The “spring check” in 2019 will be the last of the regular formal checks at this location; hereafter, we will visit only when we are informed in a log or personal e-mail that maintenance is needed - so please keep us informed!
Equation Solving
Median is the thirteenth in a series of fifteen puzzle caches of increasing difficulty based on solving equations. It is recommended that one start with Abacus (GC6A006) and continue in alphabetical sequence through Brackets (GC6AR81), Compass (GC6AWD0), Decagon (GC6C4EE), Ellipse (GC6CFW3), FOIL (GC6HDNE), Googol (GC751JK), Hectare (GC75BD3), Index (GC75EAM), Jargon (GC75X8W), Kite (GC76010), LessThan (GC763HV ), Median (GC7DH17), Nonagon (GC7DWC4) and Obelus (GC7DWCD).
Median - The Name
In statistics, the median is the middle number of any group of numbers that are placed in sequence from smallest to largest. If there is an odd number of elements in the sequence - for example, 1, 4, 7, 9, 13, 16, 18 - the median would be 9. If there is an even number of elements in the sequence - for example, 2, 5, 7, 10, 12, 15, 17, 21 - the median would be the average of the two middle numbers, that being 11 in this case. With a large sample of numbers, the median is a good measure of the middle of the sample.
In geometry, the median is the line from the vertex of a triangle to the mid-point of the opposite side. The three medians meet at one point (called the centroid) and the centroid cuts each median in a ratio of 2:1, the longer portion being that closest to its vertex.
Neither of those meanings has anything to do with this cache . . . but who knows when you might be asked . . . or might need a topic at an uncomfortable moment at a cocktail party!?!?
Median - the Cache
Introduction
“LessThan” provided a break from equations. But, in “Kite,” we learned to solve two equations in two unknowns and so, in “Median,” we are going to put those skills to work in solving some tricky - hence the difficulty rating of 4.0 - problems . . . but we’ll provide some examples to help you along.
The Steps in Solving a Word Problem Using Two Unknowns
(a) Represent the quantities that you’re after by letters. The choice is often “x” and “y” but it could be anything - even symbols like # or % if you wished!
(b) Step by step, translate the words of the problem into equation form. If you have two unknowns, there will be enough information to construct two equations.
(c) Ensure that the equations are in a form which you can handle; so far, we have used the form: 2x + 3y = 17 so we will stick with that for now - we’ll refer to it as “standard form.”
(c) Solve the equations for one unknown.
(d) Substitute in one of the equations for the found unknown in order to determine the other.
(e) State, in words, the answers to the problem.
And Now Some Examples
Example 1
Paul and Ian are father and son respectively. The difference between their ages is one third of what the sum of their ages will be ten years from now. The sum of their ages fifteen years ago was one half of what it will be in five years. How old are they now?
Let x represent the age of Paul now.
Let y represent the age of Ian now.
The difference between their ages is x - y.
In ten years, the sum of their ages will be (x + 10) + (y + 10) or x + y + 20.
So the first equation will be:
x - y = (x + y + 20)/3 . . . the 3 represents the "one third" in the question's wording.
Multiply both sides by 3 to eliminate the fraction:
3x - 3y = x + y + 20
Re-arrange into “standard form.”
2x - 4y = 20 or, dividing both sides by 2,
x - 2y = 10 . . . (1)
The sum of their ages fifteen years ago was (x - 15) + (y - 15) or x + y - 30.
The sum of their ages in five years will be (x + 5) + (y + 5) or x + y + 10
So the second equation will be:
x + y - 30 = (x + y + 10)/2 . . . the 2 represents the "one half" in the question's wording.
Multiply both sides by 2 to eliminate the fraction.
2x + 2y - 60 = x + y + 10
Re-arrange in “standard form.”
x + y = 70 . . . (2)
Solution:
Re-write the equations so that they are together.
x - 2y = 10 . . . (1)
x + y = 70 . . . (2)
It can be seen that by subtracting the equations, the “x” terms will disappear so:
(1) - (2) ➝ - 3y = - 60 and dividing both sides by - 3, we get
y = 20
Substitute for y = 20 into (2)
x + 20 = 70 and so
x = 50
So Paul’s age is 50 and Ian’s age is 20.
Example 2
If a two digit number is doubled, the result is one more than the original number with its digits reversed. If the original number is reduced by 9 and the result tripled, the resulting number will be the same as the result of adding 11 to the original number with its digits reversed. What is the original number?
Note: The trick here is to recognize what a number in our decimal system means. The number 58 means, as they used to teach us in the primary grades, “5 groups of ten and 8 ones.” To represent that number, then, to work with it in puzzles such as this, we have to think of 58 as being “(10)(5) + (1)(8).” If the number had its digits reversed, it would be 85 and would be expressed as (10)(8) + (1)(5). In the same way, 367 would be expressed as (100)(3) + (10)(6) + (1)(7).
So, if one is asked to express an unknown three-digit number one would first define x, y and z as the hundreds, tens and ones digits respectively and then express the number as (100)x + (10) y + (1)z or 100x + 10y + z. And now to the problem.
Let x represent the tens digits.
Let y represent the ones digit.
Hence the original number can be expressed as 10x + y and the number with its digits reversed can be expressed as 10y + x.
We are now ready to translate the statements in the problem into equations and then simplify them.
From the First Statement:
2(10x + y) = (10y + x) + l
20x + 2y = 10y + x + 1
Transpose into what we have referred to as the “standard form.”
20x - x + 2y - 10y = 1 or
19x - 8y = 1 . . . (1)
From the Second Statement:
3(10x + y - 9) = 10y + x + 11
30x + 3y - 27 = 10y + x + 11
Transpose into “standard form.”
30x - x + 3y - 10y = 11 + 27
29x - 7y = 38 . . . (2)
Solution:
Re-write the equations so that they are together.
19x - 8y = 1 . . . (1)
29x - 7y = 38 . . . (2)
Looking at the two, one can see that probably the easiest thing to do would be to multiply both equations in order to get 56 in front of the “y” terms so we do that:
(1) X 7 ➝ 133x - 56y = 7 . . . (3)
(2) X 8 ➝ 232x - 56y = 304 . . . (4)
We could subtract (4) from (3) or (3) from (4) and we choose to do the latter because it results in a positive number in front of the x term . . . but it doesn’t matter . . . either will do.
(4) - (3) ➝ 99x = 297
x = 297/99
x = 3.
Now substitute in (1) for x = 3 to find y:
(19)(3) - 8y = 1
57 - 8y = 1
Transpose:
- 8y = 1 - 57
- 8y = - 56
y = ( -56)/(-8)
y = 7.
So x = 3 and y = 7 and our original number is 37.
(Applying the statements of the problem now to 37 and 73, one can see that our solution "fits!")
Example 3
Heather says to Sarah: “Give me $45 and I will have the same amount as you will have.” Sarah responds: “You give me $45 and then I will have five and a half times as much as you will have.” How much money has each of them to start with?
Let a represent the amount of money Heather has to start with.
Let b represent the amount of money Sarah has to start with.
"Translate” Heather’s statement:
a + 45 = b - 45
a - b = - 90 . . . (1)
Now “translate” Sarah’s statement:
5.5(a - 45) = b + 45
Change the decimal to a fraction - unnecessary but a little less work.
11(a - 45)/2 = b + 45
Multiply both sides by 2 to eliminate the denominator.
11(a - 45) = 2b + 90
Simplify.
11a - 495 = 2b + 90
11a - 2b = 585 . . . (2)
Solution:
Re-write the two equations so that they are together.
a - b = - 90 . . . (1)
11a - 2b = 585 . . . (2)
(1) X (- 2) ➝ - 2a + 2b = 180 . . . (3)
(2) ➝11a - 2b = 585
(3) + (2) ➝ 9a = 765
a = 765/9
a = 85
Substitute in (1) for a = 85
a - b = - 90
85 - b = - 90
- b = - 90 - 85
- b = - 175
b = -175/(- 1)
b - 175
Heather has $85 and Sarah has $175
Your Turn - the Way to the Cache!
Question 1
Catherine and Bill had never disclosed their ages to Jocelyn, their curious daughter, who wanted to know just how old her mother and father were. “Well,” Catherine said, “here’s a way for you to figure out how old I am. In twelve years, I’ll be six times what your age was ten years ago. And, twelve years ago, I was twice as old as you were two years ago. Figure it out!” At least, Jocelyn knew her own age so had no problem . . . but you do! How old are Catherine and Jocelyn now?
Catherine’s age is A B or ___ ___; Jocelyn’s age is C D or ___ ___.
Question 2.
If a two-digit number is doubled, the result is the same as if the original number were reversed, the tens digit increased by two, the ones digit increased by one and the result tripled. If the ones digit of the original number is quadrupled and the result doubled, the result is six more than if the original number were reversed, the ones digit decreased by two and the result multiplied by six. What is the original number?
The original number is E F or ___ ___.
Question 3.
Belinda and Anne were comparing their bank accounts before heading out for a little jaunt. Belinda scribbled for a time and then noted that if she tripled the amount she had, she would have eight dollars more than Anne would have if Anne divided the amount in her account by three and then multiplied it by five! Anne took a few moments to come back with the statement that if she gave eighty dollars to Belinda, then Belinda would still be better off than Anne by eight dollars. How much did each girl have in her bank account?
Anne had $ G H J or $ ___ ___ ___ ; Belinda had $ K L M or $ ___ ___ ___ .
The Location of the Cache
A = ___ , B = ___ , C = ___ , D = ___ , E = ___ , F = ___ ,
G = ___ , H ___ , J = ___, K = ___, L = ___, M = ___.
The co-ordinates of the cache are N 44° ab.cde’ and W 078° fg.hij’ where none of the lower case letters is related to the upper case letters above except as defined below:
a = M - H = ___ ; b = G - C = ___ ; c = L - F = ___ ; d = D + E = ___ ;
e = L - J - G + F = ___ ; f = G + K = ___ ; g = K + A = ___ ; h = B - C + K = ___;
i = E + F = ___ ; j = D[(H + J + E) - (A + B + M)] = ___ .
You Will Find the Cache at:
N 44° __ __ . __ __ __’ and W 078° __ __ . __ __ __’
Additional Comments
- Please bring your own writing utensil;
- Be aware of the possibility of others in the area;
- Please replace the cache as hidden;
- You are welcome to confirm your answer on GeoChecker.com.