51,61 divisible by 1
Six numbers: 12,22,32,42,52,62 divisible by 2
Five numbers: 15,25,35,45,55 divisible by 5
Six numbers: 33,63,24,44,36,48 divisible by their respective units digit
Total 9+6+6+5+6=32

Answer: 32

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Sum of 7’s

In the sum of the expression 7+77+777+7777+\cdots+7,\!777,\!777,\!777,\!777,\!777,\!777, what digit will be in the tens place?
Source: NCTM Mathematics Teacher, February 2008

Solution
We are adding nineteen numbers from the smallest 7 to the largest 7,\!777,\!777,\!777,\!777,\!777,\!777. Note that there are nineteen 7\mathrm{'s} in the ones place and eighteen 7\mathrm{'s} in the tens place. First, we add the nineteen 7\mathrm{'s} in the ones place and get 19\times 7=133. Consider the number 133. The 3 in the ones place of 133 will be the ones place digit of the final sum. The 3 in the tens place of 133 means 30 and if we regroup 30 with the eighteen 70\mathrm{'s}, we get 30+18\times 70=1290. The digit in the tens place of the final sum is 9.

Answer: 9

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Numerical Puzzle

The five-square numerical puzzle below requires the vertical column to be filled with the digits from a three-digit integral power of 5 and the horizontal row to be filled with the digits from a three-digit integral power of 2. What digit will be in the shaded square?
image
Source: NCTM Mathematics Teacher, February 2008

Solution
5^2=25\quad 5^3=125\quad 5^4=625\quad 5^5=3125
2^6=64\quad 2^7=128\quad 2^8=256\quad 2^9=512\quad 2^{10}=1024
image
Answer: 6

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4-Digit Numbers Divisible by 7

Given the digits 1,3,6, and 9, find the probability that a four-digit number formed by using each of them only once is divisible by 7.
Source: NCTM Mathematics Teacher, February 2008

Solution
There are 4\times 3\times 2\times 1=24 possible four-digit numbers using 1,3,6,9 each only once
1369,1396,1639,1693,1936,1963
3169,3196,\underline{3619},3691,3916,3961
\underline{6139},6193,6319,\underline{6391},6913,6931
9136,\underline{9163},9316,9361,9613,9631
Numbers divisible by 7 are 3619,6139,6391,9163
Probability = 4/24=1/6.

Answer: 1/6

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Two-mile Long Train

How long will it take a two-mile long train traveling 12 miles per hour to travel completely through a mile-long tunnel?
Source: NCTM Mathematics Teacher, February 2008

Solution
image
The diagram shows that for the train to clear the tunnel, point C must travel three mile-long segments each of which takes 1/12 hour or 5 minutes to finish. The train will clear the tunnel in 3\times 5=15 minutes.

Answer: 15 minutes

Alternative solution
When the head of the train enters the tunnel, the end of the train is 3 miles away from the exit. The rear of the train will have to travel 3 miles at 12 miles per hour.

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Triangular Array of Numbers

If you continue the triangular array of numbers shown in the figure, what number would be directly below 122?
|\qquad\qquad\qquad\:\: 1
|\quad\quad\quad\quad\:\: 2\quad\:\: 3\quad\:\: 4
|\quad\quad\:\: 5\quad\:\:\: 6\quad\:\: 7\quad\:\: 8\quad\:\: 9
|\:10\quad 11\quad12\quad 13\quad 14\quad 15\quad 16
Source: NCTM Mathematics Teacher, February 2008

Solution
|\qquad\qquad\qquad\qquad\:\: 1
|\qquad\quad\quad\quad\quad\:\: 2\quad\:\: 3\quad\:\: 4
|\qquad\quad\quad\: 5\quad\:\:\: 6\quad\:\: 7\quad\:\: 8\quad\:\: 9
|\qquad\: 10\quad 11\quad12\quad 13\quad 14\quad 15\quad 16
|\:17\quad 18\quad 19\quad 20\quad 21\quad 22\quad 23\quad 24\quad 25
|\:\cdots
Note that the leading numbers increase by 1,3,5,7, etc. If we follow this pattern, the leading numbers of the first 13 rows are 1,2,5,10,17,26,37,50,65,82,101,122,145, which means that the number directly below 122 is 146.

Answer: 146

Alternative solution 1
Note that the ending number of each row is a perfect square 1,4,9,16,25, etc. If we follow this pattern, 121=11^2 is the ending number of row 11 and 144=12^2 is the ending number of row 12.

Alternative solution 2
Let x=0,1,2,3,\cdots represent the row number. We want to find a function f(x) that relates row x to the leading number of that row. We already know a few values of f(x)
x\quad f(x)
0\quad 1
1\quad 2
2\quad 5
3\quad 10
4\quad 17
Is f(x) a linear or quadratic function or neither? The first differences (FD) and second differences (SD) in the values of f(x) will tell us what it is
x\quad f(x)\:\mathrm{FD}\:\mathrm{SD}
0\quad\:\: 1
1\quad\:\: 2\quad\: 1
2\quad\:\: 5\quad\: 3\quad\: 2
3\quad 10\quad\: 5\quad\: 2
4\quad 17\quad\: 7\quad\: 2
f(x) is a quadratic function because the second differences are a constant 2
f(x)=ax^2+bx+c
a=\mathrm{SD}/2=2/2=1
c=f(0)=1
b=0 because the term bx does not contribute anything to the function
f(x)=x^2+1
f(11)=11^2+1=122
f(12)=12^2+1=145

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Smaller Factor

Given a three-digit number, interchange the units digit and the hundreds digit. The product of the original number and the new number is 65125. What is the smaller three-digit factor?
Source: NCTM Mathematics Teacher, February 2008

Solution
Since the product of the original number and the new number 65125 is odd, the two numbers must be odd. Furthermore, their units digits must multiply to 5. These restrictions limit the possibilities to 1\!-\!5,3\!-\!5,5\!-\!7, and 5\!-\!9. Let’s examine a few numbers to see if we can narrow down the choices starting with 5\!-\!9
519\times 915=474885 – not possible, too big a product
517\times 715=369655 – not possible
315\times 513=161595 – not possible
115\times 511=58765
125\times 521=65125
The smaller three-digit factor is 125.

Answer: 125

Alternative solution
The prime factorization of 65125=5^3\times 521.
2\times 521=1042
3\times 521=1563
\cdots
521 is a prime number and any multiple of 521 is a 4-digit number, thus the two factors must be 5^3=125 and 521.

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Non-Congruent Triangles

Among the triangles of the previous problem (repeated below), how many distinct (non-congruent) triangles exist? Previous Problem: How many distinct triangles can be constructed by choosing three vertices from among the corners of a unit cube?
Source: NCTM Mathematics Teacher, February 2008

Solution
image
24 triangles on the faces of the cube have side lengths of 1,1,\sqrt 2.
image
24 triangles formed by two diagonals and one side have side lengths of 1,\sqrt 2,\sqrt 3.
image
8 triangles that are bases of tetrahedrons formed by connecting three corners of the cube have side lengths of \sqrt 2,\sqrt 2,\sqrt 2.

There are 3 distinct (non-congruent) triangles.

Answer: 3

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