Problem solving

Question

Place the following numbers on the lines below, so that the product will become as big as possible:
5,6,7,8,9

_ _ _ * _ _

..................................................................................................................

How am I supposed to calculate this?

Answer 1

use trial and error

here are some numbers I came up with...

875*96=84,000     875-96=779

965*87=83,955     965-87=878

975*86=83,850     975-86=889

976*85=82,960     976-85=891

985*76=74,860     985-76=909

it appears that the smaller the difference between the numbers, the greater the product, up to a point and then the products start to get smaller

for example, 578*69=39,882 and 578-69=509 but you want much larger numbers, numbers in the 800s and 900s

and the other number in the 80s or 90s

 

Answer 2

Hi Fredrik,

 

Lets start by asking what we are given:

1. There are 5 numbers specifically 5, 6, 7, 8 & 9
2. The numbers can be is any order.
3. The answer requires 2 numbers, one of 3 digits and the other of 2 digits, to be multiplied together.
4. We want the largest number possible as a result.

Next, lets ask how we can figure out an answer.

1. Obviously, we can take every combination of 3 digit and 2 digit numbers, multiply them and see which number is bigger.  This is simple but laborious, boring and disregards any knowledge we might have about the situation. So the question becomes, what else do we know about the situation and how can we use that knowledge to find an easier answer?

What we know about the situation:

1. We are doing multiplication.  This means we are adding a number to itself repeatedly a given number of times.
2. Multiplication is commutative. That is, for any two numbers X and Y X*Y = Y*X, or in numbers 95 * 67 = 67 * 95.
3. Multiplying two large numbers gives a larger result than multiplying a large number with a smaller one.
4. This means that 654 * 32 will be larger than 546 * 32.  Note neither situation is the largest possible value when combining the given digits, it is just an illustration that the bigger the numbers the bigger the answer. 
5. The biggest numbers are obtained by arranging the digits to go from the highest digit on the left down to the lowest digit on the right.

Hopefully everything above is fairly obvious but there is one final thing we need to know that is what digits to place in which number. The largest numbers are 9 and 8 so we will want to start one number with 8 and the other with 9 and the following numbers will be arranged highest to lowest.  This gives you a answers to try out, for instance 965*87, 865*97 and so on 

 

Hope this helps

Mark