Touching Exponentials

An exponential function and its inverse, the logarithm, are touching in one point only, as in this graph. Find the appropriate base for the functions to accomplish this. See solution here.

Touching Quadratics

A quadratic and its inverse are touching in one point only, as in this graph. Find the appropriate coefficients for the quadratic to accomplish this. See solution here.


Solution a27264a - British Gentleman.

Problem: A British gentleman and his Canadian grandson were renting a car in the United States and were told that the car averages 26 miles (mi) per gallon (US gal). Grandpa wanted to know how much this is in miles per imperial gallon (imp gal), and the grandson wanted to know the fuel consumption in liters (L) per 100 km. Find the figures each are looking for, using these facts:
Solution:
To get the conversion the other way around, just equation solve on each of the facts given above:



To get the miles per imperial gallon, using the conversion factors we were given, we need to go via liters:
For the fuel consumption we need to have volume per distance, which is the reciprocal of fuel economy, or in other words, the fraction up side down:
So, the answers are that the 26 mi/US gal is approximately the same as 31 mi/Imp gal or 9 L/100km.
Comment:
We note that the units (US gal, Imp gal, L, mi, km) can be treated as variables for the purposes of equation solving, simplifications, substitutions, and calculations, in order to convert from one unit to another.

Problem a27264a - British Gentleman.


A British gentleman and his Canadian grandson were renting a car in the United States and were told that the car averages 26 miles (mi) per gallon (US gal). Grandpa wanted to know how much this is in miles per imperial gallon (imp gal), and the grandson wanted to know the fuel consumption in liters (L) per 100 km. Find the figures each are looking for, using these facts:

Factor Table.


The Factor Table, is the multiplication tables backwards, and more!
Knowing the multiplication tables makes the numbers your friends, and you start to be familiar with characteristics of the most commonly used numbers.
The 36 items, highlighted in green in the factor table below are from the single- digit multiplication tables and they are useful to know by heart. The numbers highlighted in yellow are the prime numbers, and those with red highlight are the squares.
The multiplication entries without highlighting are not required knowledge, but are still useful and actually easily determined using the single digit multiplication tables forwards and backwards repeatedly.
When faced with 29/42 for example, we know that since 29 does not appear in any of the single digit multiplication tables, it must be a prime number. And since 42 is not a multiple of 29, we can approximate with 28 instead for an easy estimate. 28/42 reduces immediately to 4/6 since both 28 and 42 are in the 7 multiplication table. 4/6 reduces to 2/3 since both 4 and 6 are in the 2 multiplication table. So we can conclude that 29/42 is approximately two thirds, which gives us a much better understanding of its value.
Alternatively we could have estimated 29 with 30 and looked at 30/42. Since both 30 and 42 are in the 6 multiplication table, we reduce it to 5/7, which is a little harder than two thirds to grasp. If we on the other hand just halved 30 and 42 to get 15/21 we could then estimate 21 with 20 to get 15/20 = 3/4, which is easy to understand. However, here we did two approximations along the way, and we are therefore probably not very accurate, so the two thirds still is the preferred guess, and actually the most correct one as well.
Here is a colorful Factor Table for the fridge door, or for the bulletin board by your desk. A PDF version for printing is available by clicking here.


Multiplication Table.


The multiplication table, is it really needed now when every student has a calculator?
It is a matter of speed, estimates, thinking on your feet, and independence!
Knowing the 36 items, framed in green in the single-digit multiplication tables shown below, is as necessary to function in daily life as the ability to speak ones own language.
One should not need to bring out the calculator to understand that you can get five apples at 59 cents each for around three dollars. Estimate 59 with 60, 6 times 5 is 30, and the total price follows.
Knowing the multiplication tables also gives the answer to 24*7/8 instantly, because 24*7=7*24 and you have 7*24/8=7*3=21, since 3*8=24, so 24/8=3.
Calculators can be useful for comparing prices per liter for example, to see which is cheaper, 5 fl oz, (US or Imperial?, they are different!), for $1.09, or 235 mL for $1.77. Yes, there is a place and time for using calculators.
But as with anything mechanical or electronic, the calculator can break, lose power, or be lost altogether, when needed the most. You do not want to be too dependent on it for daily chores and conversations.
Here is a colorful multiplication table for the fridge door, or for the bulletin board by your desk. A PDF version for printing is available here.

Solution a27237a - Arithmetic Sum.

Problem: Find the sum of the sequence of terms less than one thousand, and starting with 37 + 50 + 63 + 76 + ... .

Solution:
This is a typical arithmetic sequence problem, and the formulas to use are easily derived by starting a table of the terms, with a the first one, and d the difference between the terms, as follows:
In our case we see that a = 37 and d = 13 so we have:
To find n, the number of terms to include, we solve the inequality:
Since n is the largest integer less than 75.1 we see that n = 75 and we get:
Hence our final answer is that the Arithmetic Sum is 38850.

Comment:
One could wonder where this formula for summing up the first consecutive natural numbers come from:
Well, if we go to n instead of n − 1 it might look a little bit more familiar:
To justify it conceptually, just add the first and the last, the second and the next to the last, etc, numbers:
Of course this only works if n is an even number. But if n is odd, start by zero instead:
In the first case, for even n, we have half of n rows all summing to n + 1 each, so the total clearly is:
In the second case, for odd n, we have added zero to make it an even number of rows, so we now have half of n + 1 rows, that is the one zero row plus half of n − 1 rows, all summing to, in this case, n each. The total therefore is again:
For those who want a more serious mathematical proof for this formula, we can use the Principle of Mathematical Induction as follows:
Step 1:
For n = 1 the formula is true since:
Step 2:
For any natural number n > 1 we assume that the formula is true for n − 1, that is, as we saw above:
We then add n to see that we get the same formula with n − 1 replaced by n, and consequently n replaced by n + 1:
Step 3:
Since the formula according to Step 1 is true for n = 1, and since according to Step 2, if it is true for some n − 1 (for example n − 1 = 1 when n = 2) then it is also true for n, the Principle of Mathematical Induction gives that it is also true for all natural numbers n. This proves the formula.