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.